Startseite Derivation of septic B-spline function in n-dimensional to solve n-dimensional partial differential equations
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Derivation of septic B-spline function in n-dimensional to solve n-dimensional partial differential equations

  • Kamal R. Raslan , Khalid K. Ali EMAIL logo und Mohamed S. Mohamed
Veröffentlicht/Copyright: 13. Juli 2023
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Nonlinear Engineering
Aus der Zeitschrift Nonlinear Engineering Band 12 Heft 1

Abstract

In this study, a new structure for the septic B-spline collocation algorithm in n-dimensional is presented as a continuation of generating B-spline functions in n-dimensional to solve mathematical models in n-dimensional. The septic B-spline collocation algorithm is displayed in three forms: one dimensional, two dimensional, and three dimensional. In various domains, these constructs are essential for solving mathematical models. The effectiveness and correctness of the suggested method are demonstrated using a few two- and three-dimensional test problems. The proposed new structure provides better results than other methods because it deals with a larger number of points than the field. To create comparisons, we use different numerical approaches accessible in the literature.

Keywords: collocation method; n-dimensional B-spline functions; mathematical models

1 Introduction

Many researchers have solved some mathematical models in different dimensions using some analytical and numerical methods such as the (3+1)-dimensional Date–Jimbo–Kashiwara–Miwa equation [1], (3+1)-dimensional Boiti–Leon–Manna–Pempinelli equation [2], (2+1)-dimensional Schrödinger’s hyperbolic equation [3], and non-Newtonian fluid models [4,5]. Most mathematical models in several fields, such as fluid mechanics and physics, are challenging to deal with analytically, prompting some researchers to consider numerical solutions. The finite differences approach, as seen in previous studies [6,7], is one of the strategies used in solving n-dimensional models. Various academics have also attempted to adapt some approaches for solving mathematical models in one dimension to solving models in n-dimensional, such as spectral methods [8,9]. However, most nonlinear models were challenging to solve using spectral approaches. Gardner and Gardner studied a two-dimensional bi-cubic B-spline finite element for solving two-dimensional problems [10]. To solve several diverse mathematical models, some researchers used the bi-cubic B-spline finite element method [11–13] and bi-quintic B-spline collocation method [14–16]. Raslan and Ali began to consider generalizing all types of B-spline functions. They talked about n-dimensional quadratic B-splines [17], new structure formulations for the cubic B-spline collocation method in three and four dimensional [18], construction of extended cubic B-splines in n-dimensional for solving n-dimensional partial differential equations [19], and a new structure for n-dimensional trigonometric cubic B-spline functions for solving n-dimensional partial differential equations [20]. The B-spline collocation method and other methods have been used in many articles to solve many mathematical models such as quintic B-splinemethod [21], cubic B-spline method [22,23], and novel collocation techniques [24–26].

The idea of solving mathematical models in different dimensional remains an idea that haunts most researchers in various fields. Although in previous articles we have presented solutions to these problems by generalizing B-spline collocation functions, we are continuing with these generalizations to deal with models that contain ranks higher than the fifth degree. In this article, we present a generalization of the septic B-spline function, where this method can deal with equations of the seventh order and below.

The structure of this article is as follows: In Section 2, n-dimensional septic B-spline formulas are presented. In Section 3, the numerical outcomes are presented. Section 4 introduces numerical examples. Finally, the conclusion of this work is presented.

2 n-dimensional septic B-spline functions

In this section, we present the n-dimensional septic B-splines.

2.1 One-dimensional septic B-spline [27,28]

Let l ≤ x ≤ m and L i ( x ) are those septic B-spline with knots at the points x ϱ . Then, the set of septic B-splines L − 3 ( x ) , L − 2 ( x ) , L 0 ( x ) , … , L N − 1 ( x ) , L N ( x ) , L N + 2 ( x ) , L N + 3 , serves as a basis for functions specified over a range of values. The H N ( x ) approximation to H ( x ) is given by:

(1) H N ( x ) = ∑ ϱ = − 3 N + 3 T ϱ L ϱ ( x ) ,

where T ϱ is the unknown term and L ϱ ( x ) is a function given by:

(2) L ϱ ( x ) = 1 h 7 a 1 = ( x − x ϱ − 4 ) 7 , x ϱ − 4 ≤ x < x ϱ − 3 a 2 = a 1 − 8 ( x − x ϱ − 3 ) 7 , x ϱ − 3 ≤ x < x ϱ − 2 a 3 = a 2 + 28 ( x − x ϱ − 2 ) 7 , x ϱ − 2 ≤ x < x ϱ − 1 a 4 = a 3 − 56 ( x − x ϱ − 1 ) 7 , x ϱ − 1 ≤ x < x ϱ b 4 = b 3 − 56 ( x ϱ − 1 − x ) 7 , x ϱ ≤ x < x ϱ + 1 b 3 = b 2 + 28 ( x ϱ − 2 − x ) 7 , x ϱ + 1 ≤ x < x ϱ + 2 b 2 = b 1 − 8 ( x ϱ − 3 − x ) 7 , x ϱ + 2 ≤ x < x ϱ + 3 b 1 = ( x ϱ − 3 − x ) 7 , x ϱ + 3 ≤ x < x ϱ + 4 0 otherwise .

We use (1) and (2) with substitution by collection points to find H ϱ , d H ϱ d x and d 2 H ϱ d x 2 as follows:

(3) H ϱ = T ϱ − 3 + 120 T ϱ − 2 + 1,191 T ϱ − 1 + 2,416 T ϱ + 1,191 T ϱ + 1 + 120 T ϱ + 2 + T ϱ + 3 , d H ϱ d x = − 7 ( T ϱ − 3 + 56 T ϱ − 2 + 245 T ϱ − 1 − 245 T ϱ + 1 − 56 T ϱ + 2 − T ϱ + 3 ) h , d 2 H ϱ d x 2 = 42 ( T ϱ − 3 + 24 T ϱ − 2 + 15 T ϱ − 1 − 80 T ϱ + 15 T ϱ + 1 + 24 T ϱ + 2 + T ϱ + 3 ) h 2 .

The aforementioned analysis yields the following theorem.

Theorem 1

From (1) the approximation formulas to H ϱ , d H ϱ d x , and d 2 H ϱ d x 2 are given in terms of T ϱ at (3).

2.2 Two-dimensional septic B-spline

This subsection shows the formula for a two-dimensional septic B-spline on a rectangular grid divided into regular rectangular finite elements on both sides. h = Δ x , k = Δ y by the knots ( x ϱ , y ς ) , where ϱ = 0 , 1 , … , N , ς = 0 , 1 , … , M . The approximation H N ( x , y ) to H ( x , y ) is given by:

(4) H N ( x , y ) = ∑ ϱ = − 3 N + 3 ∑ ς = − 3 M + 3 T ϱ , ς Θ ϱ , ς ( x , y ) ,

where T ϱ , ς are the amplitudes of the septic B-splines. Θ ϱ , ς ( x , y ) are given by:

Θ ϱ , ς ( x , y ) = L ϱ ( x ) L ς ( y ) .

Moreover, L ς ( x ) , L n ( y ) have the same shape as septic B-splines in one dimension. Then, the formulations of H ϱ , ς , ∂ H ϱ , ς ∂ x , ∂ H ϱ , ς ∂ y , ∂ 2 H ϱ , ς ∂ x 2 , ∂ 2 H ϱ , ς ∂ y 2 , ∂ 2 H ϱ , ς ∂ x ∂ y , … are given by:

(5) H ϱ , ς = T ϱ − 3 , ς − 3 + 120 T ϱ − 3 , ς − 2 + 1,191 T ϱ − 3 , ς − 1 + 2,416 T ϱ − 3 , ς + 1,191 T ϱ − 3 , ς + 1 + 120 T ϱ − 3 , ς + 2 + T ϱ − 3 , ς + 3 + 120 T ϱ − 2 , ς − 3 + 14,400 T ϱ − 2 , ς − 2 + 142,920 T ϱ − 2 , ς − 1 + 289,920 T ϱ − 2 , ς + 142,920 T ϱ − 2 , ς + 1 + 14,400 T ϱ − 2 , ς + 2 + 120 T ϱ − 2 , ς + 3 + 1,191 T ϱ − 1 , ς − 3 + 142,920 T ϱ − 1 , ς − 2 + 1,418,481 T ϱ − 1 , ς − 1 + 2,877,456 T ϱ − 1 , ς + 1,418,481 T ϱ − 1 , ς + 1 + 142,920 T ϱ − 1 , ς + 2 + 1,191 T ϱ − 1 , ς + 3 + 2,416 T ϱ , ς − 3 + 289,920 T ϱ , ς − 2 + 2,877,456 T ϱ , ς − 1 + 5,837,056 T ϱ , ς + 2,877,456 T ϱ , ς + 1 + 289,920 T ϱ , ς + 2 + 2,416 T ϱ , ς + 3 + 1,191 T ϱ + 1 , ς − 3 + 142,920 T ϱ + 1 , ς − 2 + 1,418,481 T ϱ + 1 , ς − 1 + 2,877,456 T ϱ + 1 , ς + 1,418,481 T ϱ + 1 , ς + 1 + 142,920 T ϱ + 1 , ς + 2 + 1,191 T ϱ + 1 , ς + 3 + 120 T ϱ + 2 , ς − 3 + 14,400 T ϱ + 2 , ς − 2 + 142,920 T ϱ + 2 , ς − 1 + 289,920 T ϱ + 2 , ς + 142,920 T ϱ + 2 , ς + 1 + 14,400 T ϱ + 2 , ς + 2 + 120 T ϱ + 2 , ς + 3 + T ϱ + 3 , ς − 3 + 120 T ϱ + 3 , ς − 2 + 1,191 T ϱ + 3 , ς − 1 + 2,416 T ϱ + 3 , ς + 1,191 T ϱ + 3 , ς + 1 + 120 T ϱ + 3 , ς + 2 + T ϱ + 3 , ς + 3 .

∂ H ϱ , ς ∂ x = − 7 h ( T ϱ − 3 , ς − 3 + 120 T ϱ − 3 , ς − 2 + 1,191 T ϱ − 3 , ς − 1 + 2,416 T ϱ − 3 , ς + 1,191 T ϱ − 3 , ς + 1 + 120 T ϱ − 3 , ς + 2 + T ϱ − 3 , ς + 3 + 56 T ϱ − 2 , ς − 3 + 6,720 T ϱ − 2 , ς − 2 + 66,696 T ϱ − 2 , ς − 1 + 135,296 T ϱ − 2 , ς + 66,696 T ϱ − 2 , ς + 1 + 6,720 T ϱ − 2 , ς + 2 + 56 T ϱ − 2 , ς + 3 + 245 T ϱ − 1 , ς − 3 + 29,400 T ϱ − 1 , ς − 2 + 291,795 T ϱ − 1 , ς − 1 + 591,920 T ϱ − 1 , ς + 291,795 T ϱ − 1 , ς + 1 + 29,400 T ϱ − 1 , ς + 2 + 245 T ϱ − 1 , ς + 3 − 245 T ϱ + 1 , ς − 3 − 29,400 T ϱ + 1 , ς − 2 − 291,795 T ϱ + 1 , ς − 1 − 591,920 T ϱ + 1 , ς − 291,795 T ϱ + 1 , ς + 1 − 29,400 T ϱ + 1 , ς + 2 − 245 T ϱ + 1 , ς + 3 − 56 T ϱ + 2 , ς − 3 − 6,720 T ϱ + 2 , ς − 2 − 66,696 T ϱ + 2 , ς − 1 − 135,296 T ϱ + 2 , ς − 66,696 T ϱ + 2 , ς + 1 − 6,720 T ϱ + 2 , ς + 2 − 56 T ϱ + 2 , ς + 3 − T ϱ + 3 , ς − 3 − 120 T ϱ + 3 , ς − 2 − 1,191 T ϱ + 3 , ς − 1 − 2,416 T ϱ + 3 , ς − 1,191 T ϱ + 3 , ς + 1 − 120 T ϱ + 3 , ς + 2 − T ϱ + 3 , ς + 3 ) ,

(6) ∂ H ϱ , ς ∂ y = − 7 k ( T ϱ − 3 , ς − 3 + 56 T ϱ − 3 , ς − 2 + 245 T ϱ − 3 , ς − 1 − 245 T ϱ − 3 , ς + 1 − 56 T ϱ − 3 , ς + 2 − T ϱ − 3 , ς + 3 + 120 T ϱ − 2 , ς − 3 + 6,720 T ϱ − 2 , ς − 2 + 29,400 T ϱ − 2 , ς − 1 − 29,400 T ϱ − 2 , ς + 1 − 6,720 T ϱ − 2 , ς + 2 − 120 T ϱ − 2 , ς + 3 + 1,191 T ϱ − 1 , ς − 3 + 66,696 T ϱ − 1 , ς − 2 + 291,795 T ϱ − 1 , ς − 1 − 291,795 T ϱ − 1 , ς + 1 − 66,696 T ϱ − 1 , ς + 2 − 1,191 T ϱ − 1 , ς + 3 + 2,416 T ϱ , ς − 3 + 135,296 T ϱ , ς − 2 + 591,920 T ϱ , ς − 1 − 591,920 T ϱ , ς + 1 − 135,296 T ϱ , ς + 2 − 2,416 T ϱ , ς + 3 + 1,191 T ϱ + 1 , ς − 3 + 66,696 T ϱ + 1 , ς − 2 + 291,795 T ϱ + 1 , ς − 1 − 291,795 T ϱ + 1 , ς + 1 − 66,696 T ϱ + 1 , ς + 2 − 1,191 T ϱ + 1 , ς + 3 + 120 T ϱ + 2 , ς − 3 + 6,720 T ϱ + 2 , ς − 2 + 29,400 T ϱ + 2 , ς − 1 − 29,400 T ϱ + 2 , ς + 1 − 6,720 T ϱ + 2 , ς + 2 − 120 T ϱ + 2 , ς + 3 + T ϱ + 3 , τ − 3 + 7 ( 8 T ϱ + 3 , τ − 2 + 35 T ϱ + 3 , τ − 1 − 35 T ϱ + 3 , τ + 1 − 8 T ϱ + 3 , τ + 2 ) − T ϱ + 3 , ς + 3 ) .

∂ 2 H ϱ , ς ∂ x 2 = 42 h 2 ( T ϱ − 3 , ς − 3 + 120 T ϱ − 3 , ς − 2 + 1,191 T ϱ − 3 , ς − 1 + 2,416 T ϱ − 3 , ς + 1,191 T ϱ − 3 , ς + 1 + 120 T ϱ − 3 , ς + 2 + T ϱ − 3 , ς + 3 + 24 T ϱ − 2 , ς − 3 + 2,880 T ϱ − 2 , ς − 2 + 28,584 T ϱ − 2 , ς − 1 + 57,984 T ϱ − 2 , ς + 28,584 T ϱ − 2 , ς + 1 + 2,880 T ϱ − 2 , ς + 2 + 24 T ϱ − 2 , ς + 3 + 15 T ϱ − 1 , ς − 3 + 1,800 T ϱ − 1 , ς − 2 + 17,865 T ϱ − 1 , ς − 1 + 36,240 T ϱ − 1 , ς + 17,865 T ϱ − 1 , ς + 1 + 1,800 T ϱ − 1 , ς + 2 + 15 T ϱ − 1 , ς + 3 − 80 T ϱ , ς − 3 − 9,600 T ϱ , ς − 2 − 95,280 T ϱ , ς − 1 − 193,280 T ϱ , ς − 95,280 T ϱ , ς + 1 − 9,600 T ϱ , ς + 2 − 80 T ϱ , ς + 3 + 15 T ϱ + 1 , ς − 3 + 1,800 T ϱ + 1 , ς − 2 + 17,865 T ϱ + 1 , ς − 1 + 36,240 T ϱ + 1 , ς + 17,865 T ϱ + 1 , ς + 1 + 1,800 T ϱ + 1 , ς + 2 + 15 T ϱ + 1 , ς + 3 + 24 T ϱ + 2 , ς − 3 + 2,880 T ϱ + 2 , ς − 2 + 28,584 T ϱ + 2 , ς − 1 + 57,984 T ϱ + 2 , ς + 28,584 T ϱ + 2 , ς + 1 + 2,880 T ϱ + 2 , ς + 2 + 24 T ϱ + 2 , ς + 3 + T ϱ + 3 , ς − 3 + 120 T ϱ + 3 , ς − 2 + 1,191 T ϱ + 3 , ς − 1 + 2,416 T ϱ + 3 , ς + 1,191 T ϱ + 3 , ς + 1 + 120 T ϱ + 3 , ς + 2 + T ϱ + 3 , ς + 3 ) .

(7) ∂ 2 H ϱ , ς ∂ y 2 = 42 k 2 ( T ϱ − 3 , ς − 3 + 24 T ϱ − 3 , ς − 2 + 15 T ϱ − 3 , ς − 1 − 80 T ϱ − 3 , ς + 15 T ϱ − 3 , ς + 1 + 24 T ϱ − 3 , ς + 2 + T ϱ − 3 , ς + 3 + 120 T ϱ − 2 , ς − 3 + 2,880 T ϱ − 2 , ς − 2 + 1,800 T ϱ − 2 , ς − 1 − 9,600 T ϱ − 2 , ς + 1,800 T ϱ − 2 , ς + 1 + 2,880 T ϱ − 2 , ς + 2 + 120 T ϱ − 2 , ς + 3 + 1,191 T ϱ − 1 , ς − 3 + 28,584 T ϱ − 1 , ς − 2 + 17,865 T ϱ − 1 , ς − 1 − 95,280 T ϱ − 1 , ς + 17,865 T ϱ − 1 , ς + 1 + 28,584 T ϱ − 1 , ς + 2 + 1,191 T ϱ − 1 , ς + 3 + 2,416 T ϱ , ς − 3 + 57,984 T ϱ , ς − 2 + 36,240 T ϱ , ς − 1 − 193,280 T ϱ , ς + 36,240 T ϱ , ς + 1 + 57,984 T ϱ , ς + 2 + 2,416 T ϱ , ς + 3 + 1,191 T ϱ + 1 , ς − 3 + 28,584 T ϱ + 1 , ς − 2 + 17,865 T ϱ + 1 , ς − 1 − 95,280 T ϱ + 1 , ς + 17,865 T ϱ + 1 , ς + 1 + 28,584 T ϱ + 1 , ς + 2 + 1,191 T ϱ + 1 , ς + 3 + 120 T ϱ + 2 , ς − 3 + 2,880 T ϱ + 2 , ς − 2 + 1,800 T ϱ + 2 , ς − 1 − 9,600 T ϱ + 2 , ς + 1,800 T ϱ + 2 , ς + 1 + 2,880 T ϱ + 2 , ς + 2 + 120 T ϱ + 2 , ς + 3 + T ϱ + 3 , ς − 3 + 24 T ϱ + 3 , ς − 2 + 15 T ϱ + 3 , ς − 1 − 80 T ϱ + 3 , ς + 15 T ϱ + 3 , ς + 1 + 24 T ϱ + 3 , ς + 2 + T ϱ + 3 , ς + 3 ) , ⋮

The aforementioned analysis yields the following theorem.

Theorem 2

From (4), the approximation formulas to H ϱ , ς , ∂ H ϱ , ς ∂ x , ∂ H ϱ , ς ∂ y , ∂ 2 H ϱ , ς ∂ x 2 , ∂ 2 H ϱ , ς ∂ y 2 , ∂ 2 H ϱ , ς ∂ x ∂ y , … are given in terms of T ϱ , ς at (5)–(7).

2.3 Three-dimensional septic B-spline

Now, we obtain the septic B-spline in three measurements approximates on a framework divided up into limited components of sides h = Δ x , k = Δ y , and q = Δ z by the knots ( x ϱ , y ς , z s ) , where ϱ = 0 , 1 , … , N , ς = 0 , 1 , … , M , s = 0 , 1 , … , R can be interpolated in terms of piece-wise septic B-splines. If H ( x , y , z ) is a function of x , y , and z , it can be shown there exists a unique approximation H N ( x , y , z ) as

(8) H N ( x , y , z ) = ∑ ϱ = − 3 N + 3 ∑ ς = − 3 M + 3 ∑ s = − 3 R + 3 T ϱ , ς , s B ϱ , ς , s ( x , y , z ) ,

where T ϱ , ς , s are the septic B-spline amplitudes B ϱ , ς , s ( x , y , z ) given by:

B ϱ , ς , s ( x , y , z ) = L ϱ ( x ) L ς ( y ) L s ( z ) .

Also, L ϱ ( x ) , L ς ( y ) , and L s ( z ) have the same shape as septic B-splines in one dimension. The compositions of H ϱ , ς , s , ∂ H ϱ , ς , s ∂ x , ∂ H ϱ , ς , s ∂ y , ∂ H ϱ , ς , s ∂ z , ∂ 2 H ϱ , ς , s ∂ x 2 , ∂ 2 H ϱ , ς , s ∂ y 2 , ∂ 2 H ϱ , ς , s ∂ z 2 , ∂ 2 H ϱ , ς , s ∂ x ∂ y , ∂ 2 H ϱ , ς , s ∂ x ∂ z , … , are given in terms of T ϱ , ς , s by:

H ϱ , ς , s = T ϱ − 3 , ς − 3 , s − 3 + 120 T ϱ − 3 , ς − 3 , s − 2 + 1,191 T ϱ − 3 , ς − 3 , s − 1 + 2,416 T ϱ − 3 , ς − 3 , s + 1,191 T ϱ − 3 , ς − 3 , s + 1 + 120 T ϱ − 3 , ς − 3 , s + 2 + T ϱ − 3 , ς − 3 , s + 3 + 120 T ϱ − 3 , ς − 2 , s − 3 + 14,400 T ϱ − 3 , ς − 2 , s − 2 + 142,920 T ϱ − 3 , ς − 2 , s − 1 + 289,920 T ϱ − 3 , ς − 2 , s + 142,920 T ϱ − 3 , ς − 2 , s + 1 + 14,400 T ϱ − 3 , ς − 2 , s + 2 + 120 T ϱ − 3 , ς − 2 , s + 3 + 1,191 T ϱ − 3 , ς − 1 , s − 3 + 142,920 T ϱ − 3 , ς − 1 , s − 2 + 1,418,481 T ϱ − 3 , ς − 1 , s − 1 + 2,877,456 T ϱ − 3 , ς − 1 , s + 1,418,481 T ϱ − 3 , ς − 1 , s + 1 + 142,920 T ϱ − 3 , ς − 1 , s + 2 + 1,191 T ϱ − 3 , ς − 1 , s + 3 + 2,416 T ϱ − 3 , ς , s − 3 + 289,920 T ϱ − 3 , ς , s − 2 + 2,877,456 T ϱ − 3 , ς , s − 1 + 5,837,056 T ϱ − 3 , ς , s + 2,877,456 T ϱ − 3 , ς , s + 1 + 289,920 T ϱ − 3 , ς , s + 2 + 2,416 T ϱ − 3 , ς , s + 3 + 1,191 T ϱ − 3 , ς + 1 , s − 3 + 142,920 T ϱ − 3 , ς + 1 , s − 2 + 1,418,481 T ϱ − 3 , ς + 1 , s − 1 + 2,877,456 T ϱ − 3 , ς + 1 , s + 1,418,481 T ϱ − 3 , ς + 1 , s + 1 + 142,920 T ϱ − 3 , ς + 1 , s + 2 + 1,191 T ϱ − 3 , ς + 1 , s + 3 + 120 T ϱ − 3 , ς + 2 , s − 3 + 14,400 T ϱ − 3 , ς + 2 , s − 2 + 142,920 T ϱ − 3 , ς + 2 , s − 1 + 289,920 T ϱ − 3 , ς + 2 , s + 142,920 T ϱ − 3 , ς + 2 , s + 1 + 14,400 T ϱ − 3 , ς + 2 , s + 2 + 120 T ϱ − 3 , ς + 2 , s + 3 + T ϱ − 3 , ς + 3 , s − 3 + 120 T ϱ − 3 , ς + 3 , s − 2 + 1,191 T ϱ − 3 , ς + 3 , s − 1 + 2,416 T ϱ − 3 , ς + 3 , s + 1,191 T ϱ − 3 , ς + 3 , s + 1 + 120 T ϱ − 3 , ς + 3 , s + 2 + T ϱ − 3 , ς + 3 , s + 3 + 120 T ϱ − 2 , ς − 3 , s − 3 + 14,400 T ϱ − 2 , ς − 3 , s − 2 + 142,920 T ϱ − 2 , ς − 3 , s − 1 + 289,920 T ϱ − 2 , ς − 3 , s + 142,920 T ϱ − 2 , ς − 3 , s + 1 + 14,400 T ϱ − 2 , ς − 3 , s + 2 + 120 T ϱ − 2 , ς − 3 , s + 3 + 14,400 T ϱ − 2 , ς − 2 , s − 3 + 1,728,000 T ϱ − 2 , ς − 2 , s − 2 + 17,150,400 T ϱ − 2 , ς − 2 , s − 1 + 34,790,400 T ϱ − 2 , ς − 2 , s + 17,150,400 T ϱ − 2 , ς − 2 , s + 1 + 1,728,000 T ϱ − 2 , ς − 2 , s + 2 + 14,400 T ϱ − 2 , ς − 2 , s + 3 + 142,920 T ϱ − 2 , ς − 1 , s − 3 + 17,150,400 T ϱ − 2 , ς − 1 , s − 2 + 170,217,720 T ϱ − 2 , ς − 1 , s − 1 + 345,294,720 T ϱ − 2 , ς − 1 , s + 170,217,720 T ϱ − 2 , ς − 1 , s + 1 + 17,150,400 T ϱ − 2 , ς − 1 , s + 2 + 142,920 T ϱ − 2 , ς − 1 , s + 3 + 289,920 T ϱ − 2 , ς , s − 3 + 34,790,400 T ϱ − 2 , ς , s − 2 + 345,294,720 T ϱ − 2 , ς , s − 1 + 700,446,720 T ϱ − 2 , ς , s + 345,294,720 T ϱ − 2 , ς , s + 1 + 34,790,400 T ϱ − 2 , ς , s + 2 + 289,920 T ϱ − 2 , ς , s + 3 + 142,920 T ϱ − 2 , ς + 1 , s − 3 + 17,150,400 T ϱ − 2 , ς + 1 , s − 2 + 170,217,720 T ϱ − 2 , ς + 1 , s − 1 + 345,294,720 T ϱ − 2 , ς + 1 , s + 170,217,720 T ϱ − 2 , ς + 1 , s + 1 + 17,150,400 T ϱ − 2 , ς + 1 , s + 2 + 142,920 T ϱ − 2 , ς + 1 , s + 3 + 14,400 T ϱ − 2 , ς + 2 , s − 3 + 1,728,000 T ϱ − 2 , ς + 2 , s − 2 + 17,150,400 T ϱ − 2 , ς + 2 , s − 1 + 34,790,400 T ϱ − 2 , ς + 2 , s + 17,150,400 T ϱ − 2 , ς + 2 , s + 1 + 1,728,000 T ϱ − 2 , ς + 2 , s + 2 + 14,400 T ϱ − 2 , ς + 2 , s + 3 + 120 T ϱ − 2 , ς + 3 , s − 3 + 14,400 T ϱ − 2 , ς + 3 , s − 2 + 142,920 T ϱ − 2 , ς + 3 , s − 1 + 289,920 T ϱ − 2 , ς + 3 , s + 142,920 T ϱ − 2 , ς + 3 , s + 1 + 14,400 T ϱ − 2 , ς + 3 , s + 2 + 120 T ϱ − 2 , ς + 3 , s + 3 + 1,191 T ϱ − 1 , ς − 3 , s − 3 + 142,920 T ϱ − 1 , ς − 3 , s − 2 + 1,418,481 T ϱ − 1 , ς − 3 , s − 1 + 2,877,456 T ϱ − 1 , ς − 3 , s + 1,418,481 T ϱ − 1 , ς − 3 , s + 1 + 142,920 T ϱ − 1 , ς − 3 , s + 2 + 1,191 T ϱ − 1 , ς − 3 , s + 3 + 142,920 T ϱ − 1 , ς − 2 , s − 3 + 17,150,400 T ϱ − 1 , ς − 2 , s − 2 + 170,217,720 T ϱ − 1 , ς − 2 , s − 1 + 345,294,720 T ϱ − 1 , ς − 2 , s + 170,217,720 T ϱ − 1 , ς − 2 , s + 1 + 17,150,400 T ϱ − 1 , ς − 2 , s + 2 + 142,920 T ϱ − 1 , ς − 2 , s + 3 + 1,418,481 T ϱ − 1 , ς − 1 , s − 3 + 170,217,720 T ϱ − 1 , ς − 1 , s − 2 + 1,689,410,871 T ϱ − 1 , ς − 1 , s − 1 + 3,427,050,096 T ϱ − 1 , ς − 1 , s + 1,689,410,871 T ϱ − 1 , ς − 1 , s + 1 + 170,217,720 T ϱ − 1 , ς − 1 , s + 2 + 1,418,481 T ϱ − 1 , ς − 1 , s + 3 + 2,877,456 T ϱ − 1 , ς , s − 3 + 345,294,720 T ϱ − 1 , ς , s − 2 + 3,427,050,096 T ϱ − 1 , ς , s − 1 + 6,951,933,696 T ϱ − 1 , ς , s + 3,427,050,096 T ϱ − 1 , ς , s + 1 + 345,294,720 T ϱ − 1 , ς , s + 2 + 2,877,456 T ϱ − 1 , ς , s + 3 + 1,418,481 T ϱ − 1 , ς + 1 , s − 3 + 170,217,720 T ϱ − 1 , ς + 1 , s − 2 + 1,689,410,871 T ϱ − 1 , ς + 1 , s − 1

(9) + 3,427,050,096 T ϱ − 1 , ς + 1 , s + 1,689,410,871 T ϱ − 1 , ς + 1 , s + 1 + 170,217,720 T ϱ − 1 , ς + 1 , s + 2 + 1,418,481 T ϱ − 1 , ς + 1 , s + 3 + 142,920 T ϱ − 1 , ς + 2 , s − 3 + 17,150,400 T ϱ − 1 , ς + 2 , s − 2 + 170,217,720 T ϱ − 1 , ς + 2 , s − 1 + 345,294,720 T ϱ − 1 , ς + 2 , s + 170,217,720 T ϱ − 1 , ς + 2 , s + 1 + 17,150,400 T ϱ − 1 , ς + 2 , s + 2 + 142,920 T ϱ − 1 , ς + 2 , s + 3 + 1,191 T ϱ − 1 , ς + 3 , s − 3 + 142,920 T ϱ − 1 , ς + 3 , s − 2 + 1,418,481 T ϱ − 1 , ς + 3 , s − 1 + 2,877,456 T ϱ − 1 , ς + 3 , s + 1,418,481 T ϱ − 1 , ς + 3 , s + 1 + 142,920 T ϱ − 1 , ς + 3 , s + 2 + 1,191 T ϱ − 1 , ς + 3 , s + 3 + 2,416 T ϱ , ς − 3 , s − 3 + 289,920 T ϱ , ς − 3 , s − 2 + 2,877,456 T ϱ , ς − 3 , s − 1 + 5,837,056 T ϱ , ς − 3 , s + 2,877,456 T ϱ , ς − 3 , s + 1 + 289,920 T ϱ , ς − 3 , s + 2 + 2,416 T ϱ , ς − 3 , s + 3 + 289,920 T ϱ , ς − 2 , s − 3 + 34,790,400 T ϱ , ς − 2 , s − 2 + 345,294,720 T ϱ , ς − 2 , s − 1 + 700,446,720 T ϱ , ς − 2 , s + 345,294,720 T ϱ , ς − 2 , s + 1 + 34,790,400 T ϱ , ς − 2 , s + 2 + 289,920 T ϱ , ς − 2 , s + 3 + 2,877,456 T ϱ , ς − 1 , s − 3 + 345,294,720 T ϱ , ς − 1 , s − 2 + 3,427,050,096 T ϱ , ς − 1 , s − 1 + 6,951,933,696 T ϱ , ς − 1 , s + 3,427,050,096 T ϱ , ς − 1 , s + 1 + 345,294,720 T ϱ , ς − 1 , s + 2 + 2,877,456 T ϱ , ς − 1 , s + 3 + 5,837,056 T ϱ , ς , s − 3 + 700,446,720 T ϱ , ς , s − 2 + 6,951,933,696 T ϱ , ς , s − 1 + 14102327296 T ϱ , ς , s + 6,951,933,696 T ϱ , ς , s + 1 + 700,446,720 T ϱ , ς , s + 2 + 5,837,056 T ϱ , ς , s + 3 + 2,877,456 T ϱ , ς + 1 , s − 3 + 345,294,720 T ϱ , ς + 1 , s − 2 + 3,427,050,096 T ϱ , ς + 1 , s − 1 + 6,951,933,696 T ϱ , ς + 1 , s + 3,427,050,096 T ϱ , ς + 1 , s + 1 + 345,294,720 T ϱ , ς + 1 , s + 2 + 2,877,456 T ϱ , ς + 1 , s + 3 + 289,920 T ϱ , ς + 2 , s − 3 + 34,790,400 T ϱ , ς + 2 , s − 2 + 345,294,720 T ϱ , ς + 2 , s − 1 + 700,446,720 T ϱ , ς + 2 , s + 345,294,720 T ϱ , ς + 2 , s + 1 + 34,790,400 T ϱ , ς + 2 , s + 2 + 289,920 T ϱ , ς + 2 , s + 3 + 2,416 T ϱ , ς + 3 , s − 3 + 289,920 T ϱ , ς + 3 , s − 2 + 2,877,456 T ϱ , ς + 3 , s − 1 + 5,837,056 T ϱ , ς + 3 , s + 2,877,456 T ϱ , ς + 3 , s + 1 + 289,920 T ϱ , ς + 3 , s + 2 + 2,416 T ϱ , ς + 3 , s + 3 + 1,191 T ϱ + 1 , ς − 3 , s − 3 + 142,920 T ϱ + 1 , ς − 3 , s − 2 + 1,418,481 T ϱ + 1 , ς − 3 , s − 1 + 2,877,456 T ϱ + 1 , ς − 3 , s + 1,418,481 T ϱ + 1 , ς − 3 , s + 1 + 142,920 T ϱ + 1 , ς − 3 , s + 2 + 1,191 T ϱ + 1 , ς − 3 , s + 3 + 142,920 T ϱ + 1 , ς − 2 , s − 3 + 17,150,400 T ϱ + 1 , ς − 2 , s − 2 + 170,217,720 T ϱ + 1 , ς − 2 , s − 1 + 345,294,720 T ϱ + 1 , ς − 2 , s + 170,217,720 T ϱ + 1 , ς − 2 , s + 1 + 17,150,400 T ϱ + 1 , ς − 2 , s + 2 + 142,920 T ϱ + 1 , ς − 2 , s + 3 + 1,418,481 T ϱ + 1 , ς − 1 , s − 3 + 170,217,720 T ϱ + 1 , ς − 1 , s − 2 + 1,689,410,871 T ϱ + 1 , ς − 1 , s − 1 + 3,427,050,096 T ϱ + 1 , ς − 1 , s + 1,689,410,871 T ϱ + 1 , ς − 1 , s + 1 + 170,217,720 T ϱ + 1 , ς − 1 , s + 2 + 1,418,481 T ϱ + 1 , ς − 1 , s + 3 + 2,877,456 T ϱ + 1 , ς , s − 3 + 345,294,720 T ϱ + 1 , ς , s − 2 + 3,427,050,096 T ϱ + 1 , ς , s − 1 + 6,951,933,696 T ϱ + 1 , ς , s + 3,427,050,096 T ϱ + 1 , ς , s + 1 + 345,294,720 T ϱ + 1 , ς , s + 2 + 2,877,456 T ϱ + 1 , ς , s + 3 + 1,418,481 T ϱ + 1 , ς + 1 , s − 3 + 170,217,720 T ϱ + 1 , ς + 1 , s − 2 + 1,689,410,871 T ϱ + 1 , ς + 1 , s − 1 + 3,427,050,096 T ϱ + 1 , ς + 1 , s + 1,689,410,871 T ϱ + 1 , ς + 1 , s + 1 + 170,217,720 T ϱ + 1 , ς + 1 , s + 2 + 1,418,481 T ϱ + 1 , ς + 1 , s + 3 + 142,920 T ϱ + 1 , ς + 2 , s − 3 + 17,150,400 T ϱ + 1 , ς + 2 , s − 2 + 170,217,720 T ϱ + 1 , ς + 2 , s − 1 + 345,294,720 T ϱ + 1 , ς + 2 , s + 170,217,720 T ϱ + 1 , ς + 2 , s + 1 + 17,150,400 T ϱ + 1 , ς + 2 , s + 2 + 142,920 T ϱ + 1 , ς + 2 , s + 3 + 1,191 T ϱ + 1 , ς + 3 , s − 3 + 142,920 T ϱ + 1 , ς + 3 , s − 2 + 1,418,481 T ϱ + 1 , ς + 3 , s − 1 + 2,877,456 T ϱ + 1 , ς + 3 , s + 1,418,481 T ϱ + 1 , ς + 3 , s + 1 + 142,920 T ϱ + 1 , ς + 3 , s + 2 + 1,191 T ϱ + 1 , ς + 3 , s + 3 + 120 T ϱ + 2 , ς − 3 , s − 3 + 14,400 T ϱ + 2 , ς − 3 , s − 2 + 142,920 T ϱ + 2 , ς − 3 , s − 1 + 289,920 T ϱ + 2 , ς − 3 , s + 142,920 T ϱ + 2 , ς − 3 , s + 1 + 14,400 T ϱ + 2 , ς − 3 , s + 2 + 120 T ϱ + 2 , ς − 3 , s + 3 + 14,400 T ϱ + 2 , ς − 2 , s − 3 + 1,728,000 T ϱ + 2 , ς − 2 , s − 2 + 17,150,400 T ϱ + 2 , ς − 2 , s − 1 + 34,790,400 T ϱ + 2 , ς − 2 , s + 17,150,400 T ϱ + 2 , ς − 2 , s + 1 + 1,728,000 T ϱ + 2 , ς − 2 , s + 2 + 14,400 T ϱ + 2 , ς − 2 , s + 3 + 142,920 T ϱ + 2 , ς − 1 , s − 3 + 17,150,400 T ϱ + 2 , ς − 1 , s − 2 + 170,217,720 T ϱ + 2 , ς − 1 , s − 1 + 345,294,720 T ϱ + 2 , ς − 1 , s + 170,217,720 T ϱ + 2 , ς − 1 , s + 1 + 17,150,400 T ϱ + 2 , ς − 1 , s + 2 + 142,920 T ϱ + 2 , ς − 1 , s + 3 + 289,920 T ϱ + 2 , ς , s − 3 + 34,790,400 T ϱ + 2 , ς , s − 2 + 345,294,720 T ϱ + 2 , ς , s − 1 + 700,446,720 T ϱ + 2 , ς , s + 345,294,720 T ϱ + 2 , ς , s + 1 + 34,790,400 T ϱ + 2 , ς , s + 2 + 289,920 T ϱ + 2 , ς , s + 3 + 142,920 T ϱ + 2 , ς + 1 , s − 3 + 17,150,400 T ϱ + 2 , ς + 1 , s − 2 + 170,217,720 T ϱ + 2 , ς + 1 , s − 1 + 345,294,720 T ϱ + 2 , ς + 1 , s + 170,217,720 T ϱ + 2 , ς + 1 , s + 1 + 17,150,400 T ϱ + 2 , ς + 1 , s + 2 + 142,920 T ϱ + 2 , ς + 1 , s + 3 + 14,400 T ϱ + 2 , ς + 2 , s − 3 + 1,728,000 T ϱ + 2 , ς + 2 , s − 2 + 17,150,400 T ϱ + 2 , ς + 2 , s − 1 + 34,790,400 T ϱ + 2 , ς + 2 , s + 17,150,400 T ϱ + 2 , ς + 2 , s + 1 + 1,728,000 T ϱ + 2 , ς + 2 , s + 2 + 14,400 T ϱ + 2 , ς + 2 , s + 3 + 120 T ϱ + 2 , ς + 3 , s − 3 + 14,400 T ϱ + 2 , ς + 3 , s − 2 + 142,920 T ϱ + 2 , ς + 3 , s − 1 + 289,920 T ϱ + 2 , ς + 3 , s + 142,920 T ϱ + 2 , ς + 3 , s + 1 + 14,400 T ϱ + 2 , ς + 3 , s + 2 + 120 T ϱ + 2 , ς + 3 , s + 3 + T ϱ + 3 , ς − 3 , s − 3 + 120 T ϱ + 3 , ς − 3 , s − 2 + 1,191 T ϱ + 3 , ς − 3 , s − 1 + 2,416 T ϱ + 3 , ς − 3 , s + 1,191 T ϱ + 3 , ς − 3 , s + 1 + 120 T ϱ + 3 , ς − 3 , s + 2 + T ϱ + 3 , ς − 3 , s + 3 + 120 T ϱ + 3 , ς − 2 , s − 3 + 14,400 T ϱ + 3 , ς − 2 , s − 2 + 142,920 T ϱ + 3 , ς − 2 , s − 1

+ 289,920 T ϱ + 3 , ς − 2 , s + 142,920 T ϱ + 3 , ς − 2 , s + 1 + 14,400 T ϱ + 3 , ς − 2 , s + 2 + 120 T ϱ + 3 , ς − 2 , s + 3 + 1,191 T ϱ + 3 , ς − 1 , s − 3 + 142,920 T ϱ + 3 , ς − 1 , s − 2 + 1,418,481 T ϱ + 3 , ς − 1 , s − 1 + 2,877,456 T ϱ + 3 , ς − 1 , s + 1,418,481 T ϱ + 3 , ς − 1 , s + 1 + 142,920 T ϱ + 3 , ς − 1 , s + 2 + 1,191 T ϱ + 3 , ς − 1 , s + 3 + 2,416 T ϱ + 3 , ς , s − 3 + 289,920 T ϱ + 3 , ς , s − 2 + 2,877,456 T ϱ + 3 , ς , s − 1 + 5,837,056 T ϱ + 3 , ς , s + 2,877,456 T ϱ + 3 , ς , s + 1 + 289,920 T ϱ + 3 , ς , s + 2 + 2,416 T ϱ + 3 , ς , s + 3 + 1,191 T ϱ + 3 , ς + 1 , s − 3 + 142,920 T ϱ + 3 , ς + 1 , s − 2 + 1,418,481 T ϱ + 3 , ς + 1 , s − 1 + 2,877,456 T ϱ + 3 , ς + 1 , s + 1,418,481 T ϱ + 3 , ς + 1 , s + 1 + 142,920 T ϱ + 3 , ς + 1 , s + 2 + 1,191 T ϱ + 3 , ς + 1 , s + 3 + 120 T ϱ + 3 , ς + 2 , s − 3 + 14,400 T ϱ + 3 , ς + 2 , s − 2 + 142,920 T ϱ + 3 , ς + 2 , s − 1 + 289,920 T ϱ + 3 , ς + 2 , s + 142,920 T ϱ + 3 , ς + 2 , s + 1 + 14,400 T ϱ + 3 , ς + 2 , s + 2 + 120 T ϱ + 3 , ς + 2 , s + 3 + T ϱ + 3 , ς + 3 , s − 3 + 120 T ϱ + 3 , ς + 3 , s − 2 + 1,191 T ϱ + 3 , ς + 3 , s − 1 + 2,416 T ϱ + 3 , ς + 3 , s + 1,191 T ϱ + 3 , ς + 3 , s + 1 + 120 T ϱ + 3 , ς + 3 , s + 2 + T ϱ + 3 , ς + 3 , s + 3 .

∂ H ϱ , ς , s ∂ x = − 7 h ( T ϱ − 3 , ς − 3 , s − 3 + 120 T ϱ − 3 , ς − 3 , s − 2 + 1,191 T ϱ − 3 , ς − 3 , s − 1 + 2,416 T ϱ − 3 , ς − 3 , s + 1,191 T ϱ − 3 , ς − 3 , s + 1 + 120 T ϱ − 3 , ς − 3 , s + 2 + T ϱ − 3 , ς − 3 , s + 3 + 120 T ϱ − 3 , ς − 2 , s − 3 + 14,400 T ϱ − 3 , ς − 2 , s − 2 + 142,920 T ϱ − 3 , ς − 2 , s − 1 + 289,920 T ϱ − 3 , ς − 2 , s + 142,920 T ϱ − 3 , ς − 2 , s + 1 + 14,400 T ϱ − 3 , ς − 2 , s + 2 + 120 T ϱ − 3 , ς − 2 , s + 3 + 1,191 T ϱ − 3 , ς − 1 , s − 3 + 142,920 T ϱ − 3 , ς − 1 , s − 2 + 1,418,481 T ϱ − 3 , ς − 1 , s − 1 + 2,877,456 T ϱ − 3 , ς − 1 , s + 1,418,481 T ϱ − 3 , ς − 1 , s + 1 + 142,920 T ϱ − 3 , ς − 1 , s + 2 + 1,191 T ϱ − 3 , ς − 1 , s + 3 + 2,416 T ϱ − 3 , ς , s − 3 + 289,920 T ϱ − 3 , ς , s − 2 + 2,877,456 T ϱ − 3 , ς , s − 1 + 5,837,056 T ϱ − 3 , ς , s + 2,877,456 T ϱ − 3 , ς , s + 1 + 289,920 T ϱ − 3 , ς , s + 2 + 2,416 T ϱ − 3 , ς , s + 3 + 1,191 T ϱ − 3 , ς + 1 , s − 3 + 142,920 T ϱ − 3 , ς + 1 , s − 2 + 1,418,481 T ϱ − 3 , ς + 1 , s − 1 + 2,877,456 T ϱ − 3 , ς + 1 , s + 1,418,481 T ϱ − 3 , ς + 1 , s + 1 + 142,920 T ϱ − 3 , ς + 1 , s + 2 + 1,191 T ϱ − 3 , ς + 1 , s + 3 + 120 T ϱ − 3 , ς + 2 , s − 3 + 14,400 T ϱ − 3 , ς + 2 , s − 2 + 142,920 T ϱ − 3 , ς + 2 , s − 1 + 289,920 T ϱ − 3 , ς + 2 , s + 142,920 T ϱ − 3 , ς + 2 , s + 1 + 14,400 T ϱ − 3 , ς + 2 , s + 2 + 120 T ϱ − 3 , ς + 2 , s + 3 + T ϱ − 3 , ς + 3 , s − 3 + 120 T ϱ − 3 , ς + 3 , s − 2 + 1,191 T ϱ − 3 , ς + 3 , s − 1 + 2,416 T ϱ − 3 , ς + 3 , s + 1,191 T ϱ − 3 , ς + 3 , s + 1 + 120 T ϱ − 3 , ς + 3 , s + 2 + T ϱ − 3 , ς + 3 , s + 3 + 56 T ϱ − 2 , ς − 3 , s − 3 + 6,720 T ϱ − 2 , ς − 3 , s − 2 + 66,696 T ϱ − 2 , ς − 3 , s − 1 + 135,296 T ϱ − 2 , ς − 3 , s + 66,696 T ϱ − 2 , ς − 3 , s + 1 + 6,720 T ϱ − 2 , ς − 3 , s + 2 + 56 T ϱ − 2 , ς − 3 , s + 3 + 6,720 T ϱ − 2 , ς − 2 , s − 3 + 806,400 T ϱ − 2 , ς − 2 , s − 2 + 8,003,520 T ϱ − 2 , ς − 2 , s − 1 + 16,235,520 T ϱ − 2 , ς − 2 , s + 8,003,520 T ϱ − 2 , ς − 2 , s + 1 + 806,400 T ϱ − 2 , ς − 2 , s + 2 + 6,720 T ϱ − 2 , ς − 2 , s + 3 + 66,696 T ϱ − 2 , ς − 1 , s − 3 + 8,003,520 T ϱ − 2 , ς − 1 , s − 2 + 79,434,936 T ϱ − 2 , ς − 1 , s − 1 + 161,137,536 T ϱ − 2 , ς − 1 , s + 79,434,936 T ϱ − 2 , ς − 1 , s + 1 + 8,003,520 T ϱ − 2 , ς − 1 , s + 2 + 66,696 T ϱ − 2 , ς − 1 , s + 3 + 135,296 T ϱ − 2 , ς , s − 3 + 16,235,520 T ϱ − 2 , ς , s − 2 + 161,137,536 T ϱ − 2 , ς , s − 1 + 326,875,136 T ϱ − 2 , ς , s + 161,137,536 T ϱ − 2 , ς , s + 1 + 16,235,520 T ϱ − 2 , ς , s + 2 + 135,296 T ϱ − 2 , ς , s + 3 + 66,696 T ϱ − 2 , ς + 1 , s − 3 + 8,003,520 T ϱ − 2 , ς + 1 , s − 2 + 79,434,936 T ϱ − 2 , ς + 1 , s − 1 + 161,137,536 T ϱ − 2 , ς + 1 , s + 79,434,936 T ϱ − 2 , ς + 1 , s + 1 + 8,003,520 T ϱ − 2 , ς + 1 , s + 2 + 66,696 T ϱ − 2 , ς + 1 , s + 3 + 6,720 T ϱ − 2 , ς + 2 , s − 3 + 806,400 T ϱ − 2 , ς + 2 , s − 2 + 8,003,520 T ϱ − 2 , ς + 2 , s − 1 + 16,235,520 T ϱ − 2 , ς + 2 , s + 8,003,520 T ϱ − 2 , ς + 2 , s + 1 + 806,400 T ϱ − 2 , ς + 2 , s + 2 + 6,720 T ϱ − 2 , ς + 2 , s + 3 + 56 T ϱ − 2 , ς + 3 , s − 3 + 6,720 T ϱ − 2 , ς + 3 , s − 2 + 66,696 T ϱ − 2 , ς + 3 , s − 1 + 135,296 T ϱ − 2 , ς + 3 , s + 66,696 T ϱ − 2 , ς + 3 , s + 1 + 6,720 T ϱ − 2 , ς + 3 , s + 2 + 56 T ϱ − 2 , ς + 3 , s + 3 + 245 T ϱ − 1 , ς − 3 , s − 3 + 29,400 T ϱ − 1 , ς − 3 , s − 2 + 291,795 T ϱ − 1 , ς − 3 , s − 1 + 591,920 T ϱ − 1 , ς − 3 , s + 291,795 T ϱ − 1 , ς − 3 , s + 1 + 29,400 T ϱ − 1 , ς − 3 , s + 2 + 245 T ϱ − 1 , ς − 3 , s + 3 + 29,400 T ϱ − 1 , ς − 2 , s − 3 + 3,528,000 T ϱ − 1 , ς − 2 , s − 2 + 35,015,400 T ϱ − 1 , ς − 2 , s − 1 + 71,030,400 T ϱ − 1 , ς − 2 , s + 35,015,400 T ϱ − 1 , ς − 2 , s + 1 + 3,528,000 T ϱ − 1 , ς − 2 , s + 2 + 29,400 T ϱ − 1 , ς − 2 , s + 3 + 291,795 T ϱ − 1 , ς − 1 , s − 3 + 35,015,400 T ϱ − 1 , ς − 1 , s − 2 + 347,527,845 T ϱ − 1 , ς − 1 , s − 1 + 704,976,720 T ϱ − 1 , ς − 1 , s + 347,527,845 T ϱ − 1 , ς − 1 , s + 1 + 35,015,400 T ϱ − 1 , ς − 1 , s + 2 + 291,795 T ϱ − 1 , ς − 1 , s + 3 + 591,920 T ϱ − 1 , ς , s − 3 + 71,030,400 T ϱ − 1 , ς , s − 2 + 704,976,720 T ϱ − 1 , ς , s − 1 + 1,430,078,720 T ϱ − 1 , ς , s + 704,976,720 T ϱ − 1 , ς , s + 1 + 71,030,400 T ϱ − 1 , ς , s + 2 + 591,920 T ϱ − 1 , ς , s + 3 + 291,795 T ϱ − 1 , ς + 1 , s − 3 + 35,015,400 T ϱ − 1 , ς + 1 , s − 2 + 347,527,845 T ϱ − 1 , ς + 1 , s − 1

(10) + 704,976,720 T ϱ − 1 , ς + 1 , s + 347,527,845 T ϱ − 1 , ς + 1 , s + 1 + 35,015,400 T ϱ − 1 , ς + 1 , s + 2 + 291,795 T ϱ − 1 , ς + 1 , s + 3 + 29,400 T ϱ − 1 , ς + 2 , s − 3 + 3,528,000 T ϱ − 1 , ς + 2 , s − 2 + 35,015,400 T ϱ − 1 , ς + 2 , s − 1 + 71,030,400 T ϱ − 1 , ς + 2 , s + 35,015,400 T ϱ − 1 , ς + 2 , s + 1 + 3,528,000 T ϱ − 1 , ς + 2 , s + 2 + 29,400 T ϱ − 1 , ς + 2 , s + 3 + 245 T ϱ − 1 , ς + 3 , s − 3 + 29,400 T ϱ − 1 , ς + 3 , s − 2 + 291,795 T ϱ − 1 , ς + 3 , s − 1 + 591,920 T ϱ − 1 , ς + 3 , s + 291,795 T ϱ − 1 , ς + 3 , s + 1 + 29,400 T ϱ − 1 , ς + 3 , s + 2 + 245 T ϱ − 1 , ς + 3 , s + 3 − 245 T ϱ + 1 , ς − 3 , s − 3 − 29,400 T ϱ + 1 , ς − 3 , s − 2 − 291,795 T ϱ + 1 , ς − 3 , s − 1 − 591,920 T ϱ + 1 , ς − 3 , s − 291,795 T ϱ + 1 , ς − 3 , s + 1 − 29,400 T ϱ + 1 , ς − 3 , s + 2 − 245 T ϱ + 1 , ς − 3 , s + 3 − 29,400 T ϱ + 1 , ς − 2 , s − 3 − 3,528,000 T ϱ + 1 , ς − 2 , s − 2 − 35,015,400 T ϱ + 1 , ς − 2 , s − 1 − 71,030,400 T ϱ + 1 , ς − 2 , s − 35,015,400 T ϱ + 1 , ς − 2 , s + 1 − 3,528,000 T ϱ + 1 , ς − 2 , s + 2 − 29,400 T ϱ + 1 , ς − 2 , s + 3 − 291,795 T ϱ + 1 , ς − 1 , s − 3 − 35,015,400 T ϱ + 1 , ς − 1 , s − 2 − 347,527,845 T ϱ + 1 , ς − 1 , s − 1 − 704,976,720 T ϱ + 1 , ς − 1 , s − 347,527,845 T ϱ + 1 , ς − 1 , s + 1 − 35,015,400 T ϱ + 1 , ς − 1 , s + 2 − 291,795 T ϱ + 1 , ς − 1 , s + 3 − 591,920 T ϱ + 1 , ς , s − 3 − 71,030,400 T ϱ + 1 , ς , s − 2 − 704,976,720 T ϱ + 1 , ς , s − 1 − 1,430,078,720 T ϱ + 1 , ς , s − 704,976,720 T ϱ + 1 , ς , s + 1 − 71,030,400 T ϱ + 1 , ς , s + 2 − 591,920 T ϱ + 1 , ς , s + 3 − 291,795 T ϱ + 1 , ς + 1 , s − 3 − 35,015,400 T ϱ + 1 , ς + 1 , s − 2 − 347,527,845 T ϱ + 1 , ς + 1 , s − 1 − 704,976,720 T ϱ + 1 , ς + 1 , s − 347,527,845 T ϱ + 1 , ς + 1 , s + 1 − 35,015,400 T ϱ + 1 , ς + 1 , s + 2 − 291,795 T ϱ + 1 , ς + 1 , s + 3 − 29,400 T ϱ + 1 , ς + 2 , s − 3 − 3,528,000 T ϱ + 1 , ς + 2 , s − 2 − 35,015,400 T ϱ + 1 , ς + 2 , s − 1 − 71,030,400 T ϱ + 1 , ς + 2 , s − 35,015,400 T ϱ + 1 , ς + 2 , s + 1 − 3,528,000 T ϱ + 1 , ς + 2 , s + 2 − 29,400 T ϱ + 1 , ς + 2 , s + 3 − 245 T ϱ + 1 , ς + 3 , s − 3 − 29,400 T ϱ + 1 , ς + 3 , s − 2 − 291,795 T ϱ + 1 , ς + 3 , s − 1 − 591,920 T ϱ + 1 , ς + 3 , s − 291,795 T ϱ + 1 , ς + 3 , s + 1 − 29,400 T ϱ + 1 , ς + 3 , s + 2 − 245 T ϱ + 1 , ς + 3 , s + 3 − 56 T ϱ + 2 , ς − 3 , s − 3 − 6,720 T ϱ + 2 , ς − 3 , s − 2 − 66,696 T ϱ + 2 , ς − 3 , s − 1 − 135,296 T ϱ + 2 , ς − 3 , s − 66,696 T ϱ + 2 , ς − 3 , s + 1 − 6,720 T ϱ + 2 , ς − 3 , s + 2 − 56 T ϱ + 2 , ς − 3 , s + 3 − 6,720 T ϱ + 2 , ς − 2 , s − 3 − 806,400 T ϱ + 2 , ς − 2 , s − 2 − 8,003,520 T ϱ + 2 , ς − 2 , s − 1 − 16,235,520 T ϱ + 2 , ς − 2 , s − 8,003,520 T ϱ + 2 , ς − 2 , s + 1 − 806,400 T ϱ + 2 , ς − 2 , s + 2 − 6,720 T ϱ + 2 , ς − 2 , s + 3 − 66,696 T ϱ + 2 , ς − 1 , s − 3 − 8,003,520 T ϱ + 2 , ς − 1 , s − 2 − 79,434,936 T ϱ + 2 , ς − 1 , s − 1 − 161,137,536 T ϱ + 2 , ς − 1 , s − 79,434,936 T ϱ + 2 , ς − 1 , s + 1 − 8,003,520 T ϱ + 2 , ς − 1 , s + 2 − 66,696 T ϱ + 2 , ς − 1 , s + 3 − 135,296 T ϱ + 2 , ς , s − 3 − 16,235,520 T ϱ + 2 , ς , s − 2 − 161,137,536 T ϱ + 2 , ς , s − 1 − 326,875,136 T ϱ + 2 , ς , s − 161,137,536 T ϱ + 2 , ς , s + 1 − 16,235,520 T ϱ + 2 , ς , s + 2 − 135,296 T ϱ + 2 , ς , s + 3 − 66,696 T ϱ + 2 , ς + 1 , s − 3 − 8,003,520 T ϱ + 2 , ς + 1 , s − 2 − 79,434,936 T ϱ + 2 , ς + 1 , s − 1 − 161,137,536 T ϱ + 2 , ς + 1 , s − 79,434,936 T ϱ + 2 , ς + 1 , s + 1 − 8,003,520 T ϱ + 2 , ς + 1 , s + 2 − 66,696 T ϱ + 2 , ς + 1 , s + 3 − 6,720 T ϱ + 2 , ς + 2 , s − 3 − 806,400 T ϱ + 2 , ς + 2 , s − 2 − 8,003,520 T ϱ + 2 , ς + 2 , s − 1 − 16,235,520 T ϱ + 2 , ς + 2 , s − 8,003,520 T ϱ + 2 , ς + 2 , s + 1 − 806,400 T ϱ + 2 , ς + 2 , s + 2 − 6,720 T ϱ + 2 , ς + 2 , s + 3 − 56 T ϱ + 2 , ς + 3 , s − 3 − 6,720 T ϱ + 2 , ς + 3 , s − 2 − 66,696 T ϱ + 2 , ς + 3 , s − 1 − 135,296 T ϱ + 2 , ς + 3 , s − 66,696 T ϱ + 2 , ς + 3 , s + 1 − 6,720 T ϱ + 2 , ς + 3 , s + 2 − 56 T ϱ + 2 , ς + 3 , s + 3 − T ϱ + 3 , ς − 3 , s − 3 − 120 T ϱ + 3 , ς − 3 , s − 2 − 1,191 T ϱ + 3 , ς − 3 , s − 1 − 2,416 T ϱ + 3 , ς − 3 , s − 1,191 T ϱ + 3 , ς − 3 , s + 1 − 120 T ϱ + 3 , ς − 3 , s + 2 − T ϱ + 3 , ς − 3 , s + 3 − 120 T ϱ + 3 , ς − 2 , s − 3 − 14,400 T ϱ + 3 , ς − 2 , s − 2 − 142,920 T ϱ + 3 , ς − 2 , s − 1 − 289,920 T ϱ + 3 , ς − 2 , s − 142,920 T ϱ + 3 , ς − 2 , s + 1 − 14,400 T ϱ + 3 , ς − 2 , s + 2 − 120 T ϱ + 3 , ς − 2 , s + 3 − 1,191 T ϱ + 3 , ς − 1 , s − 3 − 142,920 T ϱ + 3 , ς − 1 , s − 2 − 1,418,481 T ϱ + 3 , ς − 1 , s − 1 − 2,877,456 T ϱ + 3 , ς − 1 , s − 1,418,481 T ϱ + 3 , ς − 1 , s + 1 − 142,920 T ϱ + 3 , ς − 1 , s + 2 − 1,191 T ϱ + 3 , ς − 1 , s + 3 − 2,416 T ϱ + 3 , ς , s − 3 − 289,920 T ϱ + 3 , ς , s − 2 − 2,877,456 T ϱ + 3 , ς , s − 1 − 5,837,056 T ϱ + 3 , ς , s − 2,877,456 T ϱ + 3 , ς , s + 1 − 289,920 T ϱ + 3 , ς , s + 2 − 2,416 T ϱ + 3 , ς , s + 3 − 1,191 T ϱ + 3 , ς + 1 , s − 3 − 142,920 T ϱ + 3 , ς + 1 , s − 2 − 1,418,481 T ϱ + 3 , ς + 1 , s − 1 − 2,877,456 T ϱ + 3 , ς + 1 , s − 1,418,481 T ϱ + 3 , ς + 1 , s + 1 − 142,920 T ϱ + 3 , ς + 1 , s + 2 − 1,191 T ϱ + 3 , ς + 1 , s + 3 − 120 T ϱ + 3 , ς + 2 , s − 3 − 14,400 T ϱ + 3 , ς + 2 , s − 2 − 142,920 T ϱ + 3 , ς + 2 , s − 1 − 289,920 T ϱ + 3 , ς + 2 , s − 142,920 T ϱ + 3 , ς + 2 , s + 1 − 14,400 T ϱ + 3 , ς + 2 , s + 2 − 120 T ϱ + 3 , ς + 2 , s + 3 − T ϱ + 3 , ς + 3 , σ − 3 − 120 T ϱ + 3 , ς + 3 , σ − 2 − 1,191 T ϱ + 3 , ς + 3 , σ − 1 − 2,416 T ϱ + 3 , ς + 3 , σ − 1,191 T ϱ + 3 , ς + 3 , σ + 1 − 120 T ϱ + 3 , ς + 3 , s + 2 − T ϱ + 3 , ς + 3 , s + 3 ) ,

(11) ∂ H ϱ , ς , s ∂ y = − 7 k ( T ϱ − 3 , ς − 3 , s − 3 + 120 T ϱ − 3 , ς − 3 , s − 2 + 1,191 T ϱ − 3 , ς − 3 , s − 1 + 2,416 T ϱ − 3 , ς − 3 , s + 1,191 T ϱ − 3 , ς − 3 , s + 1 + 120 T ϱ − 3 , ς − 3 , s + 2 + T ϱ − 3 , ς − 3 , s + 3 + 56 T ϱ − 3 , ς − 2 , s − 3 + 6,720 T ϱ − 3 , ς − 2 , s − 2 + 66,696 T ϱ − 3 , ς − 2 , s − 1 + 135,296 T ϱ − 3 , ς − 2 , s + 66,696 T ϱ − 3 , ς − 2 , s + 1 + 6,720 T ϱ − 3 , ς − 2 , s + 2 + 56 T ϱ − 3 , ς − 2 , s + 3 + 245 T ϱ − 3 , ς − 1 , s − 3 + 29,400 T ϱ − 3 , ς − 1 , s − 2 + 291,795 T ϱ − 3 , ς − 1 , s − 1 + 591,920 T ϱ − 3 , ς − 1 , s + 291,795 T ϱ − 3 , ς − 1 , s + 1 + 29,400 T ϱ − 3 , ς − 1 , s + 2 + 245 T ϱ − 3 , ς − 1 , s + 3 − 245 T ϱ − 3 , ς + 1 , s − 3 − 29,400 T ϱ − 3 , ς + 1 , s − 2 − 291,795 T ϱ − 3 , ς + 1 , s − 1 − 591,920 T ϱ − 3 , ς + 1 , s − 291,795 T ϱ − 3 , ς + 1 , s + 1 − 29,400 T ϱ − 3 , ς + 1 , s + 2 − 245 T ϱ − 3 , ς + 1 , s + 3 − 56 T ϱ − 3 , ς + 2 , s − 3 − 6,720 T ϱ − 3 , ς + 2 , s − 2 − 66,696 T ϱ − 3 , ς + 2 , s − 1 − 135,296 T ϱ − 3 , ς + 2 , s − 66,696 T ϱ − 3 , ς + 2 , s + 1 − 6,720 T ϱ − 3 , ς + 2 , s + 2 − 56 T ϱ − 3 , ς + 2 , s + 3 − T ϱ − 3 , ς + 3 , s − 3 − 120 T ϱ − 3 , ς + 3 , s − 2 − 1,191 T ϱ − 3 , ς + 3 , s − 1 − 2,416 T ϱ − 3 , ς + 3 , s − 1,191 T ϱ − 3 , ς + 3 , s + 1 − 120 T ϱ − 3 , ς + 3 , s + 2 − T ϱ − 3 , ς + 3 , s + 3 + 120 T ϱ − 2 , ς − 3 , s − 3 + 14,400 T ϱ − 2 , ς − 3 , s − 2 + 142,920 T ϱ − 2 , ς − 3 , s − 1 + 289,920 T ϱ − 2 , ς − 3 , s + 6,720 T ϱ − 2 , ς − 2 , s − 3 + 806,400 T ϱ − 2 , ς − 2 , s − 2 + 8,003,520 T ϱ − 2 , ς − 2 , s − 1 + 16,235,520 T ϱ − 2 , ς − 2 , s + 8,003,520 T ϱ − 2 , ς − 2 , s + 1 + 806,400 T ϱ − 2 , ς − 2 , s + 2 + 6,720 T ϱ − 2 , ς − 2 , s + 3 + 29,400 T ϱ − 2 , ς − 1 , s − 3 + 3,528,000 T ϱ − 2 , ς − 1 , s − 2 + 35,015,400 T ϱ − 2 , ς − 1 , s − 1 + 71,030,400 T ϱ − 2 , ς − 1 , s + 35,015,400 T ϱ − 2 , ς − 1 , s + 1 + 3,528,000 T ϱ − 2 , ς − 1 , s + 2 + 29,400 T ϱ − 2 , ς − 1 , s + 3 − 29,400 T ϱ − 2 , ς + 1 , s − 3 − 3,528,000 T ϱ − 2 , ς + 1 , s − 2 − 35,015,400 T ϱ − 2 , ς + 1 , s − 1 − 71,030,400 T ϱ − 2 , ς + 1 , s − 35,015,400 T ϱ − 2 , ς + 1 , s + 1 − 3,528,000 T ϱ − 2 , ς + 1 , s + 2 − 29,400 T ϱ − 2 , ς + 1 , s + 3 − 6,720 T ϱ − 2 , ς + 2 , s − 3 − 806,400 T ϱ − 2 , ς + 2 , s − 2 − 8,003,520 T ϱ − 2 , ς + 2 , s − 1 − 16,235,520 T ϱ − 2 , ς + 2 , s − 8,003,520 T ϱ − 2 , ς + 2 , s + 1 − 806,400 T ϱ − 2 , ς + 2 , s + 2 − 6,720 T ϱ − 2 , ς + 2 , s + 3 − 120 T ϱ − 2 , ς + 3 , s − 3 − 14,400 T ϱ − 2 , ς + 3 , s − 2 − 142,920 T ϱ − 2 , ς + 3 , s − 1 − 289,920 T ϱ − 2 , ς + 3 , s − 142,920 T ϱ − 2 , ς + 3 , s + 1 − 14,400 T ϱ − 2 , ς + 3 , s + 2 − 120 T ϱ − 2 , ς + 3 , s + 3 + 1,191 T ϱ − 1 , ς − 3 , s − 3 + 142,920 T ϱ − 1 , ς − 3 , s − 2 + 1,418,481 T ϱ − 1 , ς − 3 , s − 1 + 2,877,456 T ϱ − 1 , ς − 3 , s + 1,418,481 T ϱ − 1 , ς − 3 , s + 1 + 142,920 T ϱ − 1 , ς − 3 , s + 2 + 1,191 T ϱ − 1 , ς − 3 , s + 3 + 66,696 T ϱ − 1 , ς − 2 , s − 3 + 8,003,520 T ϱ − 1 , ς − 2 , s − 2 + 79,434,936 T ϱ − 1 , ς − 2 , s − 1 + 161,137,536 T ϱ − 1 , ς − 2 , s + 79,434,936 T ϱ − 1 , ς − 2 , s + 1 + 8,003,520 T ϱ − 1 , ς − 2 , s + 2 + 66,696 T ϱ − 1 , ς − 2 , s + 3 + 291,795 T ϱ − 1 , ς − 1 , s − 3 + 35,015,400 T ϱ − 1 , ς − 1 , s − 2 + 347,527,845 T ϱ − 1 , ς − 1 , s − 1 + 704,976,720 T ϱ − 1 , ς − 1 , s + 347,527,845 T ϱ − 1 , ς − 1 , s + 1 + 35,015,400 T ϱ − 1 , ς − 1 , s + 2 + 291,795 T ϱ − 1 , ς − 1 , s + 3 − 291,795 T ϱ − 1 , ς + 1 , s − 3 − 35,015,400 T ϱ − 1 , ς + 1 , s − 2 − 347,527,845 T ϱ − 1 , ς + 1 , s − 1 − 704,976,720 T ϱ − 1 , ς + 1 , s − 347,527,845 T ϱ − 1 , ς + 1 , s + 1 − 35,015,400 T ϱ − 1 , ς + 1 , s + 2 − 291,795 T ϱ − 1 , ς + 1 , s + 3 − 66,696 T ϱ − 1 , ς + 2 , s − 3 − 8,003,520 T ϱ − 1 , ς + 2 , s − 2 − 79,434,936 T ϱ − 1 , ς + 2 , s − 1 − 161,137,536 T ϱ − 1 , ς + 2 , s − 79,434,936 T ϱ − 1 , ς + 2 , s + 1 − 8,003,520 T ϱ − 1 , ς + 2 , s + 2 − 66,696 T ϱ − 1 , ς + 2 , s + 3 − 1,191 T ϱ − 1 , ς + 3 , s − 3 − 142,920 T ϱ − 1 , ς + 3 , s − 2 − 1,418,481 T ϱ − 1 , ς + 3 , s − 1 − 2,877,456 T ϱ − 1 , ς + 3 , s − 1,418,481 T ϱ − 1 , ς + 3 , s + 1 − 142,920 T ϱ − 1 , ς + 3 , s + 2 − 1,191 T ϱ − 1 , ς + 3 , s + 3 + 2,416 T ϱ , ς − 3 , s − 3 + 289,920 T ϱ , ς − 3 , s − 2 + 2,877,456 T ϱ , ς − 3 , s − 1 + 5,837,056 T ϱ , ς − 3 , s + 2,877,456 T ϱ , ς − 3 , s + 1 + 289,920 T ϱ , ς − 3 , s + 2 + 2,416 T ϱ , ς − 3 , s + 3 + 135,296 T ϱ , ς − 2 , s − 3 + 16,235,520 T ϱ , ς − 2 , s − 2 + 161,137,536 T ϱ , ς − 2 , s − 1 + 326,875,136 T ϱ , ς − 2 , s + 161,137,536 T ϱ , ς − 2 , s + 1 + 16,235,520 T ϱ , ς − 2 , s + 2 + 135,296 T ϱ , ς − 2 , s + 3 + 591,920 T ϱ , ς − 1 , s − 3 + 71,030,400 T ϱ , ς − 1 , s − 2 + 704,976,720 T ϱ , ς − 1 , s − 1 + 1,430,078,720 T ϱ , ς − 1 , s + 704,976,720 T ϱ , ς − 1 , s + 1 + 71,030,400 T ϱ , ς − 1 , s + 2 + 591,920 T ϱ , ς − 1 , s + 3 − 591,920 T ϱ , ς + 1 , s − 3 − 71,030,400 T ϱ , ς + 1 , s − 2 − 704,976,720 T ϱ , ς + 1 , s − 1 − 1,430,078,720 T ϱ , ς + 1 , s − 704,976,720 T ϱ , ς + 1 , s + 1 − 71,030,400 T ϱ , ς + 1 , s + 2 − 591,920 T ϱ , ς + 1 , s + 3 − 135,296 T ϱ , ς + 2 , s − 3 − 16,235,520 T ϱ , ς + 2 , s − 2 − 161,137,536 T ϱ , ς + 2 , s − 1 − 326,875,136 T ϱ , ς + 2 , s − 161,137,536 T ϱ , ς + 2 , s + 1 − 16,235,520 T ϱ , ς + 2 , s + 2 − 135,296 T ϱ , ς + 2 , s + 3 − 2,416 T ϱ , ς + 3 , s − 3 − 289,920 T ϱ , ς + 3 , s − 2 − 2,877,456 T ϱ , ς + 3 , s − 1 − 5,837,056 T ϱ , ς + 3 , s − 2,877,456 T ϱ , ς + 3 , s + 1 − 289,920 T ϱ , ς + 3 , s + 2 − 2,416 T ϱ , ς + 3 , s + 3 + 1,191 T ϱ + 1 , ς − 3 , s − 3 + 142,920 T ϱ + 1 , ς − 3 , s − 2 + 1,418,481 T ϱ + 1 , ς − 3 , s − 1 + 2,877,456 T ϱ + 1 , ς − 3 , s + 1,418,481 T ϱ + 1 , ς − 3 , s + 1 + 142,920 T ϱ + 1 , ς − 3 , s + 2 + 1,191 T ϱ + 1 , ς − 3 , s + 3 + 66,696 T ϱ + 1 , ς − 2 , s − 3 + 8,003,520 T ϱ + 1 , ς − 2 , s − 2 + 79,434,936 T ϱ + 1 , ς − 2 , s − 1 + 161,137,536 T ϱ + 1 , ς − 2 , s + 79,434,936 T ϱ + 1 , ς − 2 , s + 1 + 8,003,520 T ϱ + 1 , ς − 2 , s + 2 + 66,696 T ϱ + 1 , ς − 2 , s + 3 + 291,795 T ϱ + 1 , ς − 1 , s − 3 + 35,015,400 T ϱ + 1 , ς − 1 , s − 2 + 347,527,845 T ϱ + 1 , ς − 1 , s − 1 + 704,976,720 T ϱ + 1 , ς − 1 , s + 347,527,845 T ϱ + 1 , ς − 1 , s + 1 + 35,015,400 T ϱ + 1 , ς − 1 , s + 2 + 291,795 T ϱ + 1 , ς − 1 , s + 3 − 291,795 T ϱ + 1 , ς + 1 , s − 3 − 35,015,400 T ϱ + 1 , ς + 1 , s − 2 − 347,527,845 T ϱ + 1 , ς + 1 , s − 1 − 704,976,720 T ϱ + 1 , ς + 1 , s − 347,527,845 T ϱ + 1 , ς + 1 , s + 1 − 35,015,400 T ϱ + 1 , ς + 1 , s + 2 − 291,795 T ϱ + 1 , ς + 1 , s + 3 − 66,696 T ϱ + 1 , ς + 2 , s − 3 − 8,003,520 T ϱ + 1 , ς + 2 , s − 2 − 79,434,936 T ϱ + 1 , ς + 2 , s − 1 − 161,137,536 T ϱ + 1 , ς + 2 , s − 79,434,936 T ϱ + 1 , ς + 2 , s + 1 − 8,003,520 T ϱ + 1 , ς + 2 , s + 2 − 66,696 T ϱ + 1 , ς + 2 , s + 3 − 1,191 T ϱ + 1 , ς + 3 , s − 3 − 142,920 T ϱ + 1 , ς + 3 , s − 2 − 1,418,481 T ϱ + 1 , ς + 3 , s − 1 − 2,877,456 T ϱ + 1 , ς + 3 , s − 1,418,481 T ϱ + 1 , ς + 3 , s + 1 − 142,920 T ϱ + 1 , ς + 3 , s + 2 − 1,191 T ϱ + 1 , ς + 3 , s + 3 + 120 T ϱ + 2 , ς − 3 , s − 3 + 14,400 T ϱ + 2 , ς − 3 , s − 2 + 142,920 T ϱ + 2 , ς − 3 , s − 1 + 289,920 T ϱ + 2 , ς − 3 , s + 142,920 T ϱ + 2 , ς − 3 , s + 1 + 14,400 T ϱ + 2 , ς − 3 , s + 2 + 120 T ϱ + 2 , ς − 3 , s + 3 + 6,720 T ϱ + 2 , ς − 2 , s − 3 + 806,400 T ϱ + 2 , ς − 2 , s − 2 + 8,003,520 T ϱ + 2 , ς − 2 , s − 1 + 16,235,520 T ϱ + 2 , ς − 2 , s + 8,003,520 T ϱ + 2 , ς − 2 , s + 1 + 806,400 T ϱ + 2 , ς − 2 , s + 2 + 6,720 T ϱ + 2 , ς − 2 , s + 3 + 29,400 T ϱ + 2 , ς − 1 , s − 3 + 3,528,000 T ϱ + 2 , ς − 1 , s − 2 + 35,015,400 T ϱ + 2 , ς − 1 , s − 1 + 71,030,400 T ϱ + 2 , ς − 1 , s + 35,015,400 T ϱ + 2 , ς − 1 , s + 1 + 3,528,000 T ϱ + 2 , ς − 1 , s + 2 + 29,400 T ϱ + 2 , ς − 1 , s + 3 − 29,400 T ϱ + 2 , ς + 1 , s − 3 − 3,528,000 T ϱ + 2 , ς + 1 , s − 2 − 35,015,400 T ϱ + 2 , ς + 1 , s − 1 − 71,030,400 T ϱ + 2 , ς + 1 , s − 35,015,400 T ϱ + 2 , ς + 1 , s + 1 − 3,528,000 T ϱ + 2 , ς + 1 , s + 2 − 29,400 T ϱ + 2 , ς + 1 , s + 3 − 6,720 T ϱ + 2 , ς + 2 , s − 3 − 806,400 T ϱ + 2 , ς + 2 , s − 2 − 8,003,520 T ϱ + 2 , ς + 2 , s − 1 − 16,235,520 T ϱ + 2 , ς + 2 , s − 8,003,520 T ϱ + 2 , ς + 2 , s + 1 − 806,400 T ϱ + 2 , ς + 2 , s + 2 − 6,720 T ϱ + 2 , ς + 2 , s + 3 − 120 T ϱ + 2 , ς + 3 , s − 3 − 14,400 T ϱ + 2 , ς + 3 , s − 2 − 142,920 T ϱ + 2 , ς + 3 , s − 1 − 289,920 T ϱ + 2 , ς + 3 , s − 142,920 T ϱ + 2 , ς + 3 , s + 1 − 14,400 T ϱ + 2 , ς + 3 , s + 2 − 120 T ϱ + 2 , ς + 3 , s + 3 + T ϱ + 3 , ς − 3 , s − 3 + 120 T ϱ + 3 , ς − 3 , s − 2 + 1,191 T ϱ + 3 , ς − 3 , s − 1 + 2,416 T ϱ + 3 , ς − 3 , s + 1,191 T ϱ + 3 , ς − 3 , s + 1 + 120 T ϱ + 3 , ς − 3 , s + 2 + T ϱ + 3 , ς − 3 , s + 3 + 56 T ϱ + 3 , ς − 2 , s − 3 + 6,720 T ϱ + 3 , ς − 2 , s − 2 + 66,696 T ϱ + 3 , ς − 2 , s − 1 + 135,296 T ϱ + 3 , ς − 2 , s + 66,696 T ϱ + 3 , ς − 2 , s + 1 + 6,720 T ϱ + 3 , ς − 2 , s + 2 + 56 T ϱ + 3 , ς − 2 , s + 3 + 245 T ϱ + 3 , ς − 1 , s − 3 + 29,400 T ϱ + 3 , ς − 1 , s − 2 + 291,795 T ϱ + 3 , ς − 1 , s − 1 + 591,920 T ϱ + 3 , ς − 1 , s + 291,795 T ϱ + 3 , ς − 1 , s + 1 + 29,400 T ϱ + 3 , ς − 1 , s + 2 + 245 T ϱ + 3 , ς − 1 , s + 3 − 245 T ϱ + 3 , ς + 1 , s − 3 − 29,400 T ϱ + 3 , ς + 1 , s − 2 − 291,795 T ϱ + 3 , ς + 1 , s − 1 − 591,920 T ϱ + 3 , ς + 1 , s − 291,795 T ϱ + 3 , ς + 1 , s + 1 − 29,400 T ϱ + 3 , ς + 1 , s + 2 − 245 T ϱ + 3 , ς + 1 , s + 3 − 56 T ϱ + 3 , ς + 2 , s − 3 − 6,720 T ϱ + 3 , ς + 2 , s − 2 − 66,696 T ϱ + 3 , ς + 2 , s − 1 − 135,296 T ϱ + 3 , ς + 2 , s − 66,696 T ϱ + 3 , ς + 2 , s + 1 − 6,720 T ϱ + 3 , ς + 2 , s + 2 − 56 T ϱ + 3 , ς + 2 , s + 3 − T ϱ + 3 , ς + 3 , s − 3 − 120 T ϱ + 3 , ς + 3 , s − 2 − 2,416 T ϱ + 3 , ς + 3 , s − 1,191 T ϱ + 3 , ς + 3 , s + 1 − 120 T ϱ + 3 , ς + 3 , s + 2 − T ϱ + 3 , ς + 3 , s + 3 ) .

(12) ∂ H ϱ , ς , s ∂ z = − 7 q ( T ϱ − 3 , ς − 3 , s − 3 + 56 T ϱ − 3 , ς − 3 , s − 2 + 245 T ϱ − 3 , ς − 3 , s − 1 − 245 T ϱ − 3 , ς − 3 , s + 1 − 56 T ϱ − 3 , ς − 3 , s + 2 − T ϱ − 3 , ς − 3 , s + 3 + 120 T ϱ − 3 , ς − 2 , s − 3 + 6,720 T ϱ − 3 , ς − 2 , s − 2 + 29,400 T ϱ − 3 , ς − 2 , s − 1 − 29,400 T ϱ − 3 , ς − 2 , s + 1 − 6,720 T ϱ − 3 , ς − 2 , s + 2 − 120 T ϱ − 3 , ς − 2 , s + 3 + 1,191 T ϱ − 3 , ς − 1 , s − 3 + 66,696 T ϱ − 3 , ς − 1 , s − 2 + 291,795 T ϱ − 3 , ς − 1 , s − 1 − 291,795 T ϱ − 3 , ς − 1 , s + 1 − 66,696 T ϱ − 3 , ς − 1 , s + 2 − 1,191 T ϱ − 3 , ς − 1 , s + 3 + 2,416 T ϱ − 3 , ς , s − 3 + 135,296 T ϱ − 3 , ς , s − 2 + 591,920 T ϱ − 3 , ς , s − 1 − 591,920 T ϱ − 3 , ς , s + 1 − 135,296 T ϱ − 3 , ς , s + 2 − 2,416 T ϱ − 3 , ς , s + 3 + 1,191 T ϱ − 3 , ς + 1 , s − 3 + 66,696 T ϱ − 3 , ς + 1 , s − 2 + 291,795 T ϱ − 3 , ς + 1 , s − 1 − 291,795 T ϱ − 3 , ς + 1 , s + 1 − 66,696 T ϱ − 3 , ς + 1 , s + 2 − 1,191 T ϱ − 3 , ς + 1 , s + 3 + 120 T ϱ − 3 , ς + 2 , s − 3 + 6,720 T ϱ − 3 , ς + 2 , s − 2 + 29,400 T ϱ − 3 , ς + 2 , s − 1 − 29,400 T ϱ − 3 , ς + 2 , s + 1 − 6,720 T ϱ − 3 , ς + 2 , s + 2 − 120 T ϱ − 3 , ς + 2 , s + 3 + T ϱ − 3 , ς + 3 , s − 3 + 56 T ϱ − 3 , ς + 3 , s − 2 + 245 T ϱ − 3 , ς + 3 , s − 1 − 245 T ϱ − 3 , ς + 3 , s + 1 − 56 T ϱ − 3 , ς + 3 , s + 2 − T ϱ − 3 , ς + 3 , s + 3 + 120 T ϱ − 2 , ς − 3 , s − 3 + 6,720 T ϱ − 2 , ς − 3 , s − 2 + 29,400 T ϱ − 2 , ς − 3 , s − 1 − 29,400 T ϱ − 2 , ς − 3 , s + 1 − 6,720 T ϱ − 2 , ς − 3 , s + 2 − 120 T ϱ − 2 , ς − 3 , s + 3 + 14,400 T ϱ − 2 , ς − 2 , s − 3 + 806,400 T ϱ − 2 , ς − 2 , s − 2 + 3,528,000 T ϱ − 2 , ς − 2 , s − 1 − 3,528,000 T ϱ − 2 , ς − 2 , s + 1 − 806,400 T ϱ − 2 , ς − 2 , s + 2 − 14,400 T ϱ − 2 , ς − 2 , s + 3 + 142,920 T ϱ − 2 , ς − 1 , s − 3 + 8,003,520 T ϱ − 2 , ς − 1 , s − 2 + 35,015,400 T ϱ − 2 , ς − 1 , s − 1 − 35,015,400 T ϱ − 2 , ς − 1 , s + 1 − 8,003,520 T ϱ − 2 , ς − 1 , s + 2 − 142,920 T ϱ − 2 , ς − 1 , s + 3 + 289,920 T ϱ − 2 , ς , s − 3 + 16,235,520 T ϱ − 2 , ς , s − 2 + 71,030,400 T ϱ − 2 , ς , s − 1 − 71,030,400 T ϱ − 2 , ς , s + 1 − 16,235,520 T ϱ − 2 , ς , s + 2 − 289,920 T ϱ − 2 , ς , s + 3 + 142,920 T ϱ − 2 , ς + 1 , s − 3 + 8,003,520 T ϱ − 2 , ς + 1 , s − 2 + 35,015,400 T ϱ − 2 , ς + 1 , s − 1 − 35,015,400 T ϱ − 2 , ς + 1 , s + 1 − 8,003,520 T ϱ − 2 , ς + 1 , s + 2 − 142,920 T ϱ − 2 , ς + 1 , s + 3 + 14,400 T ϱ − 2 , ς + 2 , s − 3 + 806,400 T ϱ − 2 , ς + 2 , s − 2 + 3,528,000 T ϱ − 2 , ς + 2 , s − 1 − 3,528,000 T ϱ − 2 , ς + 2 , s + 1 − 806,400 T ϱ − 2 , ς + 2 , s + 2 − 14,400 T ϱ − 2 , ς + 2 , s + 3 + 120 T ϱ − 2 , ς + 3 , s − 3 + 6,720 T ϱ − 2 , ς + 3 , s − 2 + 29,400 T ϱ − 2 , ς + 3 , s − 1 − 29,400 T ϱ − 2 , ς + 3 , s + 1 − 6,720 T ϱ − 2 , ς + 3 , s + 2 − 120 T ϱ − 2 , ς + 3 , s + 3 + 1,191 T ϱ − 1 , ς − 3 , s − 3 + 66,696 T ϱ − 1 , ς − 3 , s − 2 + 291,795 T ϱ − 1 , ς − 3 , s − 1 − 291,795 T ϱ − 1 , ς − 3 , s + 1 − 66,696 T ϱ − 1 , ς − 3 , s + 2 − 1,191 T ϱ − 1 , ς − 3 , s + 3 + 142,920 T ϱ − 1 , ς − 2 , s − 3 + 8,003,520 T ϱ − 1 , ς − 2 , s − 2 + 35,015,400 T ϱ − 1 , ς − 2 , s − 1 − 35,015,400 T ϱ − 1 , ς − 2 , s + 1 − 8,003,520 T ϱ − 1 , ς − 2 , s + 2 − 142,920 T ϱ − 1 , ς − 2 , s + 3 + 1,418,481 T ϱ − 1 , ς − 1 , s − 3 + 79,434,936 T ϱ − 1 , ς − 1 , s − 2 + 347,527,845 T ϱ − 1 , ς − 1 , s − 1 − 347,527,845 T ϱ − 1 , ς − 1 , s + 1 − 79,434,936 T ϱ − 1 , ς − 1 , s + 2 − 1,418,481 T ϱ − 1 , ς − 1 , s + 3 + 2,877,456 T ϱ − 1 , ς , s − 3 + 161,137,536 T ϱ − 1 , ς , s − 2 + 704,976,720 T ϱ − 1 , ς , s − 1 − 704,976,720 T ϱ − 1 , ς , s + 1 − 161,137,536 T ϱ − 1 , ς , s + 2 − 2,877,456 T ϱ − 1 , ς , s + 3 + 1,418,481 T ϱ − 1 , ς + 1 , s − 3 + 79,434,936 T ϱ − 1 , ς + 1 , s − 2 + 347,527,845 T ϱ − 1 , ς + 1 , s − 1 − 347,527,845 T ϱ − 1 , ς + 1 , s + 1 − 79,434,936 T ϱ − 1 , ς + 1 , s + 2 − 1,418,481 T ϱ − 1 , ς + 1 , s + 3 + 142,920 T ϱ − 1 , ς + 2 , s − 3 + 8,003,520 T ϱ − 1 , ς + 2 , s − 2 + 35,015,400 T ϱ − 1 , ς + 2 , s − 1 − 35,015,400 T ϱ − 1 , ς + 2 , s + 1 − 8,003,520 T ϱ − 1 , ς + 2 , s + 2 − 142,920 T ϱ − 1 , ς + 2 , s + 3 + 1,191 T ϱ − 1 , ς + 3 , s − 3 + 66,696 T ϱ − 1 , ς + 3 , s − 2 + 291,795 T ϱ − 1 , ς + 3 , s − 1 − 291,795 T ϱ − 1 , ς + 3 , s + 1 − 66,696 T ϱ − 1 , ς + 3 , s + 2 − 1,191 T ϱ − 1 , ς + 3 , s + 3 + 2,416 T ϱ , ς − 3 , s − 3 + 135,296 T ϱ , ς − 3 , s − 2 + 591,920 T ϱ , ς − 3 , s − 1 − 591,920 T ϱ , ς − 3 , s + 1 − 135,296 T ϱ , ς − 3 , s + 2 − 2,416 T ϱ , ς − 3 , s + 3 + 289,920 T ϱ , ς − 2 , s − 3 + 16,235,520 T ϱ , ς − 2 , s − 2 + 71,030,400 T ϱ , ς − 2 , s − 1 − 71,030,400 T ϱ , ς − 2 , s + 1 − 16,235,520 T ϱ , ς − 2 , s + 2 − 289,920 T ϱ , ς − 2 , s + 3 + 2,877,456 T ϱ , ς − 1 , s − 3 + 161,137,536 T ϱ , ς − 1 , s − 2 + 704,976,720 T ϱ , ς − 1 , s − 1 − 704,976,720 T ϱ , ς − 1 , s + 1 − 161,137,536 T ϱ , ς − 1 , s + 2 − 2,877,456 T ϱ , ς − 1 , s + 3 + 5,837,056 T ϱ , ς , s − 3 + 326,875,136 T ϱ , ς , s − 2 + 1,430,078,720 T ϱ , ς , s − 1 − 1,430,078,720 T ϱ , ς , s + 1 − 326,875,136 T ϱ , ς , s + 2 − 5,837,056 T ϱ , ς , s + 3 + 2,877,456 T ϱ , ς + 1 , s − 3 + 161,137,536 T ϱ , ς + 1 , s − 2 + 704,976,720 T ϱ , ς + 1 , s − 1 − 704,976,720 T ϱ , ς + 1 , s + 1 − 161,137,536 T ϱ , ς + 1 , s + 2 − 2,877,456 T ϱ , ς + 1 , s + 3 + 289,920 T ϱ , ς + 2 , s − 3 + 16,235,520 T ϱ , ς + 2 , s − 2 + 71,030,400 T ϱ , ς + 2 , s − 1 − 71,030,400 T ϱ , ς + 2 , s + 1 − 16,235,520 T ϱ , ς + 2 , s + 2 − 289,920 T ϱ , ς + 2 , s + 3 + 2,416 T ϱ , ς + 3 , s − 3 + 135,296 T ϱ , ς + 3 , s − 2 + 591,920 T ϱ , ς + 3 , s − 1 − 591,920 T ϱ , ς + 3 , s + 1 − 135,296 T ϱ , ς + 3 , s + 2 − 2,416 T ϱ , ς + 3 , s + 3 + 1,191 T ϱ + 1 , ς − 3 , s − 3 + 66,696 T ϱ + 1 , ς − 3 , s − 2 + 291,795 T ϱ + 1 , ς − 3 , s − 1 − 291,795 T ϱ + 1 , ς − 3 , s + 1 − 66,696 T ϱ + 1 , ς − 3 , s + 2 − 1,191 T ϱ + 1 , ς − 3 , s + 3 + 142,920 T ϱ + 1 , ς − 2 , s − 3 + 8,003,520 T ϱ + 1 , ς − 2 , s − 2 + 35,015,400 T ϱ + 1 , ς − 2 , s − 1 − 35,015,400 T ϱ + 1 , ς − 2 , s + 1 − 8,003,520 T ϱ + 1 , ς − 2 , s + 2 − 142,920 T ϱ + 1 , ς − 2 , s + 3 + 1,418,481 T ϱ + 1 , ς − 1 , s − 3 + 79,434,936 T ϱ + 1 , ς − 1 , s − 2 + 347,527,845 T ϱ + 1 , ς − 1 , s − 1 − 347,527,845 T ϱ + 1 , ς − 1 , s + 1 − 79,434,936 T ϱ + 1 , ς − 1 , s + 2 − 1,418,481 T ϱ + 1 , ς − 1 , s + 3 + 2,877,456 T ϱ + 1 , ς , s − 3 + 161,137,536 T ϱ + 1 , ς , s − 2 + 704,976,720 T ϱ + 1 , ς , s − 1 − 704,976,720 T ϱ + 1 , ς , s + 1 − 161,137,536 T ϱ + 1 , ς , s + 2 − 2,877,456 T ϱ + 1 , ς , s + 3 + 1,418,481 T ϱ + 1 , ς + 1 , s − 3 + 79,434,936 T ϱ + 1 , ς + 1 , s − 2 + 347,527,845 T ϱ + 1 , ς + 1 , s − 1 − 347,527,845 T ϱ + 1 , ς + 1 , s + 1 − 79,434,936 T ϱ + 1 , ς + 1 , s + 2 − 1,418,481 T ϱ + 1 , ς + 1 , s + 3 + 142,920 T ϱ + 1 , ς + 2 , s − 3 + 8,003,520 T ϱ + 1 , ς + 2 , s − 2 + 35,015,400 T ϱ + 1 , ς + 2 , s − 1 − 35,015,400 T ϱ + 1 , ς + 2 , s + 1 − 8,003,520 T ϱ + 1 , ς + 2 , s + 2 − 142,920 T ϱ + 1 , ς + 2 , s + 3 + 1,191 T ϱ + 1 , ς + 3 , s − 3 + 66,696 T ϱ + 1 , ς + 3 , s − 2 + 291,795 T ϱ + 1 , ς + 3 , s − 1 − 291,795 T ϱ + 1 , ς + 3 , s + 1 − 66,696 T ϱ + 1 , ς + 3 , s + 2 − 1,191 T ϱ + 1 , ς + 3 , s + 3 + 120 T ϱ + 2 , ς − 3 , s − 3 + 6,720 T ϱ + 2 , ς − 3 , s − 2 + 29,400 T ϱ + 2 , ς − 3 , s − 1 − 29,400 T ϱ + 2 , ς − 3 , s + 1 − 6,720 T ϱ + 2 , ς − 3 , s + 2 − 120 T ϱ + 2 , ς − 3 , s + 3 + 14,400 T ϱ + 2 , ς − 2 , s − 3 + 806,400 T ϱ + 2 , ς − 2 , s − 2 + 3,528,000 T ϱ + 2 , ς − 2 , s − 1 − 3,528,000 T ϱ + 2 , ς − 2 , s + 1 − 806,400 T ϱ + 2 , ς − 2 , s + 2 − 14,400 T ϱ + 2 , ς − 2 , s + 3 + 142,920 T ϱ + 2 , ς − 1 , s − 3 + 8,003,520 T ϱ + 2 , ς − 1 , s − 2 + 35,015,400 T ϱ + 2 , ς − 1 , s − 1 − 35,015,400 T ϱ + 2 , ς − 1 , s + 1 − 8,003,520 T ϱ + 2 , ς − 1 , s + 2 − 142,920 T ϱ + 2 , ς − 1 , s + 3 + 289,920 T ϱ + 2 , ς , s − 3 + 16,235,520 T ϱ + 2 , ς , s − 2 + 71,030,400 T ϱ + 2 , ς , s − 1 − 71,030,400 T ϱ + 2 , ς , s + 1 − 16,235,520 T ϱ + 2 , ς , s + 2 − 289,920 T ϱ + 2 , ς , s + 3 + 142,920 T ϱ + 2 , ς + 1 , s − 3 + 8,003,520 T ϱ + 2 , ς + 1 , s − 2 + 35,015,400 T ϱ + 2 , ς + 1 , s − 1 − 35,015,400 T ϱ + 2 , ς + 1 , s + 1 − 8,003,520 T ϱ + 2 , ς + 1 , s + 2 − 142,920 T ϱ + 2 , ς + 1 , s + 3 + 14,400 T ϱ + 2 , ς + 2 , s − 3 + 806,400 T ϱ + 2 , ς + 2 , s − 2 + 3,528,000 T ϱ + 2 , ς + 2 , s − 1 − 3,528,000 T ϱ + 2 , ς + 2 , s + 1 − 806,400 T ϱ + 2 , ς + 2 , s + 2 − 14,400 T ϱ + 2 , ς + 2 , s + 3 + 120 T ϱ + 2 , ς + 3 , s − 3 + 6,720 T ϱ + 2 , ς + 3 , s − 2 + 29,400 T ϱ + 2 , ς + 3 , s − 1 − 29,400 T ϱ + 2 , ς + 3 , s + 1 − 6,720 T ϱ + 2 , ς + 3 , s + 2 − 120 T ϱ + 2 , ς + 3 , s + 3 + T ϱ + 3 , ς − 3 , s − 3 + 56 T ϱ + 3 , ς − 3 , s − 2 + 245 T ϱ + 3 , ς − 3 , s − 1 − 245 T ϱ + 3 , ς − 3 , s + 1 − 56 T ϱ + 3 , ς − 3 , s + 2 − T ϱ + 3 , ς − 3 , s + 3 + 120 T ϱ + 3 , ς − 2 , s − 3 + 6,720 T ϱ + 3 , ς − 2 , s − 2 + 29,400 T ϱ + 3 , ς − 2 , s − 1 − 29,400 T ϱ + 3 , ς − 2 , s + 1 − 6,720 T ϱ + 3 , ς − 2 , s + 2 − 120 T ϱ + 3 , ς − 2 , s + 3 + 1,191 T ϱ + 3 , ς − 1 , s − 3 + 66,696 T ϱ + 3 , ς − 1 , s − 2 + 291,795 T ϱ + 3 , ς − 1 , s − 1 − 291,795 T ϱ + 3 , ς − 1 , s + 1 − 66,696 T ϱ + 3 , ς − 1 , s + 2 − 1,191 T ϱ + 3 , ς − 1 , s + 3 + 2,416 T ϱ + 3 , ς , s − 3 + 135,296 T ϱ + 3 , ς , s − 2 + 591,920 T ϱ + 3 , ς , s − 1 − 591,920 T ϱ + 3 , ς , s + 1 − 135,296 T ϱ + 3 , ς , s + 2 − 2,416 T ϱ + 3 , ς , s + 3 + 1,191 T ϱ + 3 , ς + 1 , s − 3 + 66,696 T ϱ + 3 , ς + 1 , s − 2 + 291,795 T ϱ + 3 , ς + 1 , s − 1 − 291,795 T ϱ + 3 , ς + 1 , s + 1 − 66,696 T ϱ + 3 , ς + 1 , s + 2 − 1,191 T ϱ + 3 , ς + 1 , s + 3 + 120 T ϱ + 3 , ς + 2 , s − 3 + 6,720 T ϱ + 3 , ς + 2 , s − 2 + 29,400 T ϱ + 3 , ς + 2 , s − 1 − 29,400 T ϱ + 3 , ς + 2 , s + 1 − 6,720 T ϱ + 3 , ς + 2 , s + 2 − 120 T ϱ + 3 , ς + 2 , s + 3 + T ϱ + 3 , ς + 3 , s − 3 + 56 T ϱ + 3 , ς + 3 , s − 2 + 245 T ϱ + 3 , ς + 3 , s − 1 − 245 T ϱ + 3 , ς + 3 , s + 1 − 56 T ϱ + 3 , ς + 3 , s + 2 − T ϱ + 3 , ς + 3 , s + 3 ) . ⋮

The aforementioned analysis yields the following theorem:

Theorem 3

From (8) the approximation formulas to H ϱ , ς , s , ∂ H ϱ , ς , s ∂ x , ∂ H ϱ , ς , s ∂ y , ∂ H ϱ , ς , s ∂ z , ∂ 2 H ϱ , ς , s ∂ x 2 , ∂ 2 H ϱ , ς , s ∂ y 2 , ∂ 2 H ϱ , ς , s ∂ z 2 , ∂ 2 H ϱ , ς , s ∂ x ∂ y , ∂ 2 H ϱ , ς , s ∂ x ∂ z , … , are given in terms of T ϱ , ς , s at (9)–(12).

3 Numerical outcomes

Now, to determine whether or not this method, which was developed by presenting its constructions in n-dimensional space, is correct and effective. In this section, we provide several numerical examples in various dimensions to demonstrate the accuracy of this method. In addition to the comparison of our results with those previously obtained, we also exhibit some of the obtained figures. We should point out that all of the examples were created using the Mathematica 12.1 package and ran on a standard computer (Intel(R) core(TM) i7-351U, CPU@1.90 Hz 2.40 GHz).

The first test problem: [8,9,13,17,18,29]

We take the two-dimensional problem in the following form:

(13) h x x ( x , y ) + h y y ( x , y ) − sin ( π x ) sin ( π y ) = 0 , x , y ∈ [ l , m ] .

The following is the exact solution to that problem:

(14) h ( x , y ) = − sin ( π x ) sin ( π y ) 2 π 2 .

We take the boundary conditions to the third problem in the following form:

(15) h ( l , y ) = h ( x , l ) = α , h ( m , y ) = h ( x , m ) = β .

By substituting from (5)–(7) into (13) with (15), we obtain the numerical results as in Table 1.

Table 1

Numerical results for the third issue are available at y = 0.4 and x , y ∈ [ 0 , 1 ]

x Numerical results Exact results Absolute error
0.2 ‒0.028320 ‒0.028320 8.07856 × 1 0 ‒ 11
0.4 ‒0.045822 ‒0.045822 1.30714 × 1 0 ‒ 10
0.6 ‒0.045822 ‒0.045822 1.30714 × 1 0 ‒ 10
0.8 ‒0.028320 ‒0.028320 8.07856 × 1 0 ‒ 11

The results of the two-dimensional septic B-spline approach at 15 × 15 are shown in Table 1. In terms of outcomes, we can presume that they are satisfactory. We compare the suggested method’s results to those of several approaches [8,9,13,17,18,29] that are presented in Table 2 and use mesh 15 × 15 grid points.

Table 2

Maximum absolute error based on the approach used to solve the problem

The proposed method Cubic B-spline approach [18] Quadratic B-spline approach [17] MCBDQM approach [13] Spline-based DQM approach [29] Haar wavelet approach [8] SCA based on Haarwavelets [9]
8.07 × 1 0 ‒ 11 1.67 × 1 0 ‒ 4 3.72 × 1 0 ‒ 5 2.11 × 1 0 ‒ 5 1.62 × 1 0 ‒ 4 3.08 × 1 0 ‒ 4 3.08 × 1 0 ‒ 4

Now, we show the numerical results and absolute errors at y = 0.4 (Figures 1 and 2). Also, Figure 3 depicts a three-dimensional graph of numerical results.

Figure 1 The graph of numerical results and absolute error at y = 0.4 y=0.4 .
Figure 1

The graph of numerical results and absolute error at y = 0.4 .

Figure 2 The graph of numerical results at x = 0.4 x=0.4 .
Figure 2

The graph of numerical results at x = 0.4 .

Figure 3 Three-dimensional graph for numerical results.
Figure 3

Three-dimensional graph for numerical results.

The second test problem:

We take the two-dimensional nonlinear problem in the following form:

(16) h x x ( x , y ) + exp ( x 2 ) h y y ( x , y ) + sin ( h ( x , y ) ) − f ( x , y ) = 0 , x , y ∈ [ l , m ] ,

where

(17) f ( x , y ) = e x 2 ( − ( 1 − e 1 − x ) x 2 ( 1 − e 1 − y ) sin ( x y ) − ( 1 − e 1 − x ) e 1 − y sin ( x y ) + 2 ( 1 − e 1 − x ) x e 1 − y cos ( x y ) ) − ( 1 − e 1 − x ) ( 1 − e 1 − y ) y 2 sin ( x y ) − e 1 − x ( 1 − e 1 − y ) sin ( x y ) + sin ( ( 1 − e 1 − x ) ( 1 − e 1 − y ) sin ( x y ) ) + 2 e 1 − x ( 1 − e 1 − y ) y cos ( x y ) .

The following is the exact solution to that problem:

(18) h ( x , y ) = sin ( y x ) ( 1 − exp ( 1 − x ) ) ( 1 − exp ( 1 − y ) ) .

We take the boundary conditions to the third problem in the following form:

(19) h ( l , y ) = h ( x , l ) = α , h ( m , y ) = h ( x , m ) = β .

By substituting from (5)–(7) into (16) with (19), we obtain the numerical results as in Table 3.

Table 3

Numerical results for the third issue are available at y = 0.5 and x , y ∈ [ 0 , 1 ]

x Numerical results Exact results Absolute error
0.2 0.079328 0.079371 4.28156 × 1 0 ‒ 5
0.4 0.105922 0.105956 3.39476 × 1 0 ‒ 5
0.6 0.094261 0.094287 2.61754 × 1 0 ‒ 5
0.8 0.055914 0.055931 1.71551 × 1 0 ‒ 5

The results of the two-dimensional septic B-spline approach at 50 × 50 are shown in Table 3. From outcomes, we can presume that the method is satisfactory. Now, we show the numerical results and absolute errors at y = 0.4 in Figures 4 and 5. Also, Figure 6 depicts a three-dimensional graph of numerical results.

Figure 4 The graph of numerical results and absolute error at y = 0.5 y=0.5 .
Figure 4

The graph of numerical results and absolute error at y = 0.5 .

Figure 5 The graph of numerical results at x = 0.5 x=0.5 .
Figure 5

The graph of numerical results at x = 0.5 .

Figure 6 Three-dimensional graph for numerical results.
Figure 6

Three-dimensional graph for numerical results.

The third test problem: [17,18]

We take the second test problem in the three-dimensional in the following form:

(20) h x x ( x , y , z ) + h y y ( x , y , z ) + h z z ( x , y , z ) − f ( x , y ) = 0 , x , y , z ∈ [ l , m ] ,

where

(21) f ( x , y ) = x y z ( e x + y + z ) ( 3 y x z + y x + z x − 5 x + z y − 5 y − 5 z + 9 ) .

The exact solution to that problem is given as follows:

(22) h ( x , y , z ) = ( x − x 2 ) ( y − y 2 ) ( z − z 2 ) e x + y + z .

We take the boundary conditions to the fourth problem in the following form:

(23) h ( l , y , z ) = h ( x , l , z ) = h ( x , y , l ) = α , h ( m , y , z ) = h ( x , m , z ) = h ( x , y , m ) = β .

By substituting from (9)–(12) into (20) with (23), we obtain the numerical results as in Table 4.

Table 4

Numerical results for test problem at z = y = 0.5 and x , y , z ∈ [ 0 , 1 ]

x Numerical solution Exact solution Absolute error Quadratic B-spline method [17] Cubic B-spline method [18]
0.1 0.016868 0.0168984 3.00566 × 1 0 ‒ 5 3.24947 × 1 0 ‒ 5 3.48009 × 1 0 ‒ 5
0.2 0.033142 0.0332012 5.91788 × 1 0 ‒ 5 6.49943 × 1 0 ‒ 5 7.05722 × 1 0 ‒ 5
0.3 0.048073 0.0481595 8.65621 × 1 0 ‒ 5 9.65554 × 1 0 ‒ 5 1.06205 × 1 0 ‒ 4
0.4 0.060716 0.0608280 1.11853 × 1 0 ‒ 4 1.27075 × 1 0 ‒ 4 1.41846 × 1 0 ‒ 4
0.5 0.069891 0.0700264 1.35497 × 1 0 ‒ 4 1.57835 × 1 0 ‒ 4 1.79618 × 1 0 ‒ 4
0.6 0.074136 0.0742955 1.59281 × 1 0 ‒ 4 1.92337 × 1 0 ‒ 4 2.24718 × 1 0 ‒ 4
0.7 0.071658 0.0718456 1.86992 × 1 0 ‒ 4 2.37433 × 1 0 ‒ 4 2.86918 × 1 0 ‒ 4
0.8 0.060271 0.0604965 2.24725 × 1 0 ‒ 4 3.04639 × 1 0 ‒ 4 3.82276 × 1 0 ‒ 4
0.9 0.050423 0.0376082 2.77977 × 1 0 ‒ 4 4.11161 × 1 0 ‒ 4 5.34402 × 1 0 ‒ 4

Table 4 shows a comparison of our results with those obtained using quadratic B-spline and cubic B-spline approaches with meshes of 20 × 20 . In terms of the results, we can see that they are acceptable based on our observations. At y = z = 0.5 , Figure 7 displays the numerical results and absolute errors. Figure 8 depicts a three-dimensional graph of numerical results.

Figure 7 The graph of numerical results and absolute error at y = z = 0.5 y=z=0.5 .
Figure 7

The graph of numerical results and absolute error at y = z = 0.5 .

Figure 8 Three-dimensional graph for numerical results.
Figure 8

Three-dimensional graph for numerical results.

The fourth test problem: [9,20]

We take the test problem in the three-dimensional in the following form:

(24) h x x ( x , y , z ) + h y y ( x , y , z ) + h z z ( x , y , z ) − sin ( π x ) sin ( π y ) sin ( π z ) = 0 , x , y , z ∈ [ l , m ] .

The following is the exact solution to that problem:

(25) h ( x , y , z ) = − sin ( π x ) sin ( π y ) sin ( π z ) 2 π 2 .

We take the boundary conditions to the third problem in the following form:

(26) h ( l , y , z ) = h ( x , l , z ) = h ( x , y , l ) = α , h ( m , y , z ) = h ( x , m , z ) = h ( x , y , m ) = β .

By substituting from (9)–(12) into (24) with (26), we obtain the numerical results as in Table 5.

Table 5

Numerical results for the test problem are available at y = z = 0.5 , x , y , z ∈ [ 0 , 1 ]

x Numerical results Exact results Absolute error Maximum absolute error of our method Maximum absolute error [20] Maximum absolute error [9]
0.2 ‒ 0.019851 ‒ 0.01985 9.983 × 1 0 ‒ 12 1.615 × 1 0 ‒ 11 4.966 × 1 0 ‒ 5 8.922 × 1 0 ‒ 4
0.4 ‒ 0.032120 ‒ 0.03212 1.615 × 1 0 ‒ 11 — — —
0.6 ‒ 0.032120 ‒ 0.03210 1.615 × 1 0 ‒ 11 — — —
0.8 ‒ 0.019851 ‒ 0.01985 9.983 × 1 0 ‒ 12 — — —

The results of the three-dimensional septic B-spline approach using mesh 15 × 15 are presented in Table 5. In terms of observation, the results appear to be acceptable. Figure 9 shows the numerical results and absolute errors at y = z = 0.5 . Figure 10 depicts a three-dimensional graph of numerical results.

Figure 9 The graph of numerical results and absolute error at y = z = 0.5 y=z=0.5 .
Figure 9

The graph of numerical results and absolute error at y = z = 0.5 .

Figure 10 Three-dimensional graph for numerical results.
Figure 10

Three-dimensional graph for numerical results.

4 Conclusion

By the end of this study, we may have made a significant contribution to addressing some of the problems that most academics in various fields have when dealing with n-dimensional mathematical models. The study object is crucial, and we believe that the majority of academics are eagerly anticipating the results. We noted how difficult it is for researchers to cope with these models as the dimension expands after seeing various scholars present their discoveries on partial differential equation solutions in one-, two-, and three-dimensional. As a result, we decided to extend the basic B-spline method, which had hitherto only been used to solve one-dimensional mathematical problems, to two- and three-dimensional. To assess the correctness and efficacy of the developed schemes, we used numerical examples of various dimensions. When the numerical results are compared to the actual solution, we see that the formulas found are effective. From this perspective, we believe that a significant contribution has been made toward addressing problems involving partial differential equations in many dimensions. The proposed new structure provides accurate results than other methods because it deals with a larger number of points than the field. We would generalize a few other B-spline shapes to serve as solutions to n-dimensional differential equations as part of our long-term research.

Acknowledgement

The researchers would to like acknowledge the deanship of scientific research, Taif University, for funding this work.

  1. Funding information: Not applicable.

  2. Author contributions: The authors declare that the study was realized in collaboration with equal responsibility. All authors read and approved the final manuscript.

  3. Conflict of interest: The authors declare that they have no competing interests.

  4. Data availability statement: Data sharing was not applicable to this article as no datasets were generated or analyzed during this study.

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Received: 2022-09-17
Revised: 2023-05-30
Accepted: 2023-06-01
Published Online: 2023-07-13

© 2023 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

Artikel in diesem Heft

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https://doi.org/10.1515/nleng-2022-0298
Schlagwörter für diesen Artikel
collocation method; n-dimensional B-spline functions; mathematical models
Creative Commons
BY 4.0

Artikel in diesem Heft

  1. Research Articles
  2. The regularization of spectral methods for hyperbolic Volterra integrodifferential equations with fractional power elliptic operator
  3. Analytical and numerical study for the generalized q-deformed sinh-Gordon equation
  4. Dynamics and attitude control of space-based synthetic aperture radar
  5. A new optimal multistep optimal homotopy asymptotic method to solve nonlinear system of two biological species
  6. Dynamical aspects of transient electro-osmotic flow of Burgers' fluid with zeta potential in cylindrical tube
  7. Self-optimization examination system based on improved particle swarm optimization
  8. Overlapping grid SQLM for third-grade modified nanofluid flow deformed by porous stretchable/shrinkable Riga plate
  9. Research on indoor localization algorithm based on time unsynchronization
  10. Performance evaluation and optimization of fixture adapter for oil drilling top drives
  11. Nonlinear adaptive sliding mode control with application to quadcopters
  12. Numerical simulation of Burgers’ equations via quartic HB-spline DQM
  13. Bond performance between recycled concrete and steel bar after high temperature
  14. Deformable Laplace transform and its applications
  15. A comparative study for the numerical approximation of 1D and 2D hyperbolic telegraph equations with UAT and UAH tension B-spline DQM
  16. Numerical approximations of CNLS equations via UAH tension B-spline DQM
  17. Nonlinear numerical simulation of bond performance between recycled concrete and corroded steel bars
  18. An iterative approach using Sawi transform for fractional telegraph equation in diversified dimensions
  19. Investigation of magnetized convection for second-grade nanofluids via Prabhakar differentiation
  20. Influence of the blade size on the dynamic characteristic damage identification of wind turbine blades
  21. Cilia and electroosmosis induced double diffusive transport of hybrid nanofluids through microchannel and entropy analysis
  22. Semi-analytical approximation of time-fractional telegraph equation via natural transform in Caputo derivative
  23. Analytical solutions of fractional couple stress fluid flow for an engineering problem
  24. Simulations of fractional time-derivative against proportional time-delay for solving and investigating the generalized perturbed-KdV equation
  25. Pricing weather derivatives in an uncertain environment
  26. Variational principles for a double Rayleigh beam system undergoing vibrations and connected by a nonlinear Winkler–Pasternak elastic layer
  27. Novel soliton structures of truncated M-fractional (4+1)-dim Fokas wave model
  28. Safety decision analysis of collapse accident based on “accident tree–analytic hierarchy process”
  29. Derivation of septic B-spline function in n-dimensional to solve n-dimensional partial differential equations
  30. Development of a gray box system identification model to estimate the parameters affecting traffic accidents
  31. Homotopy analysis method for discrete quasi-reversibility mollification method of nonhomogeneous backward heat conduction problem
  32. New kink-periodic and convex–concave-periodic solutions to the modified regularized long wave equation by means of modified rational trigonometric–hyperbolic functions
  33. Explicit Chebyshev Petrov–Galerkin scheme for time-fractional fourth-order uniform Euler–Bernoulli pinned–pinned beam equation
  34. NASA DART mission: A preliminary mathematical dynamical model and its nonlinear circuit emulation
  35. Nonlinear dynamic responses of ballasted railway tracks using concrete sleepers incorporated with reinforced fibres and pre-treated crumb rubber
  36. Two-component excitation governance of giant wave clusters with the partially nonlocal nonlinearity
  37. Bifurcation analysis and control of the valve-controlled hydraulic cylinder system
  38. Engineering fault intelligent monitoring system based on Internet of Things and GIS
  39. Traveling wave solutions of the generalized scale-invariant analog of the KdV equation by tanh–coth method
  40. Electric vehicle wireless charging system for the foreign object detection with the inducted coil with magnetic field variation
  41. Dynamical structures of wave front to the fractional generalized equal width-Burgers model via two analytic schemes: Effects of parameters and fractionality
  42. Theoretical and numerical analysis of nonlinear Boussinesq equation under fractal fractional derivative
  43. Research on the artificial control method of the gas nuclei spectrum in the small-scale experimental pool under atmospheric pressure
  44. Mathematical analysis of the transmission dynamics of viral infection with effective control policies via fractional derivative
  45. On duality principles and related convex dual formulations suitable for local and global non-convex variational optimization
  46. Study on the breaking characteristics of glass-like brittle materials
  47. The construction and development of economic education model in universities based on the spatial Durbin model
  48. Homoclinic breather, periodic wave, lump solution, and M-shaped rational solutions for cold bosonic atoms in a zig-zag optical lattice
  49. Fractional insights into Zika virus transmission: Exploring preventive measures from a dynamical perspective
  50. Rapid Communication
  51. Influence of joint flexibility on buckling analysis of free–free beams
  52. Special Issue: Recent trends and emergence of technology in nonlinear engineering and its applications - Part II
  53. Research on optimization of crane fault predictive control system based on data mining
  54. Nonlinear computer image scene and target information extraction based on big data technology
  55. Nonlinear analysis and processing of software development data under Internet of things monitoring system
  56. Nonlinear remote monitoring system of manipulator based on network communication technology
  57. Nonlinear bridge deflection monitoring and prediction system based on network communication
  58. Cross-modal multi-label image classification modeling and recognition based on nonlinear
  59. Application of nonlinear clustering optimization algorithm in web data mining of cloud computing
  60. Optimization of information acquisition security of broadband carrier communication based on linear equation
  61. A review of tiger conservation studies using nonlinear trajectory: A telemetry data approach
  62. Multiwireless sensors for electrical measurement based on nonlinear improved data fusion algorithm
  63. Realization of optimization design of electromechanical integration PLC program system based on 3D model
  64. Research on nonlinear tracking and evaluation of sports 3D vision action
  65. Analysis of bridge vibration response for identification of bridge damage using BP neural network
  66. Numerical analysis of vibration response of elastic tube bundle of heat exchanger based on fluid structure coupling analysis
  67. Establishment of nonlinear network security situational awareness model based on random forest under the background of big data
  68. Research and implementation of non-linear management and monitoring system for classified information network
  69. Study of time-fractional delayed differential equations via new integral transform-based variation iteration technique
  70. Exhaustive study on post effect processing of 3D image based on nonlinear digital watermarking algorithm
  71. A versatile dynamic noise control framework based on computer simulation and modeling
  72. A novel hybrid ensemble convolutional neural network for face recognition by optimizing hyperparameters
  73. Numerical analysis of uneven settlement of highway subgrade based on nonlinear algorithm
  74. Experimental design and data analysis and optimization of mechanical condition diagnosis for transformer sets
  75. Special Issue: Reliable and Robust Fuzzy Logic Control System for Industry 4.0
  76. Framework for identifying network attacks through packet inspection using machine learning
  77. Convolutional neural network for UAV image processing and navigation in tree plantations based on deep learning
  78. Analysis of multimedia technology and mobile learning in English teaching in colleges and universities
  79. A deep learning-based mathematical modeling strategy for classifying musical genres in musical industry
  80. An effective framework to improve the managerial activities in global software development
  81. Simulation of three-dimensional temperature field in high-frequency welding based on nonlinear finite element method
  82. Multi-objective optimization model of transmission error of nonlinear dynamic load of double helical gears
  83. Fault diagnosis of electrical equipment based on virtual simulation technology
  84. Application of fractional-order nonlinear equations in coordinated control of multi-agent systems
  85. Research on railroad locomotive driving safety assistance technology based on electromechanical coupling analysis
  86. Risk assessment of computer network information using a proposed approach: Fuzzy hierarchical reasoning model based on scientific inversion parallel programming
  87. Special Issue: Dynamic Engineering and Control Methods for the Nonlinear Systems - Part I
  88. The application of iterative hard threshold algorithm based on nonlinear optimal compression sensing and electronic information technology in the field of automatic control
  89. Equilibrium stability of dynamic duopoly Cournot game under heterogeneous strategies, asymmetric information, and one-way R&D spillovers
  90. Mathematical prediction model construction of network packet loss rate and nonlinear mapping user experience under the Internet of Things
  91. Target recognition and detection system based on sensor and nonlinear machine vision fusion
  92. Risk analysis of bridge ship collision based on AIS data model and nonlinear finite element
  93. Video face target detection and tracking algorithm based on nonlinear sequence Monte Carlo filtering technique
  94. Adaptive fuzzy extended state observer for a class of nonlinear systems with output constraint
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