Startseite A new structure formulations for cubic B-spline collocation method in three and four-dimensions
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A new structure formulations for cubic B-spline collocation method in three and four-dimensions

  • K. R. Raslan und Khalid K. Ali EMAIL logo
Veröffentlicht/Copyright: 31. Dezember 2020
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Nonlinear Engineering
Aus der Zeitschrift Nonlinear Engineering Band 9 Heft 1

Abstract

In this work, we introduce a new construct to the cubic B-spline collocation method in the three and four-dimensions. The cubic B-splines method format is displayed in one, two, three, and four-dimensions format. These constructions are of utmost importance in solving differential equations in their various dimensions, which have applications in many fields of science. The efficiency and accuracy of the proposed methods are demonstrated by its application to a few test problems in two, three, and four dimensions. Also, comparing the exact solutions and with the results obtained by using other numerical methods available in the literature as much as possible.

Keywords: Collocation Method; 3-dimensions cubic B-Splines; 4-dimensions cubic B-Splines

1 Introduction

There is no doubt that obtaining solutions to partial differential equations is of the utmost importance. Most of these equations have applications in different fields such as physics, chemistry, engineering, and others. These solutions were divided into analytical solutions and numerical solutions [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]. The researchers have long tended to use various methods to find these solutions, whether analytical or numerical. With different applications in the degree of difficulty of being in one or two dimensions or in three dimensions, work continued on how to find these solutions. Over time, some researchers found it difficult to find analytical solutions in different dimensions, so they considered finding numerical solutions. Many researchers have used different numerical methods to find solutions to these equations in different dimensions [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21]. Now, we are continuing to work on developing a cubic B-spline collocation method to find numerical solutions for partial differential equations in three dimensions and four-dimensions. This work is a continuation of the work Rajni Arora et al. [22]. The collocation method was first used by Frazer et al. [23] in 1937. Later, collocation method along with the least-squares method and Galerkin method was used by Bickley [24] to study unsteady heat condition problems. Later on, in 1975, the collocation method along with B-splines was applied to unsteady heat conduction and boundary layer flows [25] and it was found that the results obtained were better when compared with results obtained with finite difference methods. Since then, the collocation method is being used over a wide range of problems [26, 27, 28, 29, 30, 31, 32, 33]. In combination with the collocation method, there has been intensive use of polynomial B-splines for solving partial differential equations. Cubic B-splines, quasi- B-splines, quartic B-splines, quintic B-splines, and so on are employed in combination with the collocation method in [26, 27, 28, 29, 30, 31] for dealing with various linear and nonlinear boundary value problems. Methods like Haar wavelet collocation method [34], a gradient reproducing kernel collocation method [35] and Newton basis functions collocation method [36] are also gaining popularity to solve differential equations.

In this work, a new approach to 3-dimensions and 4-dimensions cubic B-splines is presented. Also, some numerical examples are presented to study the effectiveness and accuracy of this approach.

This paper is organized as follows: Section two presents n-dimensions cubic B-spline formulations. Section three introduces numerical examples. Finally, the conclusion section is displayed.

2 Construct cubic B-spline formulas

The formes for n-dimensions cubic B-splines are presented in this section.

2.1 One dimension cubic B-spline [25]

Let x ∈ [a, b] and ϕi(x) are those cubic B-splines with knots at the points xi. Then the set of cubic B-splines ϕ−1(x) , ϕ0(x) , ..., ϕN−1(x) , ϕN(x), ϕN+1, forms a basis for functions defined over the interval. The approximation UN(x) to U(x) which uses these splines as:

(1)UN(x)=∑i=−1N+1χiϕi(x),

where χi unknown term. Each cubic B-spline is non-zero order four adjacent elements so that four cubic B-splines cover each finite elements ϕm−1(x), ϕm(x), ϕm+1, ϕm+2. In termes of local coordinate systems ζ given by hζ = x−xm where 0 ⩽ ζ ⩽ 1, expressions for variation of the cubic B-splines over the elements [xm, xm+1] can be expressed as

(2)ϕm−1=1−3ζ+3ζ2−ζ3,ϕm=4−6ζ2+3ζ3,ϕm+1=1+3ζ+3ζ2−3ζ3,ϕm+2=ζ3.

All other cubic B-splines are zero over [xm, xm+1]. The formulations of Ui , dUidx,d2Uidx2at node x = xm are given by

(3)Ui=χi−1+4χi+χi+1dUidx=−3h(χi−1−χi+1),d2Uidx2=6h2(χi−1−2χi+χi+1).

2.2 Two dimensions cubic B-spline [22]

In this section, we show the formula of cubic B-splines in two dimensions on a rectangular grid divided into regular rectangular finite elements on both sides. h = Δ x, m = Δ y by the knots (xi , yj) where i = 0, 1, ..., N, j = 0, 1, ..., M. The approximation UN(x, y) to U(x, y) given by:

(4)UN(x,y)=∑i=−1N+1∑j=−1M+1χi,jBi,j(x,y),

where χi, j are the amplitudes of the cubic B-splines Bi, j(x, y) given by

Bi,j(x,y)=ϕi(x)ϕj(y).

Which peaks on the knot (xi , yj) and ϕi(x), ϕj(y) are identical in form to the one dimension cubic B-splines. By generalized (2) in two dimensions then the formulations of

Ui,j , ∂Ui,j∂x,∂Ui,j∂y,∂2Ui,j∂x2,∂2Ui,j∂y2,∂2Ui,j∂x∂y,…are given by:

(5)Ui,j=χi−1,j−1+4χi−1,j+χi−1,j+1+4χi,j−1+16χi,j+4χi,j+1+χi+1,j−1+4χi+1,j+χi+1,j+1.
(6)∂Ui,j∂x=−3hχi−1,j−1+4χi−1,j+χi−1,j+1−χi+1,j−1−4χi+1,j−χi+1,j+1,∂Ui,j∂y=−3mχi−1,j−1−χi−1,j+1+4χi,j−1−4χi,j+1+χi+1,j−1−χi+1,j+1.
(7)∂2Ui,j∂x2=6h2(χi−1,j−1+4χi−1,j+χi−1,j+1−2χi,j−18χi,j−2χi,j+1+χi+1,j−1+4χi+1,j+χi+1,j+1),∂2Ui,j∂y2=6m2(χi−1,j−1−2χi−1,j+χi−1,j+1+4χi,j−18χi,j+4χi,j+1+χi+1,j−1−2χi+1,j+χi+1,j+1)
(8)∂2Ui,j∂x∂y=9hmχi−1,j−1−χi−1,j+1−χi+1,j−1+χi+1,j+1,⋮

2.3 The three dimensions cubic B-spline element

In this section we construct the cubic B-splines in three dimensions approximates on a grid subdivided into finite elements of sides h = Δ x, m = Δ y , s = Δ z by the knots (xi , yj , zj) where i = 0, 1, ..., N, j = 0, 1, ..., M, k = 0, 1, ..., L can be interpolated in terms of piecewise cubic B-splines. If U(x, y, z) is a function of x, y and z, it can be shown there exists a unique approximation UN(x, y, z) as

(9)UN(x,y,z)=∑i=−1N+1∑j=−1M+1∑k=−1L+1χi,j,kBi,j,k(x,y,z),

where χi, j, k are the amplitudes of the cubic B-splines Bi, j, k(x, y, z) given by

Bi,j,k(x,y,z)=ϕi(x)ϕj(y)ϕk(z).

Also, ϕi(x), ϕj(y) and ϕk(z) have the same form as the one dimension cubic B-splines. The formulations of Ui,j,k , ∂Ui,j,k∂x,∂Ui,j,k∂y,∂Ui,j,k∂z,∂2Ui,j,k∂x2,∂2Ui,j,k∂y2,∂2Ui,j,k∂z2,∂2Ui,j,k∂x∂y,∂2Ui,j,k∂x∂z,…,are given in terms of the χi,j,k by:

(10)Ui,j,k=χi−1,j−1,k−1+4χi−1,j−1,k+χi−1,j−1,k+1+4χi−1,j,k−1+16χi−1,j,k+4χi−1,j,k+1+χi−1,j+1,k−1+4χi−1,j+1,k+χi−1,j+1,k+1+4χi,j−1,k−1+16χi,j−1,k+4χi,j−1,k+1+16χi,j,k−1+64χi,j,k+4χi,j+1,k−1+16χi,j+1,k+4χi,j+1,k+1+χi+1,j−1,k−1+4χi+1,j−1,k+χi+1,j−1,k+1+4χi+1,j,k−1+16χi+1,j,k+4χi+1,j,k+1+χi+1,j+1,k−1+4χi+1,j+1,k+χi+1,j+1,k+1+16χi,j,k+1.
(11)∂Ui,j,k∂x=−3h(χi−1,j−1,k−1+4χi−1,j−1,k+χi−1,j−1,k+1+4χi−1,j,k−1+16χi−1,j,k+4χi−1,j,k+1+χi−1,j+1,k−1+χi−1,j+1,k+1−χi+1,j−1,k−1−4χi+1,j−1,k−χi+1,j−1,k+1−4χi+1,j,k−1−4χi+1,j,k+1−χi+1,j+1,k−1−4χi+1,j+1,k−χi+1,j+1,k+1+4di−1,j+1,k−16di+1,j,k),∂Ui,j,k∂y=−3m(χi−1,j−1,k−1+4χi−1,j−1,k−χi−1,j+1,k−1−4χi−1,j+1,k−χi−1,j+1,k+1+4χi,j−1,k−1+16χi,j−1,k+4χi,j−1,k+1−4χi,j+1,k−1−16χi,j+1,k−4χi,j+1,k+1+χi+1,j−1,k−1+4χi+1,j−1,k+χi+1,j−1,k+1−χi+1,j+1,k−1−4χi+1,j+1,k−χi+1,j+1,k+1+χi−1,n−1,k+1),∂Ui,j,k∂z=−3s(χi−1,j−1,k−1−χi−1,j−1,k+1+4χi−1,j,k−1−4χi−1,j,k+1+χi−1,j+1,k−1−χi−1,j+1,k+1+4χi,j−1,k−1−4χi,j−1,k+1+16χi,j,k−1−16χi,j,k+1+4χi,j+1,k−1−4χi,j+1,k+1+χi+1,j−1,k−1−χi+1,j−1,k+1+4χi+1,j,k−1−4χi+1,j,k+1+χi+1,j+1,k−1−χi+1,j+1,k+1).
(12)∂2Ui,j,k∂x∂y=9hm(χi−1,j−1,k−1+4χi−1,j−1,k+χi−1,j−1,k+1−χi−1,j+1,k−1−4χi−1,j+1,k−χi−1,j+1,k+1−χi+1,j−1,k−1−4χi+1,j−1,k−χi+1,j−1,k+1+χi+1,j+1,k−1+4χi+1,j+1,k+χi+1,j+1,k+1),∂2Ui,j,k∂x∂z=9hs(χi−1,j−1,k−1−χi−1,j−1,k+1+4χi−1,j,k−1−4χi−1,j,k+1+χi−1,j+1,k−1−χi−1,j+1,k+1−χi+1,j−1,k−1+χi+1,j−1,k+1−4χi+1,j,k−1+4χi+1,j,k+1−χi+1,j+1,k−1+χi+1,j+1,k+1),∂2Ui,j,k∂y∂z=9ms(χi−1,j−1,k−1−χi−1,j−1,k+1−χi−1,j+1,k−1+χi−1,j+1,k+1+4χi,j−1,k−1−4χi,j−1,k+1−4χi,j+1,k−1+4χi,j+1,k+1+χi+1,j−1,k−1−χi+1,j−1,k+1−χi+1,j+1,k−1+χi+1,j+1,k+1),⋮∂2Ui,j,k∂x2=6h2(χi−1,j−1,k−1+4χi−1,j−1,k+χi−1,j−1,k+1+4χi−1,j,k−1+16χi−1,j,k+4χi−1,j,k+1+χi−1,j+1,k−1+4χi−1,j+1,k+χi−1,j+1,k+12χi,j−1,k−1−8χi,j−1,k−2χi,j−1,k+1−8χi,j,k−1−8χi,j,k+1−2χi,j+1,k−18χi,j+1,k−2χi,j+1,k+1+χi+1,j−1,k−1+4χi+1,j−1,k+χi+1,j−1,k+1+4χi+1,j,k−1+16χi+1,j,k+4χi+1,j,k+1+χi+1,j+1,k−1+4χi+1,j+1,k+χi+1,j+1,k+1−32χi,j,k),
(13)∂2Ui,j,k∂y2=6m2(χi−1,j−1,k−1+4χi−1,j−1,k+χi−1,j−1,k+1−2χi−1,j,k−1−8χi−1,j,k−2χi−1,j,k+1+χi−1,j+1,k−1+4χi−1,j+1,k+χi−1,j+1,k+1+4χi,j−1,k−1+16χi,j−1,k+4χi,j−1,k+1−8χi,j,k−1−32χi,j,k+4χi,j+1,k−1+16χi,j+1,k+4χi,j+1,k+1+χi+1,j−1,k−1+4χi+1,j−1,k+χi+1,j−1,k+12χi+1,j,k−1−8χi+1,j,k−2χi+1,j,k+1+χi+1,j+1,k−1+4χi+1,j+1,k+χi+1,j+1,k+1−8χm,n,k+1),∂2Ui,j,k∂z2=6s2(χi−1,j−1,k−1−2χi−1,j−1,k+4χi−1,j,k−1−8χi−1,j,k+4χi−1,j,k+1+χi−1,j+1,k−1−2χi−1,j+1,k+χi−1,j+1,k+1+4χi,j−1,k−1−8χi,j−1,k+4χi,j−1,k+1+16χi,j,k−1−32χi,j,k+16χi,j,k+1+4χi,j+1,k−1−8χi,j+1,k+4χi,j+1,k+1+χi+1,j−1,k−1−2χi+1,j−1,k+χi+1,j−1,k+1+4χi+1,j,k−1−8χi+1,j,k+4χi+1,j,k+1+χi+1,j+1,k−1−2χi+1,j+1,k+χi+1,j+1,k+1+χi−1,j−1,k+1).

2.4 The four dimensions cubic B-spline element

In this section we construct the cubic B-splines in four dimensions on a rectangular grid subdivided into finite elements of sides h = Δ x, k = Δ y , s = Δ z , g = Δ d by the knots (xm, yn , zr1, dq1) where m = 0, 1, ..., M, n = 0, 1, ..., N, r1 = 0, 1, ..., R1, q1 = 0, 1, ..., N1 can be interpolated in terms of piecewise cubic B-splines. If U(x, y, z, d) is a function of x, y, z and d, it can be shown there exists a unique approximation UN(x, y, z, d) as

(14)UN(x,y,z,d)=∑m=−1M+1∑n=−1N+1∑r1=−1R1+1∑q1=−1N1+1χm,n,r1,q1Bm,n,r1,q1(x,y,z,d),

where χm,n,r1,q1 are the amplitudes of the cubic B-splines Bm,n,r1,q1(x, y, z, d) given by

Bm,n,r1,q1(x,y,z,d)=ϕm(x)ϕn(y)ϕr1(z)ϕq1(d).

Also, ϕm(x), ϕn(y), ϕr1(z) and ϕq1(d) are identical in form to the one dimension cubic B-splines in the above.

The nodal values of Um,n,r1,q1 , ∂Um,n,r1,q1∂x,∂Um,n,r1,q1∂y,∂Um,n,r1,q1∂z,∂Um,n,r1,q1∂d,∂2Um,n,r1,q1∂x2,∂2Um,n,r1,q1∂y2,∂2Um,n,r1,q1∂z2,∂2Um,n,r1,q1∂d2,…,are given in terms of the χm,n,r1,q1 by:

(15)Um,n,r1,q1=χm−1,n−1,r1−1,q1−1+4χm−1,n−1,r1−1,q1+χm−1,n−1,r1−1,q1+1+4χm−1,n−1,r1,q1−1+16χm−1,n−1,r1,q1+4χm−1,n−1,r1,q1+1+χm−1,n−1,r1+1,q1−1+4χm−1,n−1,r1+1,q1+χm−1,n−1,r1+1,q1+1+4χm−1,n,r1−1,q1−1+16χm−1,n,r1−1,q1+4χm−1,n,r1−1,q1+1+16χm−1,n,r1,q1−1+64χm−1,n,r1,q1+16χm−1,n,r1,q1+1+4χm−1,n,r1+1,q1−1+16χm−1,n,r1+1,q1+4χm−1,n,r1+1,q1+1+χm−1,n+1,r1−1,q1−1+4χm−1,n+1,r1−1,q1+χm−1,n+1,r1−1,q1+1+4χm−1,n+1,r1,q1−1+16χm−1,n+1,r1,q1+4χm−1,n+1,r1,q1+1+χm−1,n+1,r1+1,q1−1+4χm−1,n+1,r1+1,q1+χm−1,n+1,r1+1,q1+1+4χm,n−1,r1−1,q1−1+16χm,n−1,r1−1,q1+4χm,n−1,r1−1,q1+1+16χm,n−1,r1,q1−1+64χm,n−1,r1,q1+16χm,n−1,r1,q1+1+4χm,n−1,r1+1,q1−1+16χm,n−1,r1+1,q1+4χm,n−1,r1+1,q1+1+16χm,n,r1−1,q1−1+64χm,n,r1−1,q1+16χm,n,r1−1,q1+1+64χm,n,r1,q1−1+256χm,n,r1,q1+64χm,n,r1,q1+1+16χm,n,r1+1,q1−1+64χm,n,r1+1,q1+16χm,n,r1+1,q1+1+4χm,n+1,r1−1,q1−1+16χm,n+1,r1−1,q1+4χm,n+1,r1−1,q1+1+16χm,n+1,r1,q1−1+64χm,n+1,r1,q1+16χm,n+1,r1,q1+1+4χm,n+1,r1+1,q1−1+16χm,n+1,r1+1,q1+4χm,n+1,r1+1,q1+1+χm+1,n−1,r1−1,q1−1+4χm+1,n−1,r1−1,q1+χm+1,n−1,r1−1,q1+1+4χm+1,n−1,r1,q1−1+16χm+1,n−1,r1,q1+4χm+1,n−1,r1,q1+1+χm+1,n−1,r1+1,q1−1+4χm+1,n−1,r1+1,q1+χm+1,n−1,r1+1,q1+1+4χm+1,n,r1−1,q1−1+16χm+1,n,r1−1,q1+4χm+1,n,r1−1,q1+1+16χm+1,n,r1,q1−1+64χm+1,n,r1,q1+16χm+1,n,r1,q1+1+4χm+1,n,r1+1,q1−1+16χm+1,n,r1+1,q1+4χm+1,n,r1+1,q1+1+χm+1,n+1,r1−1,q1−1+4χm+1,n+1,r1−1,q1+χm+1,n+1,r1−1,q1+1+4χm+1,n+1,r1,q1−1+16χm+1,n+1,r1,q1+4χm+1,n+1,r1,q1+1+χm+1,n+1,r1+1,q1−1+4χm+1,n+1,r1+1,q1+χm+1,n+1,r1+1,q1+1,
(16)∂Um,n,r1,q1∂x=−3h(χm−1,n−1,r1−1,q1−1+4χm−1,n−1,r1−1,q1+χm−1,n−1,r1−1,q1+1+4χm−1,n−1,r1,q1−1+16χm−1,n−1,r1,q1+4χm−1,n−1,r1,q1+1+χm−1,n−1,r1+1,q1−1+4χm−1,n−1,r1+1,q1+χm−1,n−1,r1+1,q1+1+4χm−1,n,r1−1,q1−1+16χm−1,n,r1−1,q1+4χm−1,n,r1−1,q1+1+16χm−1,n,r1,q1−1+64χm−1,n,r1,q1+16χm−1,n,r1,q1+1+4χm−1,n,r1+1,q1−1+16χm−1,n,r1+1,q1+4χm−1,n,r1+1,q1+1+χm−1,n+1,r1−1,q1−1+4χm−1,n+1,r1−1,q1+χm−1,n+1,r1−1,q1+1+4χm−1,n+1,r1,q1−1+16χm−1,n+1,r1,q1+4χm−1,n+1,r1,q1+1+χm−1,n+1,r1+1,q1−1+4χm−1,n+1,r1+1,q1+χm−1,n+1,r1+1,q1+1−χm+1,n−1,r1−1,q1−1−4χm+1,n−1,r1−1,q1−χm+1,n−1,r1−1,q1+1−4χm+1,n−1,r1,q1−1−16χm+1,n−1,r1,q1−4χm+1,n−1,r1,q1+1−χm+1,n−1,r1+1,q1−1−4χm+1,n−1,r1+1,q1−χm+1,n−1,r1+1,q1+1−4χm+1,n,r1−1,q1−1−16χm+1,n,r1−1,q1−4χm+1,n,r1−1,q1+1−16χm+1,n,r1,q1−1−64χm+1,n,r1,q1−16χm+1,n,r1,q1+1−4χm+1,n,r1+1,q1−1−16χm+1,n,r1+1,q1−4χm+1,n,r1+1,q1+1−χm+1,n+1,r1−1,q1−1−4χm+1,n+1,r1−1,q1−χm+1,n+1,r1−1,q1+1−4χm+1,n+1,r1,q1−1−16χm+1,n+1,r1,q1−4χm+1,n+1,r1,q1+1−χm+1,n+1,r1+1,q1−1−4χm+1,n+1,r1+1,q1−χm+1,n+1,r1+1,q1+1),
(17)∂Um,n,r1,q1∂y=−3k(χm−1,n−1,r1−1,q1−1+4χm−1,n−1,r1−1,q1+χm−1,n−1,r1−1,q1+1+4χm−1,n−1,r1,q1−1+16χm−1,n−1,r1,q1+4χm−1,n−1,r1,q1+1+χm−1,n−1,r1+1,q1−1+4χm−1,n−1,r1+1,q1+χm−1,n−1,r1+1,q1+1−χm−1,n+1,r1−1,q1−1−4χm−1,n+1,r1−1,q1−χm−1,n+1,r1−1,q1+1−4χm−1,n+1,r1,q1−1−16χm−1,n+1,r1,q1−4χm−1,n+1,r1,q1+1−χm−1,n+1,r1+1,q1−1−4χm−1,n+1,r1+1,q1−χm−1,n+1,r1+1,q1+1+4χm,n−1,r1−1,q1−1+16χm,n−1,r1−1,q1+4χm,n−1,r1−1,q1+1+16χm,n−1,r1,q1−1+64χm,n−1,r1,q1+16χm,n−1,r1,q1+1+4χm,n−1,r1+1,q1−1+16χm,n−1,r1+1,q1+4χm,n−1,r1+1,q1+1−4χm,n+1,r1−1,q1−1−16χm,n+1,r1−1,q1−4χm,n+1,r1−1,q1+1−16χm,n+1,r1,q1−1−64χm,n+1,r1,q1−16χm,n+1,r1,q1+1−4χm,n+1,r1+1,q1−1−16χm,n+1,r1+1,q1−4χm,n+1,r1+1,q1+1+χm+1,n−1,r1−1,q1−1+4χm+1,n−1,r1−1,q1+χm+1,n−1,r1−1,q1+1+4χm+1,n−1,r1,q1−1+16χm+1,n−1,r1,q1+4χm+1,n−1,r1,q1+1+χm+1,n−1,r1+1,q1−1+4χm+1,n−1,r1+1,q1+χm+1,n−1,r1+1,q1+1−χm+1,n+1,r1−1,q1−1−4χm+1,n+1,r1−1,q1−χm+1,n+1,r1−1,q1+1−4χm+1,n+1,r1,q1−1−16χm+1,n+1,r1,q1−4χm+1,n+1,r1,q1+1−χm+1,n+1,r1+1,q1−1−4χm+1,n+1,r1+1,q1−χm+1,n+1,r1+1,q1+1),
(18)∂Um,n,r1,q1∂z=−3s(χm−1,n−1,r1−1,q1−1+4χm−1,n−1,r1−1,q1+χm−1,n−1,r1−1,q1+1−χm−1,n−1,r1+1,q1−1−4χm−1,n−1,r1+1,q1−χm−1,n−1,r1+1,q1+1+4χm−1,n,r1−1,q1−1+16χm−1,n,r1−1,q1+4χm−1,n,r1−1,q1+1−4χm−1,n,r1+1,q1−1−16χm−1,n,r1+1,q1−4χm−1,n,r1+1,q1+1+χm−1,n+1,r1−1,q1−1+4χm−1,n+1,r1−1,q1+χm−1,n+1,r1−1,q1+1−χm−1,n+1,r1+1,q1−1−4χm−1,n+1,r1+1,q1−χm−1,n+1,r1+1,q1+1+4χm,n−1,r1−1,q1−1+16χm,n−1,r1−1,q1+4χm,n−1,r1−1,q1+1−4χm,n−1,r1+1,q1−1−16χm,n−1,r1+1,q1−4χm,n−1,r1+1,q1+1+16χm,n,r1−1,q1−1+64χm,n,r1−1,q1+16χm,n,r1−1,q1+1−16χm,n,r1+1,q1−1−64χm,n,r1+1,q1−16χm,n,r1+1,q1+1+4χm,n+1,r1−1,q1−1+16χm,n+1,r1−1,q1+4χm,n+1,r1−1,q1+1−4χm,n+1,r1+1,q1−1−16χm,n+1,r1+1,q1−4χm,n+1,r1+1,q1+1+χm+1,n−1,r1−1,q1−1+4χm+1,n−1,r1−1,q1+χm+1,n−1,r1−1,q1+1−χm+1,n−1,r1+1,q1−1−4χm+1,n−1,r1+1,q1−χm+1,n−1,r1+1,q1+1+4χm+1,n,r1−1,q1−1+16χm+1,n,r1−1,q1+4χm+1,n,r1−1,q1+1−4χm+1,n,r1+1,q1−1−16χm+1,n,r1+1,q1−4χm+1,n,r1+1,q1+1+χm+1,n+1,r1−1,q1−1+4χm+1,n+1,r1−1,q1+χm+1,n+1,r1−1,q1+1−χm+1,n+1,r1+1,q1−1−4χm+1,n+1,r1+1,q1−χm+1,n+1,r1+1,q1+1),
(19)∂Um,n,r1,q1∂d=−3g(χm−1,n−1,r1−1,q1−1−χm−1,n−1,r1−1,q1+1+4χm−1,n−1,r1,q1−1−4χm−1,n−1,r1,q1+1+χm−1,n−1,r1+1,q1−1−χm−1,n−1,r1+1,q1+1+4χm−1,n,r1−1,q1−1−4χm−1,n,r1−1,q1+1+16χm−1,n,r1,q1−1−16χm−1,n,r1,q1+1+4χm−1,n,r1+1,q1−1−4χm−1,n,r1+1,q1+1+χm−1,n+1,r1−1,q1−1−χm−1,n+1,r1−1,q1+1+4χm−1,n+1,r1,q1−1−4χm−1,n+1,r1,q1+1+χm−1,n+1,r1+1,q1−1−χm−1,n+1,r1+1,q1+1+4χm,n−1,r1−1,q1−1−4χm,n−1,r1−1,q1+1+16χm,n−1,r1,q1−1−16χm,n−1,r1,q1+1+4χm,n−1,r1+1,q1−1−4χm,n−1,r1+1,q1+1+16χm,n,r1−1,q1−1−16χm,n,r1−1,q1+1+64χm,n,r1,q1−1−64χm,n,r1,q1+1+16χm,n,r1+1,q1−1−16χm,n,r1+1,q1+1+4χm,n+1,r1−1,q1−1−4χm,n+1,r1−1,q1+1+16χm,n+1,r1,q1−1−16χm,n+1,r1,q1+1+4χm,n+1,r1+1,q1−1−4χm,n+1,r1+1,q1+1+χm+1,n−1,r1−1,q1−1−χm+1,n−1,r1−1,q1+1+4χm+1,n−1,r1,q1−1−4χm+1,n−1,r1,q1+1+χm+1,n−1,r1+1,q1−1−χm+1,n−1,r1+1,q1+1+4χm+1,n,r1−1,q1−1−4χm+1,n,r1−1,q1+1+16χm+1,n,r1,q1−1−16χm+1,n,r1,q1+1+4χm+1,n,r1+1,q1−1−4χm+1,n,r1+1,q1+1+χm+1,n+1,r1−1,q1−1−χm+1,n+1,r1−1,q1+1+4χm+1,n+1,r1,q1−1−4χm+1,n+1,r1,q1+1+χm+1,n+1,r1+1,q1−1−χm+1,n+1,r1+1,q1+1),
(20)∂2Um,n,r1,q1∂x2=6h2(χm−1,n−1,r1−1,q1−1+4χm−1,n−1,r1−1,q1+χm−1,n−1,r1−1,q1+1+4χm−1,n−1,r1,q1−1+16χm−1,n−1,r1,q1+4χm−1,n−1,r1,q1+1+χm−1,n−1,r1+1,q1−1+4χm−1,n−1,r1+1,q1+χm−1,n−1,r1+1,q1+1+4χm−1,n,r1−1,q1−1+16χm−1,n,r1−1,q1+4χm−1,n,r1−1,q1+1+16χm−1,n,r1,q1−1+64χm−1,n,r1,q1+16χm−1,n,r1,q1+1+4χm−1,n,r1+1,q1−1+16χm−1,n,r1+1,q1+4χm−1,n,r1+1,q1+1+χm−1,n+1,r1−1,q1−1+4χm−1,n+1,r1−1,q1+χm−1,n+1,r1−1,q1+1+4χm−1,n+1,r1,q1−1+16χm−1,n+1,r1,q1+4χm−1,n+1,r1,q1+1+χm−1,n+1,r1+1,q1−1+4χm−1,n+1,r1+1,q1+χm−1,n+1,r1+1,q1+1−2χm,n−1,r1−1,q1−1−8χm,n−1,r1−1,q1−2χm,n−1,r1−1,q1+1−8χm,n−1,r1,q1−1−32χm,n−1,r1,q1−8χm,n−1,r1,q1+1−2χm,n−1,r1+1,q1−1−8χm,n−1,r1+1,q1−2χm,n−1,r1+1,q1+1−8χm,n,r1−1,q1−1−32χm,n,r1−1,q1−8χm,n,r1−1,q1+1−32χm,n,r1,q1−1−128χm,n,r1,q1−32χm,n,r1,q1+1−8χm,n,r1+1,q1−1−32χm,n,r1+1,q1−8χm,n,r1+1,q1+1−2χm,n+1,r1−1,q1−1−8χm,n+1,r1−1,q1−2χm,n+1,r1−1,q1+1−8χm,n+1,r1,q1−1−32χm,n+1,r1,q1−8χm,n+1,r1,q1+1−2χm,n+1,r1+1,q1−1−8χm,n+1,r1+1,q1−2χm,n+1,r1+1,q1+1+χm+1,n−1,r1−1,q1−1+4χm+1,n−1,r1−1,q1+χm+1,n−1,r1−1,q1+1+4χm+1,n−1,r1,q1−1+16χm+1,n−1,r1,q1+4χm+1,n−1,r1,q1+1+χm+1,n−1,r1+1,q1−1+4χm+1,n−1,r1+1,q1+χm+1,n−1,r1+1,q1+1+4χm+1,n,r1−1,q1−1+16χm+1,n,r1−1,q1+4χm+1,n,r1−1,q1+1+16χm+1,n,r1,q1−1+64χm+1,n,r1,q1+16χm+1,n,r1,q1+1+4χm+1,n,r1+1,q1−1+16χm+1,n,r1+1,q1+4χm+1,n,r1+1,q1+1+χm+1,n+1,r1−1,q1−1+4χm+1,n+1,r1−1,q1+χm+1,n+1,r1−1,q1+1+4χm+1,n+1,r1,q1−1+16χm+1,n+1,r1,q1+4χm+1,n+1,r1,q1+1+χm+1,n+1,r1+1,q1−1+4χm+1,n+1,r1+1,q1+χm+1,n+1,r1+1,q1+1),
(21)∂2Um,n,r1,q1∂y2=(χm−1,n−1,r1−1,q1−1+4χm−1,n−1,r1−1,q1+χm−1,n−1,r1−1,q1+1+4χm−1,n−1,r1,q1−1+16χm−1,n−1,r1,q1+4χm−1,n−1,r1,q1+1+χm−1,n−1,r1+1,q1−1+4χm−1,n−1,r1+1,q1+χm−1,n−1,r1+1,q1+1−2χm−1,n,r1−1,q1−1−8χm−1,n,r1−1,q1−2χm−1,n,r1−1,q1+1−8χm−1,n,r1,q1−1−32χm−1,n,r1,q1−8χm−1,n,r1,q1+1−2χm−1,n,r1+1,q1−1−8χm−1,n,r1+1,q1−2χm−1,n,r1+1,q1+1+χm−1,n+1,r1−1,q1−1+4χm−1,n+1,r1−1,q1+χm−1,n+1,r1−1,q1+1+4χm−1,n+1,r1,q1−1+16χm−1,n+1,r1,q1+4χm−1,n+1,r1,q1+1+χm−1,n+1,r1+1,q1−1+4χm−1,n+1,r1+1,q1+χm−1,n+1,r1+1,q1+1+4χm,n−1,r1−1,q1−1+16χm,n−1,r1−1,q1+4χm,n−1,r1−1,q1+1+16χm,n−1,r1,q1−1+64χm,n−1,r1,q1+16χm,n−1,r1,q1+1+4χm,n−1,r1+1,q1−1+16χm,n−1,r1+1,q1+4χm,n−1,r1+1,q1+1−8χm,n,r1−1,q1−1−32χm,n,r1−1,q1−8χm,n,r1−1,q1+1−32χm,n,r1,q1−1−128χm,n,r1,q1−32χm,n,r1,q1+1−8χm,n,r1+1,q1−1−32χm,n,r1+1,q1−8χm,n,r1+1,q1+1+4χm,n+1,r1−1,q1−1+16χm,n+1,r1−1,q1+4χm,n+1,r1−1,q1+1+16χm,n+1,r1,q1−1+64χm,n+1,r1,q1+16χm,n+1,r1,q1+1+4χm,n+1,r1+1,q1−1+16χm,n+1,r1+1,q1+4χm,n+1,r1+1,q1+1+χm+1,n−1,r1−1,q1−1+4χm+1,n−1,r1−1,q1+χm+1,n−1,r1−1,q1+1+4χm+1,n−1,r1,q1−1+16χm+1,n−1,r1,q1+4χm+1,n−1,r1,q1+1+χm+1,n−1,r1+1,q1−1+4χm+1,n−1,r1+1,q1+χm+1,n−1,r1+1,q1+1−2χm+1,n,r1−1,q1−1−8χm+1,n,r1−1,q1−2χm+1,n,r1−1,q1+1−8χm+1,n,r1,q1−1−32χm+1,n,r1,q1−8χm+1,n,r1,q1+1−2χm+1,n,r1+1,q1−1−8χm+1,n,r1+1,q1−2χm+1,n,r1+1,q1+1+χm+1,n+1,r1−1,q1−1+4χm+1,n+1,r1−1,q1+χm+1,n+1,r1−1,q1+1+4χm+1,n+1,r1,q1−1+16χm+1,n+1,r1,q1+4χm+1,n+1,r1,q1+1+χm+1,n+1,r1+1,q1−1+4χm+1,n+1,r1+1,q1+χm+1,n+1,r1+1,q1+1),
(22)∂2Um,n,r1,q1∂z2=(χm−1,n−1,r1−1,q1−1+4χm−1,n−1,r1−1,q1+χm−1,n−1,r1−1,q1+1−2χm−1,n−1,r1,q1−1−8χm−1,n−1,r1,q1−2χm−1,n−1,r1,q1+1+χm−1,n−1,r1+1,q1−1+4χm−1,n−1,r1+1,q1+χm−1,n−1,r1+1,q1+1+4χm−1,n,r1−1,q1−1+16χm−1,n,r1−1,q1+4χm−1,n,r1−1,q1+1−8χm−1,n,r1,q1−1−32χm−1,n,r1,q1−8χm−1,n,r1,q1+1+4χm−1,n,r1+1,q1−1+16χm−1,n,r1+1,q1+4χm−1,n,r1+1,q1+1+χm−1,n+1,r1−1,q1−1+4χm−1,n+1,r1−1,q1+χm−1,n+1,r1−1,q1+1−2χm−1,n+1,r1,q1−1−8χm−1,n+1,r1,q1−2χm−1,n+1,r1,q1+1+χm−1,n+1,r1+1,q1−1+4χm−1,n+1,r1+1,q1+χm−1,n+1,r1+1,q1+1+4χm,n−1,r1−1,q1−1+16χm,n−1,r1−1,q1+4χm,n−1,r1−1,q1+1−8χm,n−1,r1,q1−1−32χm,n−1,r1,q1−8χm,n−1,r1,q1+1+4χm,n−1,r1+1,q1−1+16χm,n−1,r1+1,q1+4χm,n−1,r1+1,q1+1+16χm,n,r1−1,q1−1+64χm,n,r1−1,q1+16χm,n,r1−1,q1+1−32χm,n,r1,q1−1−128χm,n,r1,q1−32χm,n,r1,q1+1+16χm,n,r1+1,q1−1+64χm,n,r1+1,q1+16χm,n,r1+1,q1+1+4χm,n+1,r1−1,q1−1+16χm,n+1,r1−1,q1+4χm,n+1,r1−1,q1+1−8χm,n+1,r1,q1−1−32χm,n+1,r1,q1−8χm,n+1,r1,q1+1+4χm,n+1,r1+1,q1−1+16χm,n+1,r1+1,q1+4χm,n+1,r1+1,q1+1+χm+1,n−1,r1−1,q1−1+4χm+1,n−1,r1−1,q1+χm+1,n−1,r1−1,q1+1−2χm+1,n−1,r1,q1−1−8χm+1,n−1,r1,q1−2χm+1,n−1,r1,q1+1+χm+1,n−1,r1+1,q1−1+4χm+1,n−1,r1+1,q1+χm+1,n−1,r1+1,q1+1+4χm+1,n,r1−1,q1−1+16χm+1,n,r1−1,q1+4χm+1,n,r1−1,q1+1−8χm+1,n,r1,q1−1−32χm+1,n,r1,q1−8χm+1,n,r1,q1+1+4χm+1,n,r1+1,q1−1+16χm+1,n,r1+1,q1+4χm+1,n,r1+1,q1+1+χm+1,n+1,r1−1,q1−1+4χm+1,n+1,r1−1,q1+χm+1,n+1,r1−1,q1+1−2χm+1,n+1,r1,q1−1−8χm+1,n+1,r1,q1−2χm+1,n+1,r1,q1+1+χm+1,n+1,r1+1,q1−1+4χm+1,n+1,r1+1,q1+χm+1,n+1,r1+1,q1+1),
(23)∂2Um,n,r1,q1∂d2=(χm−1,n−1,r1−1,q1−1−2χm−1,n−1,r1−1,q1+χm−1,n−1,r1−1,q1+1+4χm−1,n−1,r1,q1−1−8χm−1,n−1,r1,q1+4χm−1,n−1,r1,q1+1+χm−1,n−1,r1+1,q1−1−2χm−1,n−1,r1+1,q1+χm−1,n−1,r1+1,q1+1+4χm−1,n,r1−1,q1−1−8χm−1,n,r1−1,q1+4χm−1,n,r1−1,q1+1+16χm−1,n,r1,q1−1−32χm−1,n,r1,q1+16χm−1,n,r1,q1+1+4χm−1,n,r1+1,q1−1−8χm−1,n,r1+1,q1+4χm−1,n,r1+1,q1+1+χm−1,n+1,r1−1,q1−1−2χm−1,n+1,r1−1,q1+χm−1,n+1,r1−1,q1+1+4χm−1,n+1,r1,q1−1−8χm−1,n+1,r1,q1+4χm−1,n+1,r1,q1+1+χm−1,n+1,r1+1,q1−1−2χm−1,n+1,r1+1,q1+χm−1,n+1,r1+1,q1+1+4χm,n−1,r1−1,q1−1−8χm,n−1,r1−1,q1+4χm,n−1,r1−1,q1+1+16χm,n−1,r1,q1−1−32χm,n−1,r1,q1+16χm,n−1,r1,q1+1+4χm,n−1,r1+1,q1−1−8χm,n−1,r1+1,q1+4χm,n−1,r1+1,q1+1+16χm,n,r1−1,q1−1−32χm,n,r1−1,q1+16χm,n,r1−1,q1+1+64χm,n,r1,q1−1−128χm,n,r1,q1+64χm,n,r1,q1+1+16χm,n,r1+1,q1−1−32χm,n,r1+1,q1+16χm,n,r1+1,q1+1+4χm,n+1,r1−1,q1−1−8χm,n+1,r1−1,q1+4χm,n+1,r1−1,q1+1+16χm,n+1,r1,q1−1−32χm,n+1,r1,q1+16χm,n+1,r1,q1+1+4χm,n+1,r1+1,q1−1−8χm,n+1,r1+1,q1+4χm,n+1,r1+1,q1+1+χm+1,n−1,r1−1,q1−1−2χm+1,n−1,r1−1,q1+χm+1,n−1,r1−1,q1+1+4χm+1,n−1,r1,q1−1−8χm+1,n−1,r1,q1+4χm+1,n−1,r1,q1+1+χm+1,n−1,r1+1,q1−1−2χm+1,n−1,r1+1,q1+χm+1,n−1,r1+1,q1+1+4χm+1,n,r1−1,q1−1−8χm+1,n,r1−1,q1+4χm+1,n,r1−1,q1+1+16χm+1,n,r1,q1−1−32χm+1,n,r1,q1+16χm+1,n,r1,q1+1+4χm+1,n,r1+1,q1−1−8χm+1,n,r1+1,q1+4χm+1,n,r1+1,q1+1+χm+1,n+1,r1−1,q1−1−2χm+1,n+1,r1−1,q1+χm+1,n+1,r1−1,q1+1+4χm+1,n+1,r1,q1−1−8χm+1,n+1,r1,q1+4χm+1,n+1,r1,q1+1+χm+1,n+1,r1+1,q1−1−2χm+1,n+1,r1+1,q1+χm+1,n+1,r1+1,q1+1),⋮

In all n-dimensions PDE’s with collocation method we get a system of algebraic equations in this form

(24)Aχ∼=b∼,

We solve the above system using newton’s method to find the unknown values of χ.

3 The error estimates

Lemma. Let Û is approximation of smoothness class C2. Then error estimates of the interpolation on a square mesh of side h are

∥U−U^∥≤β0h4,∂U∂χ−∂U^∂x≤β1h3,∂U∂z−∂U^∂z≤β2h3,∂U∂y−∂U^∂y≤β3h3,∂2U∂x2−∂2U^∂x2≤β4h2∂2U∂y2−∂2U^∂y2≤β5h2,∂2U∂z2−∂2U^∂z2≤β6h2,∂2U∂χ∂y−∂2U^∂χ∂y≤β7h2,∂2U∂x∂z−∂2U^∂x∂z≤β8h2,∂2U∂y∂z−∂2U^∂y∂z≤β9h2,

where the βi are constants.

The proof of above lemma see [14].

4 The numerical results

Now, we offer some numerical examples through which we test the correctness and accuracy of n-dimensions cubic B-splines method.

The first problem

Let us take the 2-dimensions partial differential equation as:

(25)uxx(x,y)+uyy(x,y)+ux(x,y)+uy(x,y)−3e2x+3y(x2(2y(9y−2)−5)+x(5−2y(4y+3))−3(y−1)y)=0,x,y∈[a,b]

With the exact solution is given by:

(26)u(x,y)=3e2xx(1−x)e3yy(1−y).

The boundary conditions are given as follows:

(27)u(a,y)=u(x,a)=α,u(b,y)=u(x,b)=β.

By substituting from (5)-(7) in (25) with the boundary conditions (27) we get the numerical solutions as follows:

In Table 1, the results of the 2-dimensions cubic B-spline method using a mesh of 50 × 50 were compared with the exact solution. we see our results are accepted with respect to the exact solutions. In Figure 1, we show the numerical solutions with the exact solutions at y = 0.5. In Figure 2, we show the numerical solutions with the exact solutions at x = 0.5. Figure 3 illustrates the absolute error at y = 0.5.

Figure 1 The numerical solutions with the exact solutions at y = 0.5.
Figure 1

The numerical solutions with the exact solutions at y = 0.5.

Table 1

The numerical resultes for first problem at y = 0.5, x, y ∈ [0, 1]

xNumerical solutionExact solutionAbsolute error
0.10.368590.369498.95178 E-4
0.20.800210.802302.09426 E-3
0.31.283041.286173.13346 E-3
0.41.791341.795354.01762 E-3
0.52.279442.284224.77327 E-3
0.62.672892.678355.46057 E-3
0.72.856242.862436.18604 E-3
0.82.656662.663757.09892 E-3
0.91.821761.830098.34224 E-3
Figure 2 The numerical solutions with the exact solutions at x = 0.5.
Figure 2

The numerical solutions with the exact solutions at x = 0.5.

Figure 3 Absolute error y = 0.5.
Figure 3

Absolute error y = 0.5.

The second problem: MHD duct flow [12, 13, 14, 21]

The cross-section of an infinitely long rectangular duct is oriented with its sides parallel to the x− and y−axes and the origin of coordinates at its center. The duct width is 2a and height 2b so that the sides of the duct have equations x = ±a and y = ±b. A conducting fluid flows in the z direction along the duct and is subjected to a constant applied magnetic field M acting in a direction lying in the xy-plane and making an angle ϕ with the y-axis. The equations governing the flow may be expressed in the normalized form [12, 20].

(28)∂P∂z=μν∇2∇z+A0xμ0∂Pz∂x′+A0yμ0∂Pz∂y′,

and the z-component of the curl of Ohm’s law,

(29)∇2Az+ξμ0(A0x∂Uz∂x′+A0y∂Uz∂y′)=0,

with the boundary conditions: U = A = 0 at x = ±α, y = ±b,

where ν, μ and ξ are, respectively, the kinematic viscosity, density and electric conductivity of the fluid; μ0 is the magnetic permeability in vacuum; dP/dz is the constant axial pressure gradient; B0x and B0y are the x and y components of the applied magnetic field; and Uz and Az are the z components of velocity and induced magnetic field, respectively. Following the notation of P. C. Lu [20], who solved this problem using the Kantorovieh method, Eqs. (28) and (29) become in non-dimensionsized form,

(30)(∂2∂x2+∂2∂y2)U+Mx∂A∂x+My∂A∂y=−1,

and

(31)(∂2∂x2+∂2∂y2)A+Mx∂U∂x+My∂U∂y=−1,

with boundary conditions U = A = 0, x = ±α, y = ±1. Distance has been scaled to the duct semi-height b so that x = x′/b, y = y′/b, and α = a/b. The following normalisations have also been used.

(32)U=Uz−b2νμdPdz,A=Az−b2νμdPdzμ0(νμξ)12,Mx=A0x′b(ξνμ)12=Msin(ϕ),My=A0y′b(ξνμ)12=Mcos(ϕ),M=Hartmannno.=(Mx2+My2)12=A0b(ξνμ)12.

The Hartmann number is the ratio of magnetic to fluid viscosity. If M = 0, the flow field is the classieal laminar pipe flow. If M ≥ 1, the flow field is determined primarily by the E × A drift. To uncouple (30) and (31), the functions

(33)H1=U+A,

and

(34)H2=U−A,
(35)(∂2∂x2+∂2∂y2)H1+Mx∂H1∂x+My∂H1∂y=−1,

and

(36)(∂2∂x2+∂2∂y2)H2−Mx∂H2∂x−My∂H2∂y=−1,

with boundary conditions H1 = H2 = 0, x = ±α, y = ±1. Thus, if H1 is solved as H1(Mx , My) from (36), then

(37)H2(Mx,My)=H1(−Mx,−My).

So that the solution is completely determined when either H1 or H2, are known. Having determined H1 the function H2 is found from (37) and hence the velocity field U from

(38)U=12(H1+H2).

Now, we will intreduce some numerical resultes for the flow in a square duct with an applied magnetic field parallel to the x−axis so that My = 0. To compare with earlier results [12, 13, 14, 19], we give to M, the following values Mx = 0, 2, 5 and 8.

By substituting from (6)-(8) in (29) and (30) we get the numerical solutions as follows:

In Table 2, the results of the 2-dimensions cubic B-spline method using a mesh of 20 ×20 were compared with those the numerical [12, 13, 14, 21] and also with the analytic solution of Shercliff [19].

Table 2

U at the centre of the duct

MxAlexan- der [12]ones and Xenop- hontos [13]Bi-cubic B-spline [14] within a Galerkin approachFinite difference method [21]2-dimensions cubic B-spline collocation methodAnalytic [19]
00.298240.298260.2946850.2941070.2935980.294686
20.263200.263190.2589070.2586250.2582820.258907
50.174300.174280.1716020.1714750.1714850.171609
80.120140.120130.1187820.1186520.1187490.118782

In Figure 4, we show The profile of velocity with Hartmann numbers 0 (top curve), to 8 (bottom curve) at [−1, 1] using a mesh of 20 × 20 .

Figure 4 The profile of velocity with Hartmann numbers 0 (top curve), to 8 (bottom curve).
Figure 4

The profile of velocity with Hartmann numbers 0 (top curve), to 8 (bottom curve).

In Table 3, some other results are presented where the period with from [-1, 1] to [-0.5, 0.5] is changed and we also compare these results with Finite difference method [21] and the analytical solution found in the research [19].

Table 3

U at the centre of the duct. finite difference and analytic simulations compared

MxFinite differenc method using a mesh of 50 × 50 [21]2-dimensions cubic B-spline 50 × 50Analytic [19]
00.0736480.07362790.0736711
20.0711090.07109080.071128
50.0608380.06082730.060846
80.0493590.04935630.0493638

In Figure 5, we show the profile of velocity with Hartmann numbers 0 (top curve), to 8 (bottom curve) at [−0.5, 0.5] using a mesh of 50 × 50.

Figure 5 The profile of velocity with Hartmann numbers 0 (top curve), to 8 (bottom curve).
Figure 5

The profile of velocity with Hartmann numbers 0 (top curve), to 8 (bottom curve).

For various values of the Hartmann number, the solution for the velocity profile along the x−axis is shown in figures 4 and 5. As would be expected, increasing the magnetic field (increasing the Hartmann number) has an effect on the velocity of the fluid where it is the velocity decreases in the near the center of the channel, this apparent effect of the magnetic field intensity is already known. Therefore, we see that the results obtained are fully compatible with the physical meaning of the effect of the magnetic field.

The third problem: [37, 38, 39, 40, 41]

Let us take the 2-dimensions partial differential equation as:

(39)uxx(x,y,z)+uyy(x,y,z)−sin⁡(πx)sin⁡(πy)=0,x,y∈[a,b]

with the exact solution is given by:

(40)u(x,y,z)=−sin⁡(πx)sin⁡(πy)2π2.

The boundary conditions are given as follows:

(41)u(a,y)=u(x,a,)=α,u(b,y)=u(x,b)=β.

By substituting from (5)-(7) in (39) with the boundary conditions (41) we get the numerical solutions as follows:

In Table 4, the results of the 2-dimensions quadratic B-spline method using a rectangular mesh of 15 × 15 were compared with the exact solution. We see our results are accepted with respect to the exact solutions. In Figures 6, 7 we show the numerical solutions with the exact solutions at y = 0.4. Figure 8 illustrates the absolute error at y = 0.5.

Figure 6 The numerical solutions with the exact solutions at y = 0.4.
Figure 6

The numerical solutions with the exact solutions at y = 0.4.

Figure 7 The numerical solutions with the exact solutions at x = 0.4.
Figure 7

The numerical solutions with the exact solutions at x = 0.4.

Figure 8 Absolute error y = 0.5.
Figure 8

Absolute error y = 0.5.

Table 4

The numerical resultes for third problem at y = 0.4, x, y ∈ [0, 1]

xNumerical solutionExact solutionAbsolute error
0.2-0.0282168-0.02832011.03294 E-4
0.4-0.0456558-0.04582291.67134 E-4
0.6-0.0456558-0.04582291.67134 E-4
0.8-0.0282168-0.02832011.03294 E-4

Considering 15 × 15 grid points, comparison between theresults of the proposed method and the results of using differentmethods is shown in Table 6 [37, 38, 39, 40, 41].

Table 5

Maximum absolute error according to the method used for third problem.

The proposed methodQuadratic B-spline method [37]MCBDQM (modified cubic B-spline differential quadrature method) [38]Spline-based DQM [39]Haar wavelet method[40]spectral collocation method based on Haarwavelets [41]
1.67 E-43.72 E-52.11 E-51.62 E-43.08 E-43.08 E-4
Table 6

The numerical resultes for third problem at y = z = 0.5, x, y, z ∈ [0, 1]

xNumerical solutionExact solutionAbsolute error
0.10.01686360.01689843.48009 E-5
0.20.03313060.03320127.05722 E-5
0.30.04805330.04815951.06205 E-4
0.40.06068620.06082791.41846 E-4
0.50.06984680.07002641.79618 E-4
0.60.07407080.07429552.24718 E-4
0.70.07155860.07184562.86918 E-4
0.80.06011420.06049653.82276 E-4
0.90.03707380.03760825.34402 E-4

The fourth problem

Let us take the 3-dimensions partial differential equation as:

(42)uxx(x,y,z)+uyy(x,y,z)+uzz(x,y,z)+xyz(−ex+y+z)(x(3yz+y+z−5)+y(z−5)−5z+9)=0,x,y,z∈[a,b]

With the exact solution is given by:

(43)u(x,y,z)=x(1−x)y(1−y)z(1−z)ex+y+z.

The boundary conditions are given as follows:

(44)u(a,y,z)=u(x,a,z)=u(x,y,a)=α,u(b,y,z)=u(x,b,z)=u(x,y,b)=β.

By substituting from (13) in (42) with the boundary conditions (44) we get the numerical solutions as follows:

In Table 6, the results of the 3-dimensions cubic B-spline method using a mesh of 20 × 20 were compared with the exact solution. we see our results are accepted with respect to the exact solutions. In Figure 9, we show the numerical solutions with the exact solutions at y = z = 0.5. Figure 10 illustrates the absolute error at y = z = 0.5.

Figure 9 The numerical solutions with the exact solutions at y = z = 0.5.
Figure 9

The numerical solutions with the exact solutions at y = z = 0.5.

Figure 10 Absolute error y = z = 0.5.
Figure 10

Absolute error y = z = 0.5.

The fifth problem

Let us take the 4-dimensions partial differential equation as:

(45)uxx(x,y,z,d)+uyy(x,y,z,d)+uzz(x,y,z,d)+udd(x,y,z,d)−(4dxyzed+x+y+z(dz(xy−1)−d(x+y−2)−x(y+z−2)+y(−z)+2y+2z−3))=0,x,y,z,d∈[a,b]

With the exact solution is given by:

(46)u(x,y,z,d)=x(1−x)y(1−y)z(1−z)d(1−d)ex+y+z+d.

The boundary conditions are given as follows:

(47)u(a,y,z,d)=u(x,a,z,d)=u(x,y,a,d)=u(x,y,z,a)=α,u(b,y,z,d)=u(x,b,z,d)=u(x,y,b,d)=u(x,y,z,b)=β.

By substituting from (20)-(23) in (44) with the boundary conditions (46) we get the numerical solutions as follows:

In Table 7, the results of the 4-dimensions cubic B-spline method using a mesh of 10 × 10 were compared with the exact solution. we see our results are accepted with respect to the exact solutions. In Figure 11, we show the numerical solutions with the exact solutions at y = z = d = 0.5. Figure 12 illustrates the absolute error at y = z = d = 0.5.

Figure 11 The numerical solutions with the exact solutions at y = z = d = 0.5.
Figure 11

The numerical solutions with the exact solutions at y = z = d = 0.5.

Figure 12 Absolute error y = z = d = 0.5.
Figure 12

Absolute error y = z = d = 0.5.

Table 7

The numerical resultes for fourth problem at y = z = d = 0.5, x, y, z, d ∈ [0, 1]

xNumerical solutionExact solutionAbsolute error
0.10.00691960.00696524.55541 E-5
0.20.01359260.01368499.22627 E-5
0.30.01971250.01985041.37949 E-4
0.40.02488940.02507211.82705 E-4
0.50.02863320.02886352.30343 E-4
0.60.03033220.03062312.90919 E-4
0.70.02922880.02961333.84553 E-4
0.80.02438920.02493555.46233 E-4
0.90.01467090.01550138.30453 E-4

5 Conclusion

In the end, we can say that we have presented a solution to a basic problem in various fields of scientific research, which is dealing with partial differential equations in their different dimensions. This solution came from we have already tracked what was done from some researchers and found that they provided solutions to deal with partial differential equations in one and two dimensions. We have developed a method of cubic B-splines that were used previously in solving mathematical problems and we were able to provide a shape for this method in three dimensions and in four dimensions. It should be testing the accuracy and effectiveness of the shapes that have been inferred, so we have provided some numerical examples in various dimensions that have the exact solutions at the beginning. The numerical results were compared with the real solution, and the inferred forms were found to be effective and accurate. We also presented after the figures that show how accurate the method used. From this standpoint, we can say that the problems of partial differential equations of different dimensions have been solved. During the future work, we will also generalize some forms of B-splines to serve as a solution to partial differential equations in n-dimensions.

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Received: 2019-06-25
Accepted: 2020-11-04
Published Online: 2020-12-31

© 2020 K. R. Raslan and Khalid K. Ali, published by De Gruyter

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

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https://doi.org/10.1515/nleng-2020-0027
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Collocation Method; 3-dimensions cubic B-Splines; 4-dimensions cubic B-Splines
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Artikel in diesem Heft

  1. Comparison of the method of variation of parameters to semi-analytical methods for solving nonlinear boundary value problems in engineering
  2. Nonlinear H-infinity control for switched reluctance machines
  3. Energy flow of a 2018 FIA F1 racing car and proposed changes to the powertrain rules
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  7. Approximate method for solving strongly fractional nonlinear problems using fuzzy transform
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  11. Two dimensional simulation of laminar flow by three- jet in a semi-confined space
  12. Influence of temperature-dependent properties on a gravity-driven thin film along inclined plate
  13. Simulation and time-frequency analysis of the longitudinal train dynamics coupled with a nonlinear friction draft gear
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  20. MHD natural convection flow of Casson fluid in an annular microchannel containing porous medium with heat generation/absorption
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