A new modified technique to study the dynamics of fractional hyperbolic-telegraph equations

Hassan Khan; Hajira; Qasim Khan; Poom Kumam; Fairouz Tchier; Gurpreet Singh; Kanokwan Sitthithakerngkiet; Ferdous Mohammed Tawfiq

doi:10.1515/phys-2022-0072

Article Open Access

A new modified technique to study the dynamics of fractional hyperbolic-telegraph equations

Hassan Khan , Hajira , Qasim Khan , Poom Kumam , Fairouz Tchier , Gurpreet Singh , Kanokwan Sitthithakerngkiet and Ferdous Mohammed Tawfiq

Published/Copyright: August 11, 2022

Published by

Become an author with De Gruyter Brill

Submit Manuscript Author Information Explore this Subject

From the journal Open Physics Volume 20 Issue 1

Abstract

Usually, to find the analytical and numerical solution of the boundary value problems of fractional partial differential equations is not an easy task; however, the researchers devoted their sincere attempt to find the solutions of various equations by using either analytical or numerical procedures. In this article, a very accurate and prominent method is developed to find the analytical solution of hyperbolic-telegraph equations with initial and boundary conditions within the Caputo operator, which has very simple calculations. This method is called a new technique of Adomian decomposition method. The obtained results are described by plots to confirm the accuracy of the suggested technique. Plots are drawn for both fractional and integer order solutions to confirm the accuracy and validity of the proposed method. Solutions are obtained at different fractional orders to discuss the useful dynamics of the targeted problems. Moreover, the suggested technique has provided the highest accuracy with a small number of calculations. The suggested technique gives results in the form of a series of solutions with easily computable and convergent components. The method is simple and straightforward and therefore preferred for the solutions of other problems with both initial and boundary conditions.

Keywords: Adomian decomposition method; initial-boundary value problems; fractional hyperbolic-telegraph equations

1 Introduction

Fractional calculus (FC) is an important branch of mathematics that studies the derivatives and integrals of fractional orders. Its history started with a question asked by L’Hospital in 1695. Since then, FC has gained much attention from researchers working in different fields. FC has various applications in science and engineering, such as optics, biological models, field theory, variational calculus, optimal control, quantum mechanics, nonlinear biological systems, fluid dynamics, stochastic dynamical systems, astrophysics, image processing, turbulence, signal analysis, pollution control, social systems, biomedicine, financial systems, controlled thermonuclear fusion, landscape evolution, bioengineering, elasticity, plasma physics [1,2,3, 4,5,6, 7,8] and so on.

In recent decades, fractional partial differential equations (FPDEs) have attracted researchers because of their important applications and uses in applied sciences [9,10,11]. FPDEs are very effective in the modeling of physical and engineering events. Some significant applications of FPDEs are image deionization [12], fractional dynamics, control theory and signal processing, fluid flow, system identification [4,13], diffusive transport, rheology, electrical network, probability [14,15], climate, social sciences like food supplements, the mechanics of materials, plasma physics [16,17], electromagnetic, controlled thermonuclear fusion, astrophysics, stochastic dynamical system, image processing, scattering, turbulent flow, chaotic dynamics, diffusion processes, electrical, and rheological materials [18].

Finding the solution of FPDEs is a hard and challenging task, with higher efforts required to perform for the harder mathematical solutions. Because the exact solutions of FPDEs are difficult to calculate, we need an easy and effective numerical and analytical algorithms. Many researchers have contributed their work to find the solutions of FPDEs and, therefore, different techniques have been developed. Some novel methods are the (G’/G) method [19], the EXP method [20], Bäcklund transformation method, Kudryashov method [21], fractional sub-equation method [22], the simplest equation method [23], Laplace transform [24], the Laplace Adomian decomposition method [25], the Elzaki transform decomposition method [26], the natural transform decomposition method [27], the Chebyshev wavelet method [28], the He’s variational iteration method [29], the homotopy perturbation method [30], q-homotopy analysis transformed method [31], the extended rational sinh–cosh method and modified Khater method [32], the reduced differential transform method [33], the meshless Kansa method [34], optimal axillary function method [35], the variable separation method [36], the tanh method [37], the sine–cosine method [38], the spectral collocation method [39], and the residual power series method [40]. Some of the authors implemented the most time discretization scheme for solving time-fractional partial differential equations [41,42,43].

Many methods are introduced by the researchers to solve fractional-order hyperbolic-telegraph equations (FHTEs). Mohanty et al. [44] used an unconditional iterative scheme, and Lakestani et al. [45] used an interpolating scaling function technique to solve the 1D hyperbolic telegraph equation. Jiwari et al. [46] and Tezer–Sezgin et al. [47] used the differential quadrature method, the homotopy analysis method [48], the fictitious time integration method [49], the Chebyshev tau method [50], the hybrid meshless method [51], and the Houbolt method [52].

In this research article, we will use a new method of ADM for the solution of FHTEs. The method was introduced in the 1980s by Adomian to solve some functional equations [53,54]. After that other researchers have shown their keen interest and several modifications to the existing methods were also introduced. For example, Hosseini et al. applied it to linear and nonlinear differential equations [55]. Fractional integro-differential equations were solved by Hamoud et al. [56]. Pue-on and Viriyapong modified third-order ordinary differential equations [57]. Then, the Klein–Gordon equations were solved by Saelao and Yokchoo [58]. Other ADM modifications can be seen in refs [59,60,61, 62,63].

This modification of ADM implemented in the current work was introduced by Elaf Jaafer Ali in ref. [64]. Furthermore, all of the aforementioned existing techniques attempt to solve fractional problems with either initial or boundary conditions, but in this work, we used both initial and boundary conditions to solve FHTEs using the current technique [64]. In ref. [65], the homotopy perturbation method is used to solve for the same problems. The same procedure is used in ref. [66] to solve initial-boundary value problems using the variational iteration method. We extended the idea to fractional initial-boundary value problems in ref. [67]. The proposed method has a higher rate of convergence toward the exact solution because of the new initial approximate solution for each term. The present method is recommended for other higher-order nonlinear problems in science and engineering.

2 Preliminaries

In this section, a few definitions related to our work are taken into consideration.

2.1 Definition

The integral operator of Reimann–Liouville having order ϑ is given by ref. [39]

( I ζ ϑ h ) ( ζ ) = 1 Γ ( ϑ ) ∫ 0 ζ h ( τ ) ( ζ − τ ) 1 − ϑ d τ , ϑ > 0 , h ( ζ ) , ϑ = 0 ,

and its fractional derivative for ϑ ≥ 0 is

( D ζ ϑ h ) ( ζ ) = d d ζ m ( I m − ϑ h ) ( ζ ) , ( ϑ > 0 , m < ϑ < m − 1 ) ,

where m is an integer.

2.2 Definition

Using Reimann–Liouville [39] definition, we have

I ζ ϑ ( ζ v ) = Γ ( v + 1 ) Γ ( v + 1 + ϑ ) ζ v + ϑ ( ϑ > 0 , m < ϑ ≤ m − 1 ) .

2.3 Definition

The Mittag–Leffler function [68] E η ( ρ ) for η > 0 is

E η ( ρ ) = ∑ n = 0 ∞ ρ n Γ ( n η + 1 ) , η > 0 , ρ ∈ C .

3 ADM [64]

The present technique was discovered by Adomian (1994) to solve linear, nonlinear differential, and integro differential equations. To understand the method, let us consider an equation of the following form:

(1) F ( ω ( ζ ) ) = g ( ζ ) ,

where F is a nonlinear differential operator and g is the known function. We will split the linear term in F ( ω ( ζ ) ) into the form L u + R u , where L is the invertible operator, chosen as the highest order derivative, R represents the linear operator, then Eq. (1) has the representation as follows:

(2) L ω + R ω + N ω = g ,

where N ω is the nonlinear term of F ( ω ( ζ ) ) . Apply the inverse operator, L − 1 , to both sides of Eq. (2)

ω = ϕ + L − 1 ( g ) − L − 1 ( R ω ) − L − 1 ( N ω ) ,

where ϕ is the constant of integration and L ϕ = 0 .

The ADM solution can be represented in the form of infinite series as

ω = ∑ n = 0 ∞ ζ n .

The nonlinear term N u is denoted by A n , and is defined as follows:

(3) N ω = ∑ n = 0 ∞ A n ,

and we can calculate A n using the following formula:

A n = 1 n ! d n d ψ n N ∑ n = 0 ∞ ( λ n ω n ) λ = 0 , n = 0 , 1 , …

The series has the following relation to represent the solution of Eq. (1),

ω 0 = μ + L − 1 ( g ) , n = 0 . ω n + 1 = L − 1 ( R u n ) − L − 1 ( A n ) , n ≥ 0 .

4 Modification of ADM

To understand the main idea of the proposed technique, we will take the following one-dimensional equation [64].

(4) ∂ ϑ ω ∂ τ ϑ + γ ∂ ω ∂ τ + η ω = ∂ 2 ω ∂ ζ 2 + f ( ζ , τ ) , 1 < ϑ ≤ 2 .

With the following initial and boundary conditions as

ω ( ζ , τ ) = f 0 ( ζ ) , ∂ ω ( ζ , 0 ) ∂ τ = f 1 ( ζ ) , 0 ≤ ζ ≤ 1 , ω ( 0 , τ ) = g 0 ( τ ) , ω ( 1 , τ ) = g 1 ( τ ) , τ > 0 ,

where f ( ζ , τ ) represents the source term.

The new initial solution ( ω n ∗ ) is calculated using the proposed technique and the following iteration for Eq. (4):

ω n ∗ = ω n ( ζ , τ ) + ( 1 − ζ ) [ g 0 ( τ ) − ω n ( 0 , τ ) ] + ζ [ g 1 ( τ ) − ω n ( 1 , τ ) ] .

Using ADM, the operator form of Eq. (4) is

(5) L u = ∂ 2 ω ( ζ , τ ) ∂ ζ 2 + f ( ζ , τ ) − γ ∂ ω ( ζ , τ ) ∂ τ − η ω ( ζ , τ ) , 1 < ϑ ≤ 2 ,

where the differential operator L is defined as

L = ∂ ϑ ∂ τ ϑ .

Hence L − 1 is defined as

L − 1 ( ⋅ ) = I ϑ ( ⋅ ) d τ .

Applying L − 1 to Eq. (5), we have

ω ( ζ , τ ) = ω ( ζ , 0 ) + L − 1 ∂ 2 ω ( ζ , τ ) ∂ ζ 2 + f ( ζ , τ ) − γ ∂ ω ( ζ , τ ) ∂ τ − η ω ( ζ , τ ) , 1 < ϑ ≤ 2 ,

where n = 0 , 1 , … .

Using the ADM solution, the initial approximation becomes

ω 0 ( ζ , τ ) = ω ( ζ , 0 ) + τ ( ω τ ( ζ , τ ) ) + L − 1 ( f ( ζ , τ ) ) ,

and using the new ADM technique, the iteration formula becomes

ω n + 1 ( ζ , τ ) = L − 1 ∂ 2 ω ( ζ , τ ) ∂ ζ 2 − γ ∂ ω ( ζ , τ ) ∂ τ − η ω ( ζ , τ ) , 1 < ϑ ≤ 2 .

It is obvious that initial solutions ω n ∗ of Eq. (4) satisfy both the initial and boundary conditions, as given below:

at τ = 0 , ω n ∗ ( ζ , 0 ) = ω n ( ζ , 0 ) , ζ = 0 , ω n ∗ ( 0 , τ ) = g 0 ( τ ) , ζ = 1 , ω n ∗ ( 1 , τ ) = g 1 ( τ ) .

The proposed technique work effectively for the two-dimensional problems.

5 Numerical results

In this section, we will present the solution of some illustrative examples by using the new technique based on ADM.

5.1 Example

Consider the case of Eq. (4), when γ = 4 , η = 2 [69]

(6) ∂ ϑ ω ( ζ , τ ) ∂ τ ϑ + 4 ∂ ω ( ζ , τ ) ∂ τ + 2 ω ( ζ , τ ) = ∂ 2 ω ( ζ , τ ) ∂ τ 2 , 0 < ζ < π , 1 < ϑ ≤ 2 ,

with the initial and boundary conditions as follows:

ω ( ζ , 0 ) = sin ( ζ ) , ω τ ( ζ , 0 ) = − sin ( ζ ) , ω ( 0 , τ ) = 0 , ω ( π , τ ) = 0 .

The problem has the exact solution at ϑ = 1 as follows:

ω ( ζ , τ ) = e − τ sin ( ζ ) .

Applying the new technique based on ADM to Eq. (6), we obtain the following result:

(7) ω n ∗ ( ζ , τ ) = ω n ( ζ , τ ) + ( 1 − ζ ) [ 0 − ω n ( 0 , τ ) ] + ζ [ 0 − ω n ( π , τ ) ] ,

where n = 0 , 1 , …

Applying L to Eq. (6), we have

(8) L ω = ∂ 2 ω ( ζ , τ ) ∂ τ 2 − 4 ∂ ω ( ζ , τ ) ∂ τ − 2 ω ( ζ , τ ) ,

where L = ∂ ϑ ∂ τ ϑ and L − 1 is defined as

L − 1 ( ⋅ ) = I ϑ ( ⋅ ) d t .

Operating Eq. (6) by L − 1 , we have

ω ( ζ , τ ) = ω ( ζ , 0 ) + L − 1 ∂ 2 ω ( ζ , τ ) ∂ τ 2 − 4 ∂ ω ( ζ , τ ) ∂ τ − 2 ω ( ζ , τ ) ,

and using the ADM solution, the initial approximation becomes

ω 0 ( ζ , τ ) = ω ( ζ , 0 ) + τ ( ω τ ( ζ , 0 ) ) , ω 0 ( ζ , τ ) = ( 1 − τ ) sin ( ζ ) .

Using the new technique of initial approximation ω n ∗ , the iteration formula becomes

(9) ω n + 1 ( ζ , τ ) = L − 1 ∂ 2 ω n ∗ ( ζ , τ ) ∂ τ 2 − 4 ∂ ω n ∗ ( ζ , τ ) ∂ τ − 2 ω n ∗ ( ζ , τ ) .

By putting initial and boundary conditions in Eq. (7), for n = 0 ,

ω 0 ∗ ( ζ , τ ) = ω 0 ( ζ , τ ) + ( 1 − ζ ) ( 0 − ω 0 ( 0 , τ ) ) + ζ ( 0 − ω 0 ( π , τ ) ) , ω 0 ∗ ( ζ , τ ) = ( 1 − τ ) sin ( ζ ) .

From Eq. (9), we have

ω 1 ( ζ , τ ) = L − 1 ∂ 2 ω 0 ∗ ( ζ , τ ) ∂ ζ 2 − 4 ∂ ω 0 ∗ ( ζ , τ ) ∂ τ − 2 ω 0 ∗ ( ζ , τ ) , = L − 1 ( − sin ( ζ ) ( 1 − τ ) − 4 ( − 1 ) sin ( ζ ) − 2 ( 1 − τ ) sin ( ζ ) ) , = L − 1 ( sin ( ζ ) ( − 1 + τ − 2 + 2 τ + 4 ) ) , = L − 1 ( sin ( ζ ) ( 1 + 3 τ ) ) , ω 1 ( ζ , τ ) = sin ( ζ ) τ ϑ Γ ( ϑ + 1 ) + 3 τ ϑ + 1 Γ ( ϑ + 2 ) .

For n = 1 , Eq. (7) becomes

ω 1 ∗ ( ζ , τ ) = ω 1 ( ζ , τ ) + ( 1 − ζ ) [ 0 − ω 1 ( 0 , τ ) ] + ζ [ 0 − ω 1 ( π , τ ) ] , ω 1 ∗ ( ζ , τ ) = sin ( ζ ) τ ϑ Γ ( ϑ + 1 ) + 3 τ ϑ + 1 Γ ( ϑ + 2 ) .

From Eq. (9), we have

ω 2 ( ζ , τ ) = L − 1 ∂ 2 ω 1 ∗ ( ζ , τ ) ∂ ζ 2 − 4 ∂ ω 1 ∗ ∂ τ − 2 ω 1 ∗ , ω 2 ( ζ , τ ) = − 3 sin ( ζ ) τ 2 ϑ Γ ( 2 ϑ + 1 ) + 3 τ 2 ϑ + 1 Γ ( 2 ϑ + 2 ) − 4 sin ( ζ ) τ 2 ϑ − 1 Γ ( 2 ϑ ) + 3 τ 2 ϑ Γ ( 2 ϑ + 1 ) .

For n = 2 , Eq. (7), becomes

ω 2 ∗ ( ζ , τ ) = ω 2 ( ζ , τ ) + ( 1 − ζ ) [ 0 − ω 2 ( 0 , τ ) ] + ζ [ 0 − ω 2 ( π , τ ) ] , ω 2 ∗ ( ζ , τ ) = − 3 sin ( ζ ) τ 2 ϑ Γ ( 2 ϑ + 1 ) + 3 τ 2 ϑ + 1 Γ ( 2 ϑ + 2 ) − 4 sin ( ζ ) τ 2 ϑ − 1 Γ ( 2 ϑ ) + 3 τ 2 ϑ Γ ( 2 ϑ + 1 ) .

From Eq. (9), we have

ω 3 ( ζ , τ ) = L − 1 ∂ 2 ω 2 ∗ ( ζ , τ ) ∂ ζ 2 − 4 ∂ 2 ω 2 ∗ ∂ τ − 2 ω 2 ∗ , ω 3 ( ζ , τ ) = 9 sin ( ζ ) τ 3 ϑ Γ ( 3 ϑ + 1 ) + 3 τ 3 ϑ + 1 Γ ( 3 ϑ + 2 ) + 24 sin ( ζ ) τ 3 ϑ − 1 Γ ( 3 ϑ ) + 3 τ 3 ϑ Γ ( 3 ϑ + 1 ) + 16 sin ( ζ ) τ 3 ϑ − 2 Γ ( 3 ϑ − 1 ) + 3 τ 3 ϑ − 1 Γ ( 3 ϑ ) . ⋮

Thus, the ADM solution in the series form is

ω ( ζ , τ ) = ω 0 ( ζ , τ ) + ω 1 ( ζ , τ ) + ω 2 ( ζ , τ ) + ω 3 ( ζ , τ ) + …

5.2 Example

Consider the case, of Eq. (4), when γ = 6 and η = 2 [69]

(10) ∂ ϑ ω ( ζ , τ ) ∂ τ ϑ + 6 ∂ ω ( ζ , τ ) ∂ τ + 2 ω ( ζ , τ ) = ∂ 2 ω ( ζ , τ ) ∂ ζ 2 − 2 e − τ sin ( ζ ) 0 < ζ < π , 1 < ϑ ≤ 2 ,

with the initial and boundary conditions as follows:

ω ( ζ , 0 ) = sin ( ζ ) , ω τ ( ζ , 0 ) = − sin ( ζ ) , ω ( 0 , τ ) = 0 , ω ( π , τ ) = 0 .

The problem has the exact solution at ϑ = 1 as follows:

ω ( ζ , τ ) = e − τ sin ( ζ ) .

Applying the new technique based on ADM to Eq. (10), we obtain the following result:

(11) ω n ∗ ( ζ , τ ) = ω n ( ζ , τ ) + ( 1 − ζ ) [ 0 − ω n ( 0 , τ ) ] + ζ [ 0 − ω n ( π , τ ) ] ,

where n = 0 , 1 , …

Applying L to Eq. (10), we have

(12) L ω = ∂ 2 ω ( ζ , τ ) ∂ τ 2 − 4 ∂ ω ( ζ , τ ) ∂ τ − 2 ω ( ζ , τ ) − 2 e − τ sin ( ζ ) ,

where L = ∂ ϑ ∂ τ ϑ and L − 1 is defined as

L − 1 ( ⋅ ) = I ϑ ( ⋅ ) d t .

Operating Eq. (10) by L − 1 , we have

ω ( ζ , τ ) = ω ( ζ , 0 ) + L − 1 ∂ 2 ω ( ζ , τ ) ∂ τ 2 − 4 ∂ ω ( ζ , τ ) ∂ τ − 2 ω ( ζ , τ ) .

Using ADM solution, the initial approximation becomes

ω 0 ( ζ , τ ) = ω ( ζ , 0 ) + τ ( ω τ ( ζ , 0 ) ) + L − 1 ( 2 e − τ sin ( ζ ) ) , ω 0 ( ζ , τ ) = 1 − τ − 2 τ ϑ Γ ( ϑ + 1 ) + 2 τ ϑ + 1 Γ ( ϑ + 2 ) − 4 τ ϑ + 2 Γ ( ϑ + 3 ) sin ( ζ ) .

Using the new technique of initial approximation ω n ∗ , the iteration formula becomes

(13) ω n + 1 ( ζ , τ ) = L − 1 ∂ 2 ω n ∗ ( ζ , τ ) ∂ τ 2 − 6 ∂ ω n ∗ ( ζ , τ ) ∂ τ − 2 ω n ∗ ( ζ , τ ) .

By putting initial and boundary conditions in Eq. (11), for n = 0 ,

ω 0 ∗ ( ζ , τ ) = ω 0 ( ζ , τ ) + ( 1 − ζ ) ( 0 − ω 0 ( 0 , τ ) ) + ζ ( 0 − ω 0 ( π , τ ) ) , ω 0 ∗ ( ζ , τ ) = 1 − τ − 2 τ ϑ Γ ( ϑ + 1 ) + 2 τ ϑ + 1 Γ ( ϑ + 2 ) − 4 τ ϑ + 2 Γ ( ϑ + 3 ) sin ( ζ ) .

From Eq. (13), we have

ω 1 ( ζ , τ ) = L − 1 ∂ 2 ω 0 ∗ ( ζ , τ ) ∂ ζ 2 − 6 ∂ ω 0 ∗ ( ζ , τ ) ∂ τ − 2 ω 0 ∗ ( ζ , τ ) . ω 1 ( ζ , τ ) = − 3 sin ( ζ ) τ ϑ Γ ( ϑ + 1 ) − τ ϑ + 1 Γ ( ϑ + 2 ) − 2 τ 2 ϑ Γ ( 2 ϑ + 1 ) + 2 τ 2 ϑ + 1 Γ ( 2 ϑ + 2 ) − 4 τ 2 ϑ + 2 Γ ( 2 ϑ + 3 ) − 6 sin ( ζ ) − τ ϑ Γ ( ϑ + 1 ) − 2 τ 2 ϑ − 1 Γ ( 2 ϑ ) + 2 τ 2 ϑ Γ ( 2 ϑ + 1 ) − 4 τ 2 ϑ + 1 Γ ( 2 ϑ + 2 ) .

For n = 1 , Eq. (11) becomes

ω 1 ∗ ( ζ , τ ) = ω 1 ( ζ , τ ) + ( 1 − ζ ) [ 0 − ω 1 ( 0 , τ ) ] + ζ [ 0 − ω 1 ( π , τ ) ] , ω 1 ∗ ( ζ , τ ) = − 3 sin ( ζ ) τ ϑ Γ ( ϑ + 1 ) − τ ϑ + 1 Γ ( ϑ + 2 ) − 2 τ 2 ϑ Γ ( 2 ϑ + 1 ) + 2 τ 2 ϑ + 1 Γ ( 2 ϑ + 2 ) − 4 τ 2 ϑ + 2 Γ ( 2 ϑ + 3 ) − 6 sin ( ζ ) − τ ϑ Γ ( ϑ + 1 ) − 2 τ 2 ϑ − 1 Γ ( 2 ϑ ) + 2 τ 2 ϑ Γ ( 2 ϑ + 1 ) − 4 τ 2 ϑ + 1 Γ ( 2 ϑ + 2 ) .

From Eq. (13), we have

ω 2 ( ζ , τ ) = L − 1 ∂ 2 ω 1 ∗ ( ζ , τ ) ∂ ζ 2 − 6 ∂ ω 1 ∗ ∂ τ − 2 ω 1 ∗ , ω 2 ( ζ , τ ) = 9 sin ( ζ ) τ 2 ϑ Γ ( 2 ϑ + 1 ) − τ 2 ϑ + 1 Γ ( 2 ϑ + 2 ) − 2 τ 3 ϑ Γ ( 3 ϑ + 1 ) + 2 τ 3 ϑ + 1 Γ ( 3 ϑ + 2 ) − 4 τ 3 ϑ + 2 Γ ( 3 ϑ + 3 ) + 18 sin ( ζ ) − t 2 ϑ Γ ( 2 ϑ + 1 ) − 2 t 3 ϑ − 1 Γ ( 3 ϑ ) + 2 τ 3 ϑ Γ ( 3 ϑ + 1 ) − 4 τ 3 ϑ + 1 Γ ( 3 ϑ + 2 ) + 18 sin ( ζ ) τ 2 ϑ − 1 Γ ( 2 ϑ ) − τ 2 ϑ Γ ( 2 ϑ + 1 ) − 2 τ 3 ϑ − 1 Γ ( 3 ϑ ) + 2 τ 3 ϑ Γ ( 3 ϑ + 1 ) − 4 τ 3 ϑ + 1 Γ ( 3 ϑ + 2 ) + 36 sin ( ζ ) − τ 2 ϑ − 1 Γ ( 2 ϑ ) − 2 τ 3 ϑ − 2 Γ ( 3 ϑ − 1 ) + 2 τ 3 ϑ − 1 Γ ( 3 ϑ ) − 4 τ 3 ϑ Γ ( 3 ϑ + 1 ) .

For n = 2 , Eq. (11) becomes

ω 2 ∗ ( ζ , τ ) = ω 2 ( ζ , τ ) + ( 1 − ζ ) [ 0 − ω 2 ( 0 , τ ) ] + ζ [ 0 − ω 2 ( π , τ ) ] , ω 2 ∗ ( ζ , τ ) = 9 sin ( ζ ) τ 2 ϑ Γ ( 2 ϑ + 1 ) − τ 2 ϑ + 1 Γ ( 2 ϑ + 2 ) − 2 τ 3 ϑ Γ ( 3 ϑ + 1 ) + 2 τ 3 ϑ + 1 Γ ( 3 ϑ + 2 ) − 4 τ 3 ϑ + 2 Γ ( 3 ϑ + 3 ) + 18 sin ( ζ ) − τ 2 ϑ Γ ( 2 ϑ + 1 ) − 2 τ 3 ϑ − 1 Γ ( 3 ϑ ) + 2 τ 3 ϑ Γ ( 3 ϑ + 1 ) − 4 τ 3 ϑ + 1 Γ ( 3 ϑ + 2 ) + 18 sin ( ζ ) τ 2 ϑ − 1 Γ ( 2 ϑ ) − τ 2 ϑ Γ ( 2 ϑ + 1 ) − 2 τ 3 ϑ − 1 Γ ( 3 ϑ ) + 2 τ 3 ϑ Γ ( 3 ϑ + 1 ) − 4 τ 3 ϑ + 1 Γ ( 3 ϑ + 2 ) + 36 sin ( ζ ) − τ 2 ϑ − 1 Γ ( 2 ϑ ) − 2 τ 3 ϑ − 2 Γ ( 3 ϑ − 1 ) + 2 τ 3 ϑ − 1 Γ ( 3 ϑ ) − 4 τ 3 ϑ Γ ( 3 ϑ + 1 ) .

From Eq. (13), we have

ω 3 ( ζ , τ ) = L − 1 ∂ 2 ω 2 ∗ ( ζ , τ ) ∂ ζ 2 − 6 ∂ 2 ω 2 ∗ ∂ τ − 2 ω 2 ∗ , ω 3 ( ζ , τ ) = − 27 sin ( ζ ) τ 3 ϑ Γ ( 3 ϑ + 1 ) − τ 3 ϑ + 1 Γ ( 3 ϑ + 2 ) − 2 τ 4 ϑ Γ ( 4 ϑ + 1 ) + 2 τ 4 ϑ + 1 Γ ( 4 ϑ + 2 ) − 4 τ 4 ϑ + 2 Γ ( 4 ϑ + 3 ) − 54 sin ( ζ ) − τ 3 ϑ Γ ( 3 ϑ + 1 ) − 2 τ 4 ϑ − 1 Γ ( 4 ϑ ) + 2 τ 4 ϑ Γ ( 4 ϑ + 1 ) − 4 τ 4 ϑ + 1 Γ ( 4 ϑ + 2 ) − 108 sin ( ζ ) τ 3 ϑ − 1 Γ ( 3 ϑ ) − τ 3 ϑ Γ ( 3 ϑ + 1 ) − 2 τ 4 ϑ − 1 Γ ( 4 ϑ ) + 2 τ 4 ϑ Γ ( 4 ϑ + 1 ) − 4 τ 4 ϑ + 1 Γ ( 4 ϑ + 2 ) − 108 sin ( ζ ) − τ 3 ϑ − 1 Γ ( 3 ϑ ) − 2 τ 4 ϑ − 2 Γ ( 4 ϑ − 1 ) + 2 τ 4 ϑ − 1 Γ ( 4 ϑ ) − 4 τ 4 ϑ Γ ( 4 ϑ + 1 ) − 108 sin ( ζ ) − τ 3 ϑ − 1 Γ ( 3 ϑ ) − 2 τ 4 ϑ Γ ( 4 ϑ + 1 ) + 2 τ 4 ϑ − 1 Γ ( 4 ϑ ) − 4 τ 4 ϑ Γ ( 4 ϑ + 1 ) − 108 sin ( ζ ) τ 3 ϑ − 2 Γ ( 3 ϑ − 1 ) − τ 3 ϑ − 1 Γ ( 3 ϑ ) − 2 τ 4 ϑ − 2 Γ ( 4 ϑ − 1 ) + 2 τ 4 ϑ − 1 Γ ( 4 ϑ ) − 4 τ 4 ϑ Γ ( 4 ϑ + 1 ) − 216 sin ( ζ ) − τ 3 ϑ − 2 Γ ( 3 ϑ − 1 ) − τ 4 ϑ − 3 Γ ( 4 ϑ − 2 ) + 2 τ 4 ϑ − 2 Γ ( 4 ϑ − 1 ) − 4 τ 4 ϑ − 1 Γ ( 4 ϑ ) . ⋮

Thus, the ADM solution in the series form is

ω ( ζ , τ ) = ω 0 ( ζ , τ ) + ω 1 ( ζ , τ ) + ω 2 ( ζ , τ ) + ω 3 ( ζ , τ ) + … ω ( ζ , τ ) = e − τ sin ( ζ ) .

5.3 Example

Consider the case of Eq. (4), when γ = 1 and η = 1 [69]

(14) ∂ ϑ ω ( ζ , τ ) ∂ τ ϑ + ∂ ω ( ζ , τ ) ∂ τ + ω ( ζ , τ ) = ∂ 2 ω ( ζ , τ ) ∂ ζ 2 + ζ 2 + τ − 1 0 < ζ < π , 1 < ϑ ≤ 2 ,

with the initial and boundary conditions as follows:

ω ( ζ , 0 ) = ζ 2 , ω τ ( ζ , 0 ) = 1 , ω ( 0 , τ ) = τ , ω ( 1 , τ ) = 1 + τ .

The problem has the exact solution at ϑ = 1 as follows:

ω ( ζ , τ ) = ζ 2 + τ .

Applying the new technique based on ADM to Eq. (14), we obtain the following result:

(15) ω n ∗ ( ζ , τ ) = ω n ( ζ , τ ) + ( 1 − ζ ) [ τ − ω n ( 0 , τ ) ] + ζ [ ( 1 + τ ) − ω n ( 1 , τ ) ] ,

where n = 0 , 1 , …

Applying L to Eq. (14), we have

(16) L ω = ∂ 2 ω ( ζ , τ ) ∂ τ 2 − ∂ ω ( ζ , τ ) ∂ τ − ω ( ζ , τ ) + ζ 2 + τ − 1 ,

where L = ∂ ϑ ∂ τ ϑ and L − 1 is defined as

L − 1 ( ⋅ ) = I ϑ ( ⋅ ) d t .

Operating Eq. (14) by L − 1 , we have

ω ( ζ , τ ) = ω ( ζ , 0 ) + L − 1 ∂ 2 ω ( ζ , τ ) ∂ τ 2 − ∂ ω ( ζ , τ ) ∂ τ − ω ( ζ , τ ) + ζ 2 + τ − 1 ) .

Using ADM solution, the initial approximation becomes

ω 0 ( ζ , τ ) = ω ( ζ , 0 ) + τ ( ω τ ( ζ , 0 ) ) + L − 1 ( ζ 2 + τ − 1 ) , ω 0 ( ζ , τ ) = ζ 2 + τ + ( ζ 2 − 1 ) τ ϑ Γ ( ϑ + 1 ) + τ ϑ + 1 Γ ( ϑ + 2 ) ,

and using the new technique of initial approximation ω n ∗ , the iteration formula becomes

(17) ω n + 1 ( ζ , τ ) = L − 1 ∂ 2 ω n ∗ ( ζ , τ ) ∂ ζ 2 − ∂ ω n ∗ ( ζ , τ ) ∂ τ − 2 ω n ∗ ( ζ , τ ) .

By putting initial and boundary conditions in Eq. (15), for n = 0

ω 0 ∗ ( ζ , τ ) = ω 0 ( ζ , τ ) + ( 1 − ζ ) ( τ − ω 0 ( 0 , τ ) ) + ζ ( ( 1 + τ ) − ω 0 ( π , τ ) ) , ω 0 ∗ ( ζ , τ ) = ζ 2 + τ + ( ζ 2 − ζ ) τ ϑ Γ ( ϑ + 1 ) .

From Eq. (17), we have

ω 1 ( ζ , τ ) = L − 1 ∂ 2 ω 0 ∗ ( ζ , τ ) ∂ ζ 2 − ∂ ω 0 ∗ ( ζ , τ ) ∂ τ − ω 0 ∗ ( ζ , τ ) , ω 1 ( ζ , τ ) = ( 1 − ζ 2 ) τ ϑ Γ ( ϑ + 1 ) + ( 2 − ζ 2 + ζ ) τ 2 ϑ Γ ( 2 ϑ + 1 ) − ( ζ 2 − ζ ) τ 2 ϑ − 1 Γ ( 2 ϑ ) − τ ϑ + 1 Γ ( ϑ + 2 ) .

For n = 1 , Eq. (15) becomes

ω 1 ∗ ( ζ , τ ) = ω 1 ( ζ , τ ) + ( 1 − ζ ) [ τ − ω 1 ( 0 , τ ) ] + ζ [ ( 1 + τ ) − ω 1 ( 1 , τ ) ] , ω 1 ∗ ( ζ , τ ) = ( ζ − ζ 2 ) τ ϑ Γ ( ϑ + 1 ) + ( ζ − ζ 2 ) τ 2 ϑ Γ ( 2 ϑ + 1 ) + ( ζ − ζ 2 ) τ 2 ϑ − 1 Γ ( 2 ϑ ) + τ + ζ .

From Eq. (17), we have

ω 2 ( ζ , τ ) = L − 1 ∂ 2 ω 1 ∗ ( ζ , τ ) ∂ ζ 2 − ∂ ω 1 ∗ ∂ τ − ω 1 ∗ , ω 2 ( ζ , τ ) = ( − 2 − ζ + ζ 2 ) τ 2 ϑ Γ ( 2 ϑ + 1 ) + ( − 2 − ζ + ζ 2 ) τ 3 ϑ Γ ( 3 ϑ + 1 ) + ( − 2 − 2 ζ + 2 ζ 2 ) τ 3 ϑ − 1 Γ ( 3 ϑ ) + ( ζ − ζ 2 ) τ 3 ϑ − 2 Γ ( 3 ϑ − 1 ) − ( ζ − ζ 2 ) τ 2 ϑ − 1 Γ ( 2 ϑ ) + ( − 1 − ζ ) τ ϑ Γ ( ϑ + 1 ) − τ ϑ + 1 Γ ( ϑ + 2 ) .

For n = 2 , Eq. (15) becomes

ω 2 ∗ ( ζ , τ ) = ω 2 ( ζ , τ ) + ( 1 − ζ ) [ τ − ω 2 ( 0 , τ ) ] + ζ [ ( 1 + τ ) − ω 2 ( 1 , τ ) ] , ω 2 ∗ ( ζ , τ ) = ( ζ 2 − ζ ) τ 2 ϑ Γ ( 2 ϑ + 1 ) + ( ζ 2 − ζ ) τ 3 ϑ Γ ( 3 ϑ + 1 ) + ( 2 ζ 2 − 2 ζ ) τ 3 ϑ − 1 Γ ( 3 ϑ ) − ( ζ − ζ 2 ) τ 3 ϑ − 2 Γ ( 3 ϑ − 1 ) − ( ζ − ζ 2 ) τ 2 ϑ − 1 Γ ( 2 ϑ ) + ζ + τ .

From Eq. (17), we have

ω 3 ( ζ , τ ) = L − 1 ∂ 2 ω 2 ∗ ( ζ , τ ) ∂ ζ 2 − ∂ ω 2 ∗ ∂ τ − ω 2 ∗ , ω 3 ( ζ , τ ) = ( 2 + ζ − ζ 2 ) τ 3 ϑ Γ ( 3 ϑ + 1 ) + ( 2 + ζ − ζ 2 ) τ 4 ϑ Γ ( 4 ϑ + 1 ) + ( 4 + 3 ζ − 3 ζ 2 ) τ 4 ϑ − 1 Γ ( 4 ϑ ) + ( 2 + 2 ζ − 2 ζ 2 ) τ 4 ϑ − 2 Γ ( 4 ϑ − 1 ) + ( 2 + 2 ζ − 2 ζ 2 ) τ 3 ϑ − 1 Γ ( 3 ϑ ) + ( ζ − ζ 2 ) τ 3 ϑ − 2 Γ ( 3 ϑ − 1 ) − ( ζ − ζ 2 ) τ 4 ϑ − 3 Γ ( 4 ϑ − 2 ) − ( 1 + ζ ) τ ϑ ϑ + 1 − τ ϑ + 1 Γ ( ϑ + 2 ) + …

Thus, the ADM solution in the series form is

ω ( ζ , τ ) = ω 0 ( ζ , τ ) + ω 1 ( ζ , τ ) + ω 2 ( ζ , τ ) + ω 3 ( ζ , τ ) + … ω ( ζ , τ ) = ζ 2 + τ .

6 Results and discussion

The solution plots show the accuracy of the method. Figure 1 shows the 3D plots of the exact and approximate solutions of Example 5.1 at ϑ = 2 . It is clear from both the graphs that there is a close contact between the exact and calculated solutions of the problem. Figure 2 shows the comparison between the exact and approximate solutions at an integer order of ϑ = 2 . In Figure 3, the 3D and 2D solution plots are drawn for different fractional orders of ϑ for Example 5.1, which shows that as the fractional orders varies, the accuracy of the method from the approximate to the exact solution increases. Figure 4 shows the plot for absolute error of Example 5.1. In Figure 5, the approximate solution at different fractional orders of ϑ is shown by 3D and 2D plots. Figure 6 shows a comparison between the exact and approximate solutions of Example 5.2. In Figure 7, the 2D plot for the absolute error of Example 5.2 is drawn. Figure 8 shows the 2D and 3D plots for the exact and approximate solutions of Example 5.3. Figure 9 shows a comparison between the exact and approximate solutions of Example 5.3. In Figure 10, the 2D and 3D plots for different fractional order of ϑ . Figure 11 is drawn for the absolute error of Example 5.3. All of the graphs provide very useful information about the dynamical behavior of the targeted problems. The solution calculated for each fractional order has shown that the fractional solutions are closely converging to the exact solution of each problem. An error table is drawn for Example 5.3, which shows the high accuracy of the proposed technique. In addition, the proposed method is an accurate way of finding solutions for FPDEs having initial and boundary conditions. The method is applied to linear problems, which gives better results. Therefore, the suggested method can be extended further to linear and nonlinear problems in future.

$Figure 1 Three-dimensional plots of exact and approximate solutions for ϑ = 2 {\vartheta }=2 of Example 5.1.$

Figure 1

Three-dimensional plots of exact and approximate solutions for ϑ = 2 of Example 5.1.

$Figure 2 Two-dimensional plots for comparison between exact and approximate solution for ϑ = 2 {\vartheta }=2 of Example 5.1.$

Figure 2

Two-dimensional plots for comparison between exact and approximate solution for ϑ = 2 of Example 5.1.

$Figure 3 Plots for different values of ϑ {\vartheta } for Example 5.1.$

Figure 3

Plots for different values of ϑ for Example 5.1.

$Figure 4 Plot of absolute error at ϑ = 2 {\vartheta }=2 for Example 5.1.$

Figure 4

Plot of absolute error at ϑ = 2 for Example 5.1.

$Figure 5 Three- and two-dimensional plots of approximate solution for different ϑ {\vartheta } of Example 5.2.$

Figure 5

Three- and two-dimensional plots of approximate solution for different ϑ of Example 5.2.

$Figure 6 Two-dimensional plots for comparison between the exact and approximate solutions ϑ = 2 {\vartheta }=2 of Example 5.2.$

Figure 6

Two-dimensional plots for comparison between the exact and approximate solutions ϑ = 2 of Example 5.2.

$Figure 7 Plot of absolute error at ϑ = 2 {\vartheta }=2 for Example 5.2.$

Figure 7

Plot of absolute error at ϑ = 2 for Example 5.2.

$Figure 8 Three-dimensional plots for comparison between the exact and approximate solutions at ϑ = 2 {\vartheta }=2 of Example 5.3.$

Figure 8

Three-dimensional plots for comparison between the exact and approximate solutions at ϑ = 2 of Example 5.3.

$Figure 9 Two-dimensional plots of comparison between the exact and approximate solutions at ϑ = 2 {\vartheta }=2 of Example 5.3.$

Figure 9

Two-dimensional plots of comparison between the exact and approximate solutions at ϑ = 2 of Example 5.3.

$Figure 10 Three- and two-dimensional plots of approximate solution at different ϑ {\vartheta } of Example 5.3.$

Figure 10

Three- and two-dimensional plots of approximate solution at different ϑ of Example 5.3.

$Figure 11 Plot of absolute error at ϑ = 2 {\vartheta }=2 for Example 5.3.$

Figure 11

Plot of absolute error at ϑ = 2 for Example 5.3.

7 Conclusion

In this article, the modified ADM is developed to solve FPDEs with initial and boundary conditions. For this purpose, the Caputo operator is used to define the fractional derivative. The present method has a two-step representation. In the first step, the solutions are approximated by using the ADM iteration formula. In the second step, these approximate solutions are further refined by using another iteration formula that utilizes the boundary conditions and increases the accuracy of the proposed technique. To verify the accuracy of the method, the solutions of few numerical examples are discussed. The solutions for the targeted problems are calculated for both fractional and integer orders of the derivatives. Figures and tables are constructed to show the accuracy and applicability of the present method. In Figures 2, 6, and 9 show the comparison of exact and approximate solutions at ϑ = 2 of Examples 5.1, 5.2, and 5.3, respectively. A very strong relation is observed between the exact and approximate solutions. Figures 3, 5, and 10 are constructed to show the 2D and 3D plots of problems 5.1, 5.2, and 5.3, respectively. The 3D solution plots for the exact and approximate solutions are shown in Figures 1, 4, and 8, respectively, for problems 5.1, 5.2, and 5.3, respectively. In Table 1, the absolute error associated with the present technique is given, which shows the higher accuracy of the suggested method. Because the current technique required fewer calculations, it can be extended for the solutions of other higher nonlinear fractional order problems.

Table 1

Absolute error for different times and fractional order of Example 5.3

τ	ζ	Absolute error at ϑ = 1.3	Absolute error at ϑ = 1.5	Absolute error at ϑ = 1.7	Absolute error at ϑ = 2
0.01	0.2	5.18455806 × 1 0 − 2	1.81143765 × 1 0 − 2	6.2045458 × 1 0 − 4	1.2033333 × 1 0 − 4
	0.4	6.0444999 × 1 0 − 2	2.1123280 × 1 0 − 2	7.235455 × 1 0 − 4	1.403333 × 1 0 − 4
	0.6	6.9056642 × 1 0 − 2	2.4132289 × 1 0 − 2	8.266362 × 1 0 − 4	1.603333 × 1 0 − 4
	0.8	7.7680510 × 1 0 − 3	2.7141404 × 1 0 − 3	9.297269 × 1 0 − 4	1.803333 × 1 0 − 4
	1	8.631659 × 1 0 − 3	3.015062 × 1 0 − 3	1.032817 × 1 0 − 3	2.00333 × 1 0 − 4
0.03	0.2	2.178994850 × 1 0 − 3	9.47609717 × 1 0 − 3	4.04099673 × 1 0 − 3	1.08900091 × 1 0 − 3
	0.4	2.53688321 × 1 0 − 2	1.10393772 × 1 0 − 2	4.7083030 × 1 0 − 3	1.2690010 × 1 0 − 3
	0.6	2.89602877 × 1 0 − 2	1.26028917 × 1 0 − 2	5.3756102 × 1 0 − 3	1.4490010 × 1 0 − 3
	0.8	3.25643152 × 1 0 − 2	1.41666408 × 1 0 − 2	6.0429183 × 1 0 − 3	1.6290010 × 1 0 − 3
	1	3.61809146 × 1 0 − 2	1.5730624 × 1 0 − 2	6.710228 × 1 0 − 3	1.809000 × 1 0 − 3
0.05	0.2	4.267824285 × 1 0 − 2	2.052902812 × 1 0 − 2	9.68948097 × 1 0 − 3	3.04167829 × 1 0 − 3
	0.4	4.96171264 × 1 0 − 2	2.38921758 × 1 0 − 2	1.12797422 × 1 0 − 2	3.5416791 × 1 0 − 3
	0.6	5.65920100 × 1 0 − 2	2.72562483 × 1 0 − 2	1.28700020 × 1 0 − 2	4.0416791 × 1 0 − 3
	0.8	6.36028935 × 1 0 − 2	3.06212458 × 1 0 − 2	1.44602605 × 1 0 − 2	4.5416783 × 1 0 − 3
	1	7.06497760 × 1 0 − 2	3.3987169 × 1 0 − 2	1.6050520 × 1 0 − 2	5.041677 × 1 0 − 3

Acknowledgments

This research was supported by Researchers Supporting Project number (RSP2022R440), King Saud University, Riyadh, Saudi Arabia.

Funding information: The authors acknowledge the financial support provided by the Center of Excellence in Theoretical and Computational Science (TaCS-CoE), KMUTT. This research project is supported by Thailand Science Research and Innovation (TSRI) Basic Research Fund: Fiscal year 2022 under project number FRB650048/0164.
Author contributions: Hassan Khan: supervision; Hajira: methodology; Qasim Khan: methodology, investigation; Fairouz Tchier: project administration; Poom Kumam: funding, draft writing; Gurpreet Singh: investigation; Kanokwan Sitthithakerngkiet: funding, draft writing. 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.

References

[1] El-Nabulsi RA, Torres DF. Necessary optimality conditions for fractional action-like integrals of variational calculus with Riemann-Liouville derivatives of order (α, ϑ). Math Meth Appl Sci. 2007;30(15):1931–9. 10.1002/mma.879Search in Google Scholar

[2] Frederico GS, Torres DF. Fractional conservation laws in optimal control theory. Nonlinear Dyn. 2008;53(3):215–22. 10.1007/s11071-007-9309-zSearch in Google Scholar

[3] Jumarie G. Fractional Hamilton-Jacobi equation for the optimal control of nonrandom fractional dynamics with fractional cost function. J Appl Math Comput. 2007;23(1):215–28. 10.1007/BF02831970Search in Google Scholar

[4] Kilbas AA, Srivastava HM, Trujillo JJ. Theory and applications of fractional differential equations. Vol. 204. Amsterdam, Netherlands: Elsevier; 2006. 10.1016/S0304-0208(06)80001-0Search in Google Scholar

[5] Herzallah M, Baleanu D., Fractional-order variational calculus with generalized boundary conditions. Adv Differ Equ. 2011;2011:1–9. 10.1155/2011/357580Search in Google Scholar

[6] Hilfer R. Ed. Applications of fractional calculus in physics. River Edge, NJ, USA: World Scientific; 2000. 10.1142/3779Search in Google Scholar

[7] Jacob JS, Priya JH, Karthika A. Applications of fractional calculus in science and engineering. J Critical Rev. 2020;7(13):4385–94. Search in Google Scholar

[8] Veeresha P. A numerical approach to the coupled atmospheric ocean model using a fractional operator. Math Modell Numer Simulat Appl. 2021;1(1):1–10. 10.53391/mmnsa.2021.01.001Search in Google Scholar

[9] Gao GH, Sun ZZ, Zhang YN. A finite difference scheme for fractional sub-diffusion equations on an unbounded domain using artificial boundary conditions. J Comput Phys. 2012;231(7):2865–79. 10.1016/j.jcp.2011.12.028Search in Google Scholar

[10] De Oliveira EC, Tenreiro Machado JA. A review of definitions for fractional derivatives and integral. Math Problems Eng. 2014;2014. Article ID: 238459. 10.1155/2014/238459Search in Google Scholar

[11] Yavuz M. European option pricing models described by fractional operators with classical and generalized Mittag-Leffler kernels. Numer Meth Partial Differ Equ. 2022;38(3):434–56. Search in Google Scholar

[12] Wielandts JY. Integration of dynamic and functional patien-specific 3D models in support of interventional electrophysiological procedures. Doctoral dissertation. Université de Bordeaux. 2016. Search in Google Scholar

[13] Podlubny I, Chechkin A, Skovranek T, Chen Y, Jara BMV. Matrix approach to discrete fractional calculus II: partial fractional differential equations. J Comput Phys. 2009;228(8):3137–53. 10.1016/j.jcp.2009.01.014Search in Google Scholar

[14] He JH. Nonlinear oscillation with fractional derivative and its applications. In: International Conference on Vibrating Engineering. Vol. 98. 1998. p. 288–91. Search in Google Scholar

[15] Gitterman M. Mean first passage time for anomalous diffusion. Phys Rev E. 2000;62(5):6065. 10.1103/PhysRevE.62.6065Search in Google Scholar PubMed

[16] Sun H, Chen W, Chen Y. Variable-order fractional differential operators in anomalous diffusion modeling. Phys A Statist Mechanics Appl. 2009;388(21):4586–92. 10.1016/j.physa.2009.07.024Search in Google Scholar

[17] Sun H, Chen W, Li C, Chen Y. Fractional differential models for anomalous diffusion. Phys A Statist Mechanics Appl. 2010;389(14):2719–24. 10.1016/j.physa.2010.02.030Search in Google Scholar

[18] Javidi M, Ahmad B. Numerical solution of fractional partial differential equations by numerical Laplace inversion technique. Adv Differ Equ. 2013;2013(1):1–18. 10.1186/1687-1847-2013-375Search in Google Scholar

[19] Shang N, Zheng B. Exact solutions for three fractional partial differential equations by the method. Int J Appl Math. 2013;43(3):1–6. 10.1186/1687-1847-2013-199Search in Google Scholar

[20] Zheng B. Exp-function method for solving fractional partial differential equations. Scientific World J. 2013;2013:448–53. Article ID 365. 10.1155/2013/465723Search in Google Scholar PubMed PubMed Central

[21] Rezazadeh H, Kumar D, Neirameh A, Eslami M, Mirzazadeh M. Applications of three methods for obtaining optical soliton solutions for the Lakshmanan-Porsezian-Daniel model with Kerr law nonlinearity. Pramana. 2020;94(1):1–11. 10.1007/s12043-019-1881-5Search in Google Scholar

[22] Tang B, He Y, Wei L, Zhang X. A generalized fractional sub-equation method for fractional differential equations with variable coefficients. Phys Lett A. 2012;376(38–39):2588–90. 10.1016/j.physleta.2012.07.018Search in Google Scholar

[23] Taghizadeh N, Mirzazadeh M, Rahimian M, Akbari M. Application of the simplest equation method to some time-fractional partial differential equations. Ain Shams Eng J. 2013;4(4):897–902. 10.1016/j.asej.2013.01.006Search in Google Scholar

[24] Sene N. Second-grade fluid with Newtonian heating under Caputo fractional derivative: analytical investigations via Laplace transforms. Math Modell Numer Simulat Appl. 2022;2(1):13–25. 10.53391/mmnsa.2022.01.002Search in Google Scholar

[25] Khan M, Hussain M. Application of Laplace decomposition method on semi-infinite domain. Numer Algorithms. 2011;56(2):211–8. 10.1007/s11075-010-9382-0Search in Google Scholar

[26] Khan H, Khan A, Kumam P, Baleanu D, Arif M. An approximate analytical solution of the Navier-Stokes equations within Caputo operator and Elzaki transform decomposition method. Adv Differ Equ. 2020;2020(1):1–23. 10.1186/s13662-020-03058-1Search in Google Scholar

[27] Shah R, Khan H, Mustafa S, Kumam P, Arif M. Analytical solutions of fractional-order diffusion equations by natural transform decomposition method. Entropy. 2019;21(6):557. 10.3390/e21060557Search in Google Scholar PubMed PubMed Central

[28] Adibi H, Assari P. Chebyshev wavelet method for numerical solution of Fredholm integral equations of the first kind. Math Problems Eng. 2010;2010:1–17. 10.1155/2010/138408Search in Google Scholar

[29] Liu Y, Gurram CS. The use of He’s variational iteration method for obtaining the free vibration of an Euler-Bernoulli beam. Math Comput Modell. 2009;50(11–12):1545–52. 10.1016/j.mcm.2009.09.005Search in Google Scholar

[30] Maitama S. Local fractional natural homotopy perturbation method for solving partial differential equations with local fractional derivative. Progress Fractional Different Appl. 2018;4(3):219–28. 10.18576/pfda/040306Search in Google Scholar

[31] Veeresha P, Yavuz M, Baishya C. A computational approach for shallow water forced Korteweg-De Vries equation on critical flow over a hole with three fractional operators. Int J Optimization Control Theories Appl (IJOCTA). 2021;11(3):52–67. 10.11121/ijocta.2021.1177Search in Google Scholar

[32] Rezazadeh H, Korkmaz A, Khater MM, Eslami M, Lu D, Attia RA. New exact traveling wave solutions of biological population model via the extended rational sinh–cosh method and the modified Khater method. Modern Phys Lett B. 2019;33(28):1950338. 10.1142/S021798491950338XSearch in Google Scholar

[33] Jafari H, Jassim HK, Moshokoa SP, Ariyan VM, Tchier F. Reduced differential transform method for partial differential equations within local fractional derivative operators. Adv Mech Eng. 2016;8(4):1687814016633013. 10.1177/1687814016633013Search in Google Scholar

[34] Haq S, Hussain M. The meshless Kansa method for time-fractional higher order partial differential equations with constant and variable coefficients. Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas. 2019;113(3):1935–54. 10.1007/s13398-018-0593-xSearch in Google Scholar

[35] Zada L, Nawaz R, Nisar KS, Tahir M, Yavuz M, Kaabar MK, et al. New approximate-analytical solutions to partial differential equations via auxiliary function method. Partial Differ Equ Appl Math. 2021;4:100045. 10.1016/j.padiff.2021.100045Search in Google Scholar

[36] Zhang S, Hong S. Variable separation method for a nonlinear time fractional partial differential equation with forcing term. J Comput Appl Math. 2018;339:297–305. 10.1016/j.cam.2017.09.045Search in Google Scholar

[37] Wazwaz A-M. A sine–cosine method for handling nonlinear wave equations. Math Comput Model. 2004;40(5):499–508. 10.1016/j.mcm.2003.12.010. Search in Google Scholar

[38] Anastassiou GA. Opial type inequalities involving fractional derivatives of two functions and applications. Comput Math Appl. 2004;48(10–11):1701–31. 10.1007/978-0-387-98128-4_6Search in Google Scholar

[39] Bhrawy AH, Doha EH, Ezz-Eldien SS, Abdelkawy MA. A numerical technique based on the shifted Legendre polynomials for solving the time-fractional coupled KdV equations. Calcolo. 2016;53(1):1–17. 10.1007/s10092-014-0132-xSearch in Google Scholar

[40] Tchier F, Inc M, Korpinar ZS, Baleanu D. Solutions of the time fractional reaction-diffusion equations with residual power series method. Adv Mech Eng. 2016;8(10):1687814016670867. 10.1177/1687814016670867Search in Google Scholar

[41] Xi Q, Fu Z, Rabczuk T, Yin D. A localized collocation scheme with fundamental solutions for long-time anomalous heat conduction analysis in functionally graded materials. Int J Heat Mass Transfer. 2021;180:121778. 10.1016/j.ijheatmasstransfer.2021.121778Search in Google Scholar

[42] Tang Z, Fu Z, Sun H, Liu X. An efficient localized collocation solver for anomalous diffusion on surfaces. Fract Calculus Appl Anal. 2021;24(3):865–94. 10.1515/fca-2021-0037Search in Google Scholar

[43] Fu ZJ, Reutskiy S, Sun HG, Ma J, Khan MA. A robust kernel-based solver for variable-order time fractional PDEs under 2D/3D irregular domains. Appl Math Lett. 2019;94:105–11. 10.1016/j.aml.2019.02.025Search in Google Scholar

[44] Mohanty RK, Jain M. An unconditionally stable alternating direction implicit scheme for the two space dimensional linear hyperbolic equation. Numer Methods Partial Differ Equ Int J. 2001;17(6):684–8. 10.1002/num.1034Search in Google Scholar

[45] Lakestani M, Saray BN. Numerical solution of telegraph equation using interpolating scaling functions. Comput Math Appl. 2010;60(7):1964–72. 10.1016/j.camwa.2010.07.030Search in Google Scholar

[46] Jiwari R, Pandit S, Mittal RC. A differential quadrature algorithm for the numerical solution of the second-order one dimensional hyperbolic telegraph equation. Int J Nonlinear Sci. 2012;13(3):259–66. Search in Google Scholar

[47] Pekmen B, Tezer-Sezgin M. Differential quadrature solution of hyperbolic telegraph equation. J Appl Math. 2012;2012. Article ID: 340752.10.1155/2012/924765Search in Google Scholar

[48] Pirkhedri A, Javadi HHS, Navidi HR. Numerical algorithm based on Haar-Sinc collocation method for solving the hyperbolic PDEs. Scientific World J. 2014;2014. Article ID: 340752.10.1155/2014/340752Search in Google Scholar PubMed PubMed Central

[49] Hashemi MS, Baleanu D. Numerical approximation of higher-order time-fractional telegraph equation by using a combination of a geometric approach and method of line. J Comput Phys. 2016;316:10–20. 10.1016/j.jcp.2016.04.009Search in Google Scholar

[50] Saadatmandi A, Dehghan M. Numerical solution of hyperbolic telegraph equation using the Chebyshev tau method. Numer Methods Partial Differ Equ Int J. 2010;26(1):239–52. 10.1002/num.20442Search in Google Scholar

[51] Zhou Y, Qu W, Gu Y, Gao H. A hybrid meshless method for the solution of the second order hyperbolic telegraph equation in two space dimensions. Eng Anal Boundary Elements. 2020;115:21–27. 10.1016/j.enganabound.2020.02.015Search in Google Scholar

[52] Aslefallah M, Rostamy D. Application of the singular boundary method to the two-dimensional telegraph equation on arbitrary domains. J Eng Math. 2019;118(1):1–14. 10.1007/s10665-019-10008-8Search in Google Scholar

[53] Adomian G. A review of the decomposition method and some recent results for nonlinear equations. Math Comput Modell. 1990;13(7):17–43. 10.1016/0895-7177(90)90125-7Search in Google Scholar

[54] Adomian G. Solving Frontier problems of physics: the decomposition method. Boston, MA: Kluwer; 1994. 10.1007/978-94-015-8289-6Search in Google Scholar

[55] Hosseini MM, Nasabzadeh H. Modified Adomian decomposition method for specific second order ordinary differential equations. Appl Math Comput. 2007;186(1):117–23. 10.1016/j.amc.2006.07.094Search in Google Scholar

[56] Hamoud AA, Ghadle K, Atshan S. The approximate solutions of fractional integro-differential equations by using modified Adomian decomposition method. Khayyam J Math. 2019;5(1):21–39. 10.15393/j3.art.2018.4350Search in Google Scholar

[57] Pue-On P, Viriyapong N. Modified Adomian decomposition method for solving particular third-order ordinary differential equations. Appl Math Sci. 2012;6(30):1463–9. Search in Google Scholar

[58] Saelao J, Yokchoo N. The solution of Klein-Gordon equation by using modified Adomian decomposition method. Math Comput Simulat. 2020;171:94–102. 10.1016/j.matcom.2019.10.010Search in Google Scholar

[59] Patel HS, Meher R. Analytical investigation of Jeffery-Hamel flow by modified Adomian decomposition method. Ain Shams Eng J. 2018;9(4):599–606. 10.1016/j.asej.2016.02.007Search in Google Scholar

[60] Alizadeh A, Effati S. Modified Adomian decomposition method for solving fractional optimal control problems. Trans Institute Measurement Control. 2018;40(6):2054–61. 10.1177/0142331217700243Search in Google Scholar

[61] Moradweysi P, Ansari R, Hosseini K, Sadeghi F. Application of modified Adomian decomposition method to pull-in instability of nano-switches using nonlocal Timoshenko beam theory. Appl Math Modell. 2018;54:594–604. 10.1016/j.apm.2017.10.011Search in Google Scholar

[62] Gahgah M, Sari MR, Kezzar M, Eid MR. Duan-Rach modified Adomian decomposition method (DRMA) for viscoelastic fluid flow between nonparallel plane walls. Europ Phys J Plus. 2020;135(2):1–17. 10.1140/epjp/s13360-020-00250-wSearch in Google Scholar

[63] Hasan YQ. Modified Adomian decomposition method for second order singular initial value problems. Adv Comput Math Appl. 2012;1(2):94–99. Search in Google Scholar

[64] Ali EJ. A new technique of initial boundary value problems using Adomian decomposition method. Int Math Forum. 2012;7(17):799–814. Search in Google Scholar

[65] Mohyud-Din ST, Noor MA Homotopy perturbation method for solving partial differential equations. Zeitschrift für Naturforschung A. 2009;64(3–4):157–70. 10.1515/zna-2009-3-402Search in Google Scholar

[66] Ali EJ, Jassim AM. Development treatment of initial boundary value problems for one dimensional heat-like and wave-like equations using homotopy perturbation method. J Basrah Res. 2013;39(1). 10.31642/JoKMC/2018/010708Search in Google Scholar

[67] Mustafa S, Khan H, Shah R, Masood S. A novel analytical approach for the solution of fractional-order diffusion-wave equations. Fractal Fractional. 2021;5(4):206. 10.3390/fractalfract5040206Search in Google Scholar

[68] Seybold H, Hilfer R. Numerical algorithm for calculating the generalized Mittag-Leffler function. SIAM J Numer Anal. 2009;47(1):69–88. 10.1137/070700280Search in Google Scholar

[69] Dehghan M, Shokri A. A numerical method for solving the hyperbolic telegraph equation. Numer Methods Partial Differ Equ Int J. 2008;24(4):1080–93. 10.1002/num.20306Search in Google Scholar

Received: 2022-01-31

Revised: 2022-05-24

Accepted: 2022-06-25

Published Online: 2022-08-11

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

Articles in the same Issue

https://doi.org/10.1515/phys-2022-0072

Keywords for this article

Adomian decomposition method; initial-boundary value problems; fractional hyperbolic-telegraph equations

Creative Commons

BY 4.0