Generalized numerical RKM method for solving sixth-order fractional partial differential equations

Murtadha A. Kadhim; AllahBakhsh Yazdani Cherati; Mohammed Sahib Mechee

doi:10.1515/nleng-2022-0321

Article Open Access

Generalized numerical RKM method for solving sixth-order fractional partial differential equations

Murtadha A. Kadhim , AllahBakhsh Yazdani Cherati and Mohammed Sahib Mechee

Published/Copyright: November 3, 2025

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From the journal Nonlinear Engineering Volume 14 Issue 1

Abstract

Fractional partial differential equations (FPDEs) are used as tools in the mathematical modeling of the natural phenomena and interpretation of many life problems in the fields of engineering and applied science. Mathematical models that include different types of partial differential equations are used in some fields of applied sciences such as biology, diffusion, electronic circuits, damping laws, and fluid mechanics. The derivation of modern analytical or numerical methods for solving FPDEs is a significant problem. In this article an efficient numerical method is proposed for solving a class of sixth-order FPDEs in which created by combining numerical Runge–Kutta–Mohammed (RKM) techniques with the method of lines. We have applied the modified approach to solve some problems involving the sixth-order FPDEs, and then, the numerical and analytical solutions for these problems have been compared. The comparisons in the implementations have proved the efficiency and accuracy of the developed RKM method. Also, the wave equation of fractional order as an application of the wave equation has been presented. Additionally, by contrasting the numerical implementations with exact answers, the effectiveness and accuracy of the suggested methodologies are shown.

Keywords: RK; RKN; Runge–Kutta–Mohammed; MOL; ODEs; sixth-order; FPDEs; finite difference method; FDEs

1 Introduction

Differential equations (DEs) of all kinds are essential instruments in the various departments of applied mathematics and its applications, particularly in the scientific and engineering fields. Different varieties of DEs, particularly fractional types, are used in a large amount of mathematical modeling in the sciences and engineering. In recent years, it has been found that models built on the theory of fractional derivatives (FDs) and integrals can very accurately describe a wide range of phenomena in chemistry, engineering, physics, and other sciences. The use of fractional DEs (FDEs) in applied science or engineering has expanded recently. For example, nonlinear seismic oscillations can be described by FDs, and the insufficiency brought on by a fluid-dynamic traffic model’s assumption of continuous traffic flow can be addressed by FDEs. Finding analytical or approximative solutions to the linear fractional partial DEs (FPDEs) can be performed in a few different ways. Analytical solutions to most FPDEs are lacking. However, since most FPDEs have precise analytical solutions, numerical and approximate methodologies should be used to solve them. For instance, two relatively new methods for approximating the solutions to various types of these problems are the Adomian decomposition method and the variational iteration method (VIM). The decomposition method has been applied in the literature to approximate solutions to a variety of DEs, particularly FDEs. Long-term research has been performed on a variety of contemporary and traditional analytical techniques for solving DEs [1,2,3,4]. Various methods have been used to determine the solution of nonlinear FPDEs. In recent years, a variety of classes of FPDE problems have appeared in various branches of mathematics, leading to the development of some techniques [5,6]. Momani and Odibat [7] provided a definition with FDs in the Caputo sense and compared the outcomes of the homotopy perturbation method and VIM. The telegraph and Laplace equations on Cantor sets were roughly analytically solved [8,9] using the local fractional Laplace decomposition approach. Numerous authors examined PDEs and FPDEs to review the literature relevant to this study. For instance, Mechee et al. [10] developed a direct numerical method for solving initial-boundary value issues that involve PDEs, Khalaf and Flayyih [11] studied the COVID-19 pandemic in Iraq, by using the standard SIR model with three unknown biological parameters and Farhood and Mohammed [12] solved the delay-variable order FPDEs approximatively using the homotopy perturbation technique. However, a numerical approach for solving some classes of the sixth-order FPDEs has been presented. The modified methods were established by combining the numerical RK-type method and the method of lines (MOL), both of which were created for solving specific types of FPDEs of different orders. The developed approaches are applied to several cases involving FPDEs of the second and sixth orders, and the numerical and analytical results are examined. A wave FPDE, or application of the wave equation, has also been presented in classical physics, namely, for mechanical waves such as sound, water, seismic, and electromagnetic waves. In order to demonstrate how adaptable and useful the method is, numerical examples are provided. With the goal to show how flexible and practical the method is, numerical examples are given. The numerical results are taken to be identical to the actual solutions of the implementations in order to show the precision and utility of the suggested approaches. Additionally, by contrasting the numerical implementations with exact answers, the effectiveness and accuracy of the suggested methodologies are shown. The proposed methods, which made use of Runge–Kutta–Mohammed (RKM) techniques, provided numerical answers to the test problems that are remarkably similar to their analytical solution. To show how flexible and practical the method is, numerical examples are given. By comparing the proposed approaches with exact solutions, one may evaluate their accuracy of them. The proposed methods, which used the use of RK methodologies, provided numerical solutions to test’s issues.

2 Preliminary

In this section, the basic background related to this study was introduced.

2.1 Basic definitions of the FDs [13,14]

Many researchers have attempted to define FDEs in famous different ways. The most popular ones are the following definitions of FDEs:

(a) The definition of Riemann–Liouville:

Definition 1. The right and left Riemann–Liouville fractional integral operator of order > 0 or α ∈ [ n − 1 ; n ) :

(1) D a α f ( t ) = 1 ɣ ( n − α ) d n d t n ∫ a t f ( t ) ( t − ӽ ) α − n − 1 d ӽ , ӽ > 0 ,

(2) D a t α y ( t ) = 1 ɣ ( n − 1 ) d n d t n ∫ a τ ( t − s ) n − α − 1 y ( s ) d s , ӽ > 0 ,

and

(b) Caputo definition. For α ∈ [ n − 1 , n ) , where n is an integer and a derivative of f is as follows:

Definition 2

(3) D a α f ( t ) = 1 ɣ ( n − α ) d n d t n ∫ a t f ( n ) ( t ) ( t − x ) α − n − 1 d ӽ , ӽ > 0 .

2.2 Classical fractional partial integrals and derivatives [15]

In this section, some properties of partial fractional calculus as well as some fundamental concepts were introduced.

2.2.1 Basic definitions of the fractional partial calculus

In the following, we have introduced the definitions and preliminaries of fractional partial calculus.

Definition 3. The space-fractional derivative operator of order α > 0 of Caputo is defined as follows:

(4) D ӽ α Ѵ ( ӽ , τ ) = ∂ α Ѵ ( ӽ , τ ) ∂ ӽ α = I ӽ m − α D ӽ m Ѵ ( ӽ , τ ) , if m − 1 < α < m , D ӽ m Ѵ ( ӽ , τ ) , if α = m , m ϵ N .

Here, the smallest integer by which m is exceeded is α ,

where

(5) I ӽ α Ѵ ( ӽ , τ ) = 1 Γ ( α ) ∫ 0 ӽ ( ӽ − σ ) α − 1 Ѵ ( σ , τ ) d σ , α > 0 , ӽ > 0 ,

and the operator for the time-fractional derivative of order α > 0 can be defined as:

(6) D τ α Ѵ ( ӽ , τ ) = ∂ α Ѵ ( ӽ , τ ) ∂ τ α = I τ m − α D τ m Ѵ ( ӽ , τ ) , if m − 1 < α < m , D τ m Ѵ ( ӽ , τ ) , if α = m , m ϵ N ,

where

(7) I τ α Ѵ ( ӽ , τ ) = 1 ɣ ( α ) ∫ 0 ӽ ( ӽ − σ ) α − 1 u ( σ , τ ) d σ , α > 0 , ӽ > 0 .

2.2.2 Properties of the operators J ӽ α and J τ α

Some of the most important properties of the operators J ӽ α and J τ α are given as follows:

I ӽ α ( ӽ ɣ ) = ɣ ( ɣ + 1 ) ɣ ( α + ɣ + 1 ) ӽ α + ɣ ,
I τ α ( τ ɣ ) = ɣ ( ɣ + 1 ) ɣ ( α + ɣ + 1 ) τ α + ɣ ,
I ӽ α ( ӽ ɣ τ β ) = τ β I ӽ α ( ӽ ɣ ) ,
I τ α ( ӽ ɣ τ β ) = ӽ ɣ I τ α ( τ β ) .

The Caputo’s fractional differentiations are the linear operators with respect to λ and η. Hence, the properties of this operator are given as follows:

(8) i . D τ α ( w ( ӽ , τ ) ) = λ D τ α Ѵ ( ӽ , τ ) + η D τ α v ( ӽ , τ ) , where w ( ӽ , τ ) = λ Ѵ ( ӽ , τ ) + η v ( ӽ , τ ) ,

(9) ii . D ӽ α ( w ( ӽ , τ ) ) = λ D ӽ α Ѵ ( ӽ , τ ) + η D ӽ α v ( ӽ , τ ) ,

where λ and η are the constants.

2.3 Quasi-linear FPDEs

The general quasi-linear fractional FPDE can be expressed as follows:

(10) D τ ( n − times ) n α ( ӽ , τ ) = f ( ӽ , τ , Ѵ ( ӽ , τ ) , Ѵ ӽ ( ӽ , τ ) , . . , Ѵ ӽ ( n − times ) ( ӽ , τ ) ) , ӽ 1 ≤ ӽ ≤ ӽ 2 , 0 < τ ≤ T ,

with the initial conditions (I.C.s):

(11) Ѵ ӽ ( i − times ) ( ӽ , 0 ) = f i ( ӽ ) , i = 0 , 1 , 2 , … , n − 1 ,

and the boundary conditions (B.C.s):

(12) Ѵ ( ӽ 1 , τ ) = φ 1 ( τ ) , Ѵ ( ӽ 2 , τ ) = φ 2 ( τ ) ,

where 0 < α ≤ 1 .

2.4 Finite difference method (FDM)

A popular approach for solving DEs for ordinary DEs (ODEs) or PDEs is the FDM. However, researchers often used the central, forward, or backward finite difference (FD) formulae to solve ODE or PDE [1]. In this study, we will use FDM to solve the issues of quasi-linear FPDE in Eq. (10) with the I.C.s in Eq. (11) and the B.C.s in Eq. (12). We divide the intervals of definitions of the variables ӽ and τ , which named as [ ӽ 1 , ӽ 2 ] and [0;T], to N and M subintervals, respectively, where ӽ i = ӽ 1 + ih , τ j = jk and h = ӽ 2 − ӽ 1 N 1 and k = T M . At the position ( ӽ , τ ) , we put ( ӽ i , τ j ) for i = 1, 2, …, N − 1 and j = 1, 2, …, M. This will be transformed FPDE into the finite difference system as follows:

(13) ∂ n Ѵ ( ӽ , τ ) ∂ τ n ( ӽ , τ ) = ( ӽ i , τ j ) = f ӽ , τ , Ѵ ( ӽ , τ ) , ∂ Ѵ ( ӽ , τ ) ∂ ӽ , ∂ 2 Ѵ ( ӽ , τ ) ∂ ӽ 2 , … , ∂ n Ѵ ( ӽ , τ ) ∂ ӽ n ( ӽ , τ ) = ( ӽ i , τ j ) .

Using the orders one, two, three, etc. of the central finite difference formulae is as follows:

(14) ∂ Ѵ ( ӽ , τ ) ∂ ӽ | ( ӽ , τ ) = ( ӽ i , τ j ) ≅ Ѵ i + 1 , j − Ѵ i − 1 , j 2 h ,

(15) ∂ 2 Ѵ ( ӽ , τ ) ∂ ӽ 2 | ( ӽ , τ ) = ( ӽ i , τ j ) ≅ Ѵ i + 1 , j − 2 Ѵ i , j + Ѵ i − 1 , j h 2 ,

(16) ∂ 3 Ѵ ( ӽ , τ ) ∂ ӽ 3 | ( ӽ , τ ) = ( ӽ i , τ j ) ≅ Ѵ i + 2 , j − 2 Ѵ i + 1 , j + 2 Ѵ i − 1 , j + Ѵ i − 2 , j 2 h 3 ,

(17) ∂ 4 Ѵ ( ӽ , τ ) ∂ ӽ 4 | ( ӽ , τ ) = ( ӽ i , τ j ) ≅ Ѵ i + 2 , j − 4 Ѵ i + 1 , j + 6 Ѵ i , j − 4 Ѵ i − 1 , j + Ѵ i − 2 , j h 4 ,

(18) ∂ 5 Ѵ ( ӽ , τ ) ∂ ӽ 5 ( ӽ , τ ) = ( ӽ i , τ j ) ≅ − Ѵ i + 3 , j + 2 Ѵ i + 2 , j − 5 Ѵ i + 1 , j + 5 Ѵ i − 1 , j − 2 Ѵ i − 2 , j + Ѵ i − 3 , j 2 h 4 ,

(19) ∂ 6 Ѵ ( ӽ , τ ) ∂ ӽ 6 ( ӽ , τ ) = ( ӽ i , τ j ) ≅ Ѵ i + 3 , j − 6 Ѵ i + 2 , j + 15 Ѵ i + 1 , j − 20 Ѵ i , j + 15 Ѵ i − 1 , j − 6 Ѵ i − 2 , j + Ѵ i + 3 , j h 6 .

3 Proposed numerical method

In this article, a numerical method for solving FPDEs has been developed by combining the MOL technique with the RKN and RKM approaches. Let the numerical solution’s Ѵ ( ӽ i , τ j ) , where ( ӽ i , τ j ) ∈ [ ӽ i , τ j ] × [ 0 ; T ] , where τ j = jk and ӽ i = ӽ 1 + ih with k = T M and h = ӽ 2 − ӽ 1 N , where N and M are the numbers of points in the directions of ӽ and τ , respectively, for i = 1, 2 , …, N − 1; j = 1, 2, …, M. However, after some adjustments, the following set of explicit FDEs are obtained in the following:

Ѵ τ ( n − times ) n α ( τ ) | ӽ = ӽ i = f ( ӽ i , τ i , Ѵ i − 2 , j , Ѵ i , j , Ѵ i + 1 , j , Ѵ i + 2 , j , Ѵ i , j + 1 , Ѵ i , j − 1 , Ѵ i , j − 2 ) ,

with the I.C.s in Eq. (11) and the B.C.s in Eq. (12). We can combine the MOL with RKM method to solve Problem (10) with given initial and boundary conditions (11), (12) by the following algorithm.

Algorithm:

1- Do steps 2–6 while 1 ≤ j ≤ m,

2- Fixing ӽ = ӽ i at the FPDE’s point ( ӽ , τ ) in Eq. (10) gives in the FDEs in the following:

(20) Ѵ τ ( n − times ) n α ( τ ) = f ( ӽ , τ , Ѵ ( τ ) , Ѵ ӽ ( τ ) , Ѵ ӽ ӽ ( τ ) , . Ѵ ӽ ( n − times ) ( τ ) ) | ӽ = ӽ i ,

where Ѵ i n α ( τ ) = D τ n α Ѵ ( ӽ , τ ) | ӽ = ӽ i for i = 1, 2, …, N.

3- The derivatives of the right-hand side of the ODE in Eq. (13) may be transformed into a system of finite difference FODEs by putting in the appropriate finite difference formula provided in Eqs (14–19):

(21) Ѵ i n α ( τ ) = f ( ӽ i , τ , Ѵ i − n + 1 ( τ ) , Ѵ i − n ( τ ) , … , Ѵ i + n − 2 , Ѵ i + n − 1 ( τ ) ) .

4- For i = 1, 2, …, n − 1. When j = 1, the I.C.s are Ѵ i ( k ) ( 0 ) = f k ( ӽ i ) , where k = 0, 1, …, n − 1, and 2 ≤ j ≤ M, the ICs are as follows:

(22) Ѵ i ( τ j − 1 ) = Ѵ ( ӽ i , τ j − 1 ) , Ѵ i ( k ) ( τ j − 1 ) = d Ѵ ( k ) ( ӽ , τ j − 1 ) d ӽ ( k ) | ӽ = ӽ i .

5- Put the B.C.s:

(23) Ѵ 0 , j = Ѵ ( ӽ 1 , τ j ) = φ 1 ( τ j ) , Ѵ N , j = Ѵ ( ӽ 2 , τ ) = φ 2 ( τ j ) .

6- Solve the system of nth-order FODEs in Eq. (21) at τ = τ _j with the I.C.s in Eq. (22) and the B.C.s in Eq. (23) using the RK-type methods. In general, this algorithm is used for solving a FPDE of nth-order in Eq. (21) with the I.C.s in Eq. (22) and the B.C.s in Eq. (23).

3.1 Quasi-linear nth-order ODEs

In general, the special nth-order quasi-linear ODEs have the following form:

(24) Ѵ ( n ) ( ӽ ) = f ( ӽ , Ѵ ( ӽ ) ) , ӽ ≥ ӽ 0 ,

with the I.C.s

(25) y ( j ) ( ӽ 0 ) = α j , where j = 0 , 1 , 2 , … , n .

Let

z n ≅ Ѵ ( ӽ n ) , n = 1 , 2 , … .

3.2 RKM numerical methods [10,16,17,18,19,20]

In this study, the problems of interest are the special value problems of different orders of ODEs, in Eq. (24). Previous studies [10,16,17,18] derived the general form of RKM methods with s-stage for solving special different orders of ODEs, in Eq. (24) with the I.C.s (25).

3.2.1 RKM method for solving sixth-order ODEs

The general formula of the RKM method for solving the sixth-order ODEs in Eq. (24) with the I.C.s. (25) can be written as follows:

z n + 1 = ∑ i = 0 5 h ˆ i z n ( i ) i ! + h ˆ 6 ∑ i = 1 α b ̇ i k i , z n + 1 ′ = ∑ i = 0 4 h ˆ i z n ( i + 1 ) i ! + h ˆ 5 ∑ i = 1 α b ̇ i k i ,

z n + 1 ″ = ∑ i = 0 3 h ˆ i z n ( i + 2 ) i ! + h ˆ 4 ∑ i = 1 α b ̇ i k i , z n + 1 ′ ″ = ∑ i = 0 2 h ˆ i z n ( i + 3 ) i ! + h ˆ 3 ∑ i = 1 α b ̇ i k i ,

z n + 1 ( 4 ) = ∑ i = 0 1 h ˆ i z n ( i + 4 ) i ! + h ˆ 2 ∑ i = 1 α b ̇ i k i , z n + 1 ( 5 ) = z n ( 5 ) + h ˆ ∑ i = 1 α b ̇ i k i ,

k 1 = F ∼ ( ӽ n , z n ) ,

k i = F ∼ ӽ n + ƈ i h ˆ , ∑ j = 0 5 h ˆ j ƈ i j z n ( j ) j ! , + h ˆ 6 ∑ j = 1 i − 1 a i j k j .

The traditional RK integrators have been generalized to solve a class of sixth -order ODEs, expanding their applicability beyond the realm of the sixth-order ODEs. Wherever derived two effective direct integrators for a certain class of ODEs, which is a significant and novel contribution. Three-stage, fifth-order and four-stage, sixth-order RKM integrators, respectively, have been developed from order conditions.

4 Application

The fractional wave equation is a second-order linear FPDE that explains mechanical waves in classical physics, such as seismic, sound, water, and electromagnetic waves (including light waves) [21]. It is defined as follows:

(26) Ѵ τ 2 α ( ӽ , τ ) = c 2 Ѵ ӽ ӽ ( ӽ , τ ) , ӽ 1 ≤ ӽ ≤ ӽ 2 , 0 < τ ≤ T ,

(27)with the ICs: Ѵ ӽ ( i − times ) ( ӽ , 0 ) = f i ( ӽ ) , i = 0 , 1

(28)and the BCs: Ѵ ( ӽ i , τ ) = φ i ( τ ) , i = 1 , 2 , 0 < α ≤ 1 .

Using the proposed algorithm in Section 3, the fractional wave equation in Eq. (26) with the I.C.s and the B.C.s in Eqs (27), (28) has been solved using the modified RKN method with n = 2 and the numerical solution of this problem has plotted in Figure 1(a) for N = 100, M = 10, T = c = 1, with I.C.s f 1 ( ӽ ) = e − ӽ and f 2 ( ӽ ) = − e − ӽ and the B.C.s Ѵ ( ӽ i , τ ) = 0 , i = 1,2. The numerical solution on the section line τ = Mh 2 has been compared with the exact solution, which is defined as: Ѵ ( ӽ , τ ) = − e − ӽ − τ .

$Figure 1 A comparison between the numerical solutions evaluated by generalized numerical methods versus the analytical solutions for the application in Section 4 in (a) and Examples 5.1, 5.2, and 5.3 in (b–d) using generalized RKN and RKM methods, for different values of t, and Example 5.2. with different values of α in ( e ) . \alpha {\rm{in}}\left({\rm{e}}).$

Figure 1

A comparison between the numerical solutions evaluated by generalized numerical methods versus the analytical solutions for the application in Section 4 in (a) and Examples 5.1, 5.2, and 5.3 in (b–d) using generalized RKN and RKM methods, for different values of t, and Example 5.2. with different values of α in ( e ) .

5 Numerical examples

We implemented the proposed algorithm using the RKM method with N = 6 in Section 3 for solving three different examples of FPDEs using MATLAB software with M = 10, N = 100, and T=1. We tested the proposed methods in this study by the comparisons for the numerical solutions versus exact solutions for three examples in which the numerical solutions on the section line τ = Mh 2 have been compared with the exact solutions and plotted in Figure 1(b–e).

Example 5.1

Consider the following FPDE problem:

Ѵ τ 6 α ( ӽ , τ ) = ( − 1 ) 6 α Ѵ ( ӽ , τ ) ; ӽ 1 ≤ ӽ ≤ ӽ 2 , τ > 0 ,

with the I.C.s:

Ѵ ( ӽ , 0 ) = ӽ 6 , Ѵ ӽ ( ӽ , 0 ) = 6 ӽ 5 , Ѵ ӽ ӽ ( ӽ , 0 ) = 30 ӽ 4 , Ѵ ӽ ӽ ӽ ( χ , 0 ) = 120 ӽ 3 , Ѵ ӽ ӽ ӽ ӽ ( ӽ , 0 ) = 360 ӽ 2 , Ѵ ӽ ӽ ӽ ӽ ӽ ( ӽ , 0 ) = 720 ӽ ,

and the BCs: Ѵ ( ӽ 1 , τ ) = ӽ 1 6 e − τ , Ѵ ( ӽ 2 , τ ) = ӽ 2 56 e − τ .

The exact solution is Ѵ ( ӽ , τ ) = ӽ 6 e − τ , with ӽ 1 = 0 and ӽ 2 = π , for α = 1 .

Example 5.2

Consider the following FPDE problem:

Ѵ τ 6 α = α − 2 ( 2 ) 6 α − 1 ( α 2 Ѵ ( ӽ , τ ) − Ѵ ӽ ӽ ( ӽ , τ ) ) , ӽ 1 ≤ ӽ ≤ ӽ 2 , τ > 0 ,

with I.C.s: Ѵ ( ӽ , 0 ) = cos ( α ӽ ) , Ѵ ӽ ( ӽ , 0 ) = − α sin ( α ӽ ) , Ѵ ӽ ӽ ( ӽ , 0 ) = − α 2 cos ( α ᶍ ) , Ѵ ӽ ӽ ӽ ( ӽ , 0 ) = α 3 sin ( α ӽ ) , Ѵ ӽ ӽ ӽ ӽ ( ӽ , 0 ) = α 4 cos ( α ᶍ ) , Ѵ ӽ ӽ ӽ ӽ ӽ ( ӽ , 0 ) = − α 5 sin ( α ӽ ) , with B.C.s:

Ѵ ( ӽ 1 , τ ) = e 2 τ cos ( α ӽ 1 ) , Ѵ ( ӽ 2 , τ ) = e 2 τ cos ( α ӽ 2 ) .

The exact solution is Ѵ ( ӽ , τ ) = e 2 τ cos ( α ӽ ) , with ӽ 1 = 0 and ӽ 2 = π .

Example 5.3

Consider the following FPDE problem:

Ѵ τ 6 α ( ӽ , τ ) = Ѵ ( ӽ , τ ) ; ӽ 1 ≤ ӽ ≤ ӽ 2 ,

with the I.C.s:

Ѵ ( ӽ , 0 ) = − Ѵ ӽ ( ӽ , 0 ) = Ѵ ӽ ӽ ( ӽ , 0 ) = − Ѵ ӽ ӽ ӽ ( ӽ , 0 ) = Ѵ ӽ ӽ ӽ ӽ ( ӽ , 0 ) = − Ѵ ӽ ӽ ӽ ӽ ӽ ( ӽ , 0 ) = e − ӽ ,

with the B.C.s:

Ѵ ( ӽ 1 , τ ) = e ӽ 1 e τ , Ѵ ( ӽ 2 , τ ) = e ӽ 2 e τ .

The exact solution is Ѵ ( ӽ , τ ) = e τ e − ӽ , with ӽ 1 = 0 and ӽ 2 = 4 , for α = 1.

A comparison between the numerical solutions z (x) evaluated by the generalized RKM method versus the exact solutions Ѵ ( ӽ , τ ) for the aforementioned problems and for 10 lines of t and α = 0.96 is shown in Figure 1.

6 Discussion and conclusion

The main goal of this study is to modify a numerical method for solving FPDEs of different orders. The proposed modified method is used to solve the sixth-order FPDEs in addition to the fractional wave equation, which has been introduced. The implementations are examined for the proposed method. Also, the numerical results are compared with the analytical solutions. From this comparison, we conclude that the proposed approach is an accurate and efficient method. In contrast, the proposed method can be applied to solve PDEs up to the n th order.

Acknowledgments

The authors would like to thank the anonymous referees for very helpful comments that have led to an improvement of the article.

Funding information: The authors state no funding involved.
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
Conflict of interest: The authors state no conflict of interest.

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Received: 2023-06-09

Revised: 2023-08-06

Accepted: 2023-08-25

Published Online: 2025-11-03

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

Articles in the same Issue

https://doi.org/10.1515/nleng-2022-0321

Keywords for this article

RK; RKN; Runge–Kutta–Mohammed; MOL; ODEs; sixth-order; FPDEs; finite difference method; FDEs

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BY 4.0