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 hyperbolictelegraph 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.
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 subequation 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], qhomotopy 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 timefractional partial differential equations [41,42,43].
Many methods are introduced by the researchers to solve fractionalorder hyperbolictelegraph 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 integrodifferential equations were solved by Hamoud et al. [56]. Pueon and Viriyapong modified thirdorder 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 initialboundary value problems using the variational iteration method. We extended the idea to fractional initialboundary 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 higherorder 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
and its fractional derivative for
where
2.2 Definition
Using Reimann–Liouville [39] definition, we have
2.3 Definition
The Mittag–Leffler function [68]
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:
where
where
where
The ADM solution can be represented in the form of infinite series as
The nonlinear term
and we can calculate
The series has the following relation to represent the solution of Eq. (1),
4 Modification of ADM
To understand the main idea of the proposed technique, we will take the following onedimensional equation [64].
With the following initial and boundary conditions as
where
The new initial solution
Using ADM, the operator form of Eq. (4) is
where the differential operator
Hence
Applying
where
Using the ADM solution, the initial approximation becomes
and using the new ADM technique, the iteration formula becomes
It is obvious that initial solutions
The proposed technique work effectively for the twodimensional 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
with the initial and boundary conditions as follows:
The problem has the exact solution at
Applying the new technique based on ADM to Eq. (6), we obtain the following result:
where
Applying
where
Operating Eq. (6) by
and using the ADM solution, the initial approximation becomes
Using the new technique of initial approximation
By putting initial and boundary conditions in Eq. (7), for
From Eq. (9), we have
For
From Eq. (9), we have
For
From Eq. (9), we have
Thus, the ADM solution in the series form is
5.2 Example
Consider the case, of Eq. (4), when
with the initial and boundary conditions as follows:
The problem has the exact solution at
Applying the new technique based on ADM to Eq. (10), we obtain the following result:
where
Applying
where
Operating Eq. (10) by
Using ADM solution, the initial approximation becomes
Using the new technique of initial approximation
By putting initial and boundary conditions in Eq. (11), for
From Eq. (13), we have
For
From Eq. (13), we have
For
From Eq. (13), we have
Thus, the ADM solution in the series form is
5.3 Example
Consider the case of Eq. (4), when
with the initial and boundary conditions as follows:
The problem has the exact solution at
Applying the new technique based on ADM to Eq. (14), we obtain the following result:
where
Applying
where
Operating Eq. (14) by
Using ADM solution, the initial approximation becomes
and using the new technique of initial approximation
By putting initial and boundary conditions in Eq. (15), for
From Eq. (17), we have
For
From Eq. (17), we have
For
From Eq. (17), we have
Thus, the ADM solution in the series form is
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
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 twostep 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


Absolute error at

Absolute error at

Absolute error at

Absolute error at


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0.03  0.2 




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0.6 





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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 (TaCSCoE), 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.
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