A Reliable Treatment of Iterative Laplace Transform Method for Fractional Telegraph Equations (original) (raw)
Related papers
2014
Abstract. In this paper a reliable algorithm for the iterative Laplace transform method (ILTM) is presented. ILTM is a combination of Laplace transform method and Iterative method to solve space- and time- fractional telegraph equations. The fractional derivatives are considered in Caputo sense. Closed form analytical expressions are derived in terms of the Mittag-Leffler functions. An illustrative numerical case study is presented for the proposed method to show the preciseness and effectiveness of the method.
An Efficient Analytical Technique, for The Solution of Fractional-Order Telegraph Equations
Mathematics, 2019
In the present article, fractional-order telegraph equations are solved by using the Laplace-Adomian decomposition method. The Caputo operator is used to define the fractional derivative. Series form solutions are obtained for fractional-order telegraph equations by using the proposed method. Some numerical examples are presented to understand the procedure of the Laplace-Adomian decomposition method. As the Laplace-Adomian decomposition procedure has shown the least volume of calculations and high rate of convergence compared to other analytical techniques, the Laplace-Adomian decomposition method is considered to be one of the best analytical techniques for solving fractional-order, non-linear partial differential equations—particularly the fractional-order telegraph equation.
An efficient new perturbative Laplace method for space-time fractional telegraph equations
Advances in Difference Equations, 2012
In this paper, we propose a new technique for solving space-time fractional telegraph equations. This method is based on perturbation theory and the Laplace transformation. Fractional Taylor series and fractional initial conditions have been introduced. However, all the previous works avoid the term of fractional initial conditions in the space-time telegraph equations. The results of introducing fractional order initial conditions and the Laplace transform for the studied cases show the high accuracy, simplicity and efficiency of the approach.
An Approximate-Analytical Solution to Analyze Fractional View of Telegraph Equations
IEEE Access, 2020
In the present research article, a modified analytical method is applied to solve time-fractional telegraph equations. The Caputo-operator is used to express the derivative of fractional-order. The present method is the combination of two well-known methods namely Mohan transformation method and Adomian decomposition method. The validity of the proposed technique is confirmed through illustrative examples. It is observed that the obtained solutions have strong contact with the exact solution of the examples. Moreover, it is investigated that the present method has the desired degree of accuracy and provided the graphs closed form solutions of all targeted examples. The graphs have verified the convergence analysis of fractional-order solutions to integer-order solution. In conclusion, the suggested method is simple, straightforward and an effective technique to solve fractional-order partial differential equations. INDEX TERMS Mohand transformation, telegraph equations, Adomian decomposition method, Caputo operator.
2023
The Pseudo-Hyperbolic Telegraph partial differential equation (PHTPDE) based on the Caputo fractional derivative is investigated in this paper. The modified double Laplace transform method (MDLTM) is constructed for the proposed model. The MDLTM was used to obtain the analytic solution for the pseudo-hyperbolic telegraph equation of fractional order defined by the Caputo derivative. The proposed method is a highly effective analytical method for the fractionalorder pseudo-hyperbolic telegraph equation. A test problem was presented as an example. Based on the results, it is clear that this method is more convenient and produces an analytic solution in fewer steps than other methods that require more steps to have an identical analytical solution. This paper claims to provide an analytic solution to the fractional order pseudohyperbolic telegraph equation order using the MDLTM. An analytical solution leads to an exact, closed-form solution that can be expressed in mathematical functions or known operations. Obtaining analytic solutions to PDEs is often challenging, especially for fractional order equations, making this achievement noteworthy.
A New Computational Approach for Solving Fractional Order Telegraph Equations
Journal of Scientific Research, 2021
In this work, a modified decomposition method namely Sumudu-Adomian Decomposition Method (SADM) is implemented to find the exact and approximate solutions of fractional order telegraph equations. The derivatives of fractional-order are expressed in terms of caputo operator. Some numerical examples are illustrated to examine the efficiency of the proposed technique. Solutions of fractional order telegraph equations are obtained in the form of a series solution. It is observed that the solutions of fractional order telegraph equations converge towards the solution of an integer-order problem, which confirmed the reliability of the suggested method.
2014
In this article, an analytical solution based on the series expansion method is proposed to solve the time-fractional telegraph equation (TFTE) in two and three dimensions using a recent and reliable semi-approximate method, namely the reduced differential transform method (RDTM) subjected to the appropriate initial condition. Using RDTM, it is possible to find exact solution or a closed approximate solution of a differential equation. The accuracy, efficiency, and convergence of the method are demonstrated through the four numerical examples.
Mathematics and Statistics, 2022
The double integral transform is a robust implementation that is important in handling scientific and engineering problems. Besides its simplicity of use and straightforward application to the issue, the ability to reduce the problems to an algebraic equation that can be easily solved is a substantial advantage of the tool. Among the several integral transforms, the double Sadik transform is acknowledged to be one of the most frequently used in solving differential and integral equations. This work deals with investigating a generalized double integral transform called the double Sadik transform. The proof of the double Sadik transforms for partial fractional derivatives in the Caputo sense is displayed, and the double Sadik transforms method is introduced. The method has been applied to solve the initial boundary value problems for linear space and timefractional telegraph equations. Moreover, the suggested strategy can be used on non-linear problems via an iterative method and a decomposition concept. Some known-solution questions are evaluated with relatively minimal computational cost. The results are represented by utilizing the Mittag-Leffler function and covering the solution of a classical telegraph equation. The obtained exact solutions not only show the accuracy and efficiency of the technique, but also reveal reliability when compared to those obtained using other methods.