Performance of GRKM-method for solving classes of ordinary and partial differential equations of sixth-orders

2.1 Quasilinear sixth-order ODEs

The general class of quasilinear sixth-order ODEs can be defined as follows:

The class one of general quasilinear ODEs of 6th-order with no explicit dependence of the derivatives w ( i ) ( ξ ) , for i ≤ 5 of second, third, fourth, and fifth orders has the following definition.

2.2 Quasilinear sixth-order PDEs

The quasilinear PDEs of sixth-order in the domain with two variables ζ , ς defined generally as follows:

The sixth-order PDE in domain with n -variables s 1 , s 2 , … , s n , is defined generally as follows.

where ς ‾ = ς 1 , ς 2 , … , ς n

The general formula of quasilinear PDE of 6th-order in n -independent variables is given in the following definition:

The following categories are used to classify the quasilinear PDEs of sixth-order.

for i 1 , i 2 , … , i 6 , i j 1 , i j 2 , … , i j m = 1 , 2 , … , n . We say that Equation (7) of classes 1–5 for m = 1 , 2 , … , 6 resp.

The general formula of PDE of sixth-order of class one in two-variables ( ζ , η ) is given as follows:

or

Nowadays, the various numerical methods are used to solve some types of PDEs in various-fields of applied mathematics, engineering, and physics. However, the solutions of these PDEs could approximate by these numerical integrators. However, using modified RKD method combining with MOL, Mechee et al. [20] solved the third-order quasilinear PDEs of class one.

In this article, the sixth-order PDE of class one is converted to a system of sixth-order ODEs and then, using the proposed GRKM integrator combining with MOL, we solved this system of ODEs. Based on the constructed GRKM, the numerical solutions of the implementations of sixth-order PDEs are compared with the exact solutions of these test problems; the comparisons show that the constructed integrator is highly accurate and efficient.

3.1 Proposed GRKM for solving ODEs

The following is the form of the proposed GRKM integrator with s-stages for solving the class one of quasilinear sixth-order ODEs in Equation (2) with ICs (3):

where

and

for j = 2 , 3 , 4 , ⋯ , ş . The coefficients of GRKM are listed as follows: b i ( l ) and c i , a 1 , i , j , a 2 , i , j for i , j ≤ 8 and l = 0 , 1 , 2 , … , 5 . GRKM is explicit-method if a 1 , i , j = a 2 , i , j = 0 , for i ≤ j , else GRKM is implicit-method. The coefficients of GRKM are expressed as in Table 1 using Butcher notation.

3.2 Modified GRKM-method for solving PDEs

Consider the following form of quasilinear sixth-order PDEs of class one: ζ , η to

with ICs

with the boundary conditions (BCs),

We present an established approach for solving class one PDEs using the GRKM and MOL that is consistent with the following algorithm.

3.3 Algorithm of modified GRKM

Consider two intervals of the domain definition in two directions of ζ and η , which are named as [ a , b ] and [ 0 , T ] with the norms of the subintervals are h = b − a n and k = T m respectively. Here, n and m are the number of partitions of intervals in the directions of ζ , η respectively, where zet a i = a + ih , and η j = jk , for j = 1 , 2 , … , m . and i = 1 , 2 , … , n − 1 . We can combine GRKM with MOL method to solve Equation (24) with ICs in Equation (25) and BCs in Equation (26) according the following-steps:

1. Apply the steps (2)–(6) while 1 ≤ k ≤ mm

2. Fix ζ = ζ l at the point ( ζ , η ) of the PDE in Equation (24) in which convert to the following system of (DEs):

   (27) w l ( 6 ) ( η ) = f ζ , η , w ′ ( η ) , w ( ζ , η ) , ∂ w ( ζ , η ) ∂ s , ∂ 2 w ( ζ , η ) ∂ ζ 2 , ∂ 3 w ( ζ , η ) ∂ ζ 3 , ∂ 4 w ( ζ , η ) ∂ ζ 4 , ∂ 5 w ( ζ , η ) ∂ ζ 5 , ∂ 6 w ( ζ , η ) ∂ ζ 6 ζ = ζ l .

3. Substituting the formulas of central finite-difference in the derivative function w ( ζ , η ) in the right-hand side of Equation (27) up to order six as follows:

   ∂ w ( ζ , η ) ∂ ζ ( ζ , η ) = ( ζ i , η j ) ≅ w l + 1 , j − w l − 1 , j 2 h , ∂ 2 w ( ζ , η ) ∂ ζ 2 ( ζ , η ) = ( ζ i , η j ) ≅ w l + 1 , j − 2 w l , j + w l − 1 , j h 2 , ∂ 3 w ( ζ , η ) ∂ ζ 3 ( ζ , η ) = ( ζ i , η j ) ≅ w l + 2 , j − 2 w l + 1 , j + 2 w l − 1 , j − w l − 2 , j h 3 ,

   ∂ 4 w ( ζ , η ) ∂ ζ 4 ( ζ , η ) = ( ζ i , η j ) ≅ w l + 2 , j − 4 w l + 1 , j + 6 w i , j − 4 w i − 1 , j + w i − 2 , j 2 h 3 , ∂ 5 w ( ζ , η ) ∂ ζ 5 ( ζ , η ) = ( ζ i , η j ) ≅ w l + 3 , j − 4 w i + 2 , j + 5 w l + 1 , j − 5 w l − 1 , j + 4 w l − 2 , j − w l + 3 , j 2 h 5 , ∂ 6 w ( ζ , η ) ∂ ζ 6 ( ζ , η ) = ( ζ i , η j ) ≅ w l + 3 , j − 6 w l + 2 , j + 15 w l + 1 , j − 20 w l , j + 15 w l − 1 , j − 6 w l − 2 , j + w l + 3 , j h 6 .

   Hence, we obtain a system of sixth -order ODEs as follows:

   (28) ψ l ( 6 ) ( t ) = f ( η l , η , ψ i − 3 ( η ) , ψ l − 2 ( η ) , ψ l − 1 ( η ) , ψ l ( η ) , ψ l ′ ( η ) , ψ l + 1 ( η ) , ψ l + 2 ( η ) , ψ l + 3 ( η ) ) ,

   for l = 1 , 2 , … , n − 1 .

4. If j = 1 then, ICs have the formula:

   (29) w i ( k ) ( 0 ) = f k + 1 ( ζ i ) .

   For 2 = < j < = m , the ICs are,

   (30) w i ( k ) ( η j − 1 ) = d k w ( ζ , η j − 1 ) d ζ k ζ = ζ i = f k ( ζ i ) ,

   for k = 0 , 1 , 2 , 3 , 4 , 5 .

5. Put the BCs using the following notation,

   (31) w 0 , j = w ( a , η j ) = g 1 ( η j ) , w n , j = w ( b , η j ) = g 2 ( η j ) .

6. Finally, the system of ODEs in Equation (27) at the line η = η j with ICs (28) and (29) and BCs (30) can be solved using the GRKM, Figure 1, The class one of quasilinear sixth-order PDE in Equation (24) with the ICs in Equation (25) and the BCs in Equation (26) in the region of definition, which are shown in Figure 1, could be solved using this algorithm.

Figure 1

The domain of PDE of class one.

4.1 Implementation of ODEs

This subsection investigates the proposed GRKM by studying the numerical solutions of several problems and then, their numerical results are presented in Figure 2.

Figure 2

The efficiency curves of GRKM and their comparisons for graphs of Log10(Absolute Errors) against Log(Computational Time) of numerical solutions for Examples 4.1–4.6.

ICs

The exact solution is w ( ξ ) = sin ⁡ ( ξ ) , b = 1 .

ICs

The exact solution is w ( ξ ) = e − ξ , b = 1 .

0 < ξ ≤ b .

ICs

The exact solution is w ( ξ ) = e − ξ 2 , b = 1 .

ICs

The exact solution is w ( ξ ) = sin ⁡ ( ξ ) , b = π .

ICs

The exact solution is w ( ξ ) = 1 1 + ξ , b = 1 .