scholarly journals A Decomposition Method for a Fractional-Order Multi-Dimensional Telegraph Equation via the Elzaki Transform

Symmetry ◽  
2020 ◽  
Vol 13 (1) ◽  
pp. 8
Author(s):  
Nehad Ali Shah ◽  
Ioannis Dassios ◽  
Jae Dong Chung
Keyword(s):  
Fractional Order ◽  
Decomposition Method ◽  
Approximate Analytical Solution ◽  
Telegraph Equation ◽  
Form Solution ◽  
Derivative Operator ◽  
Analytical Strategy ◽  
Fast Convergence Rate ◽  
Computational Work ◽  
Telegraph Equations

In this article, the Elzaki decomposition method is used to evaluate the solution of fractional-order telegraph equations. The approximate analytical solution is obtained within the Caputo derivative operator. The examples are provided as a solution to illustrate the feasibility of the proposed methodology. The result of the proposed method and the exact solution is shown and analyzed with figures help. The analytical strategy generates the series form solution, with less computational work and a fast convergence rate to the exact solutions. The obtained results have shown a useful and straightforward procedure to analyze the problems in related areas of science and technology.

scholarly journals New approximate analytical technique for the solution of time fractional fluid flow models

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Umar Farooq ◽  
Hassan Khan ◽  
Fairouz Tchier ◽  
Evren Hincal ◽  
Dumitru Baleanu ◽  
...  
Keyword(s):  
Fluid Flow ◽  
Analytical Solution ◽  
Approximate Analytical Solution ◽  
Form Solution ◽  
Navier Stokes ◽  
Analytical Strategy ◽  
Flow Models ◽  
Analytical Technique ◽  
Fast Convergence Rate ◽  
Computational Work

AbstractIn this note, we broaden the utilization of an efficient computational scheme called the approximate analytical method to obtain the solutions of fractional-order Navier–Stokes model. The approximate analytical solution is obtained within Liouville–Caputo operator. The analytical strategy generates the series form solution, with less computational work and fast convergence rate to the exact solutions. The obtained results have shown a simple and useful procedure to analyze complex problems in related areas of science and technology.

scholarly journals An Efficient Analytical Technique, for The Solution of Fractional-Order Telegraph Equations

Mathematics ◽  
2019 ◽  
Vol 7 (5) ◽  
pp. 426 ◽  
Author(s):  
Hassan Khan ◽  
Rasool Shah ◽  
Poom Kumam ◽  
Dumitru Baleanu ◽  
Muhammad Arif
Keyword(s):  
Fractional Order ◽  
Decomposition Method ◽  
Adomian Decomposition Method ◽  
Analytical Techniques ◽  
Telegraph Equation ◽  
High Rate ◽  
Analytical Technique ◽  
Decomposition Procedure ◽  
Adomian Decomposition ◽  
Telegraph Equations

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.

scholarly journals Analytical Solution of Fractional-Order Hyperbolic Telegraph Equation, Using Natural Transform Decomposition Method

Electronics ◽  
2019 ◽  
Vol 8 (9) ◽  
pp. 1015 ◽  
Author(s):  
Hassan Khan ◽  
Rasool Shah ◽  
Dumitru Baleanu ◽  
Poom Kumam ◽  
Muhammad Arif
Keyword(s):  
Applied Sciences ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Fractional Order ◽  
Decomposition Method ◽  
Telegraph Equation ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Present Technique ◽  
Telegraph Equations

In the current paper, fractional-order hyperbolic telegraph equations are considered for analytical solutions, using the decomposition method based on natural transformation. The fractional derivative is defined by the Caputo operator. The present technique is implemented for both fractional- and integer-order equations, showing that the current technique is an accurate analytical instrument for the solution of partial differential equations of fractional-order arising in all branches of applied sciences. For this purpose, several examples related to hyperbolic telegraph models are presented to explain the procedure of the suggested method. It is noted that the procedure of the present technique is simple, straightforward, accurate, and found to be a better mathematical technique to solve non-linear fractional partial differential equations.

Numerical Solution of Space and Time Fractional Telegraph Equation: A Meshless Approach

2019 ◽  
Vol 20 (3-4) ◽  
pp. 325-337 ◽  
Author(s):  
Hitesh Bansu ◽  
Sushil Kumar
Keyword(s):  
Fractional Order ◽  
Kronecker Product ◽  
Fractional Derivatives ◽  
Telegraph Equation ◽  
New Approach ◽  
Numerical Examples ◽  
Novel Approach ◽  
Fractional Telegraph Equation ◽  
Precise Analysis ◽  
Telegraph Equations

AbstractIn recent years, there has been an incredible enthusiasm towards fractional order partial differential equations because of their incessant presence alongside different fields. Fractional derivatives offer an in-depth and precise analysis of the models of the systems. Particularly, fractional order telegraph equations (FOTE) have been taken into consideration and solved by plenty of researchers, using different techniques. In this paper, we present a novel approach and technique to solve fractional telegraph equation by fusion of cubic radial basis function and Chebyshev polynomials with the aid of Kronecker product. The numerical examples have been considered to verify the accuracy and also to demonstrate the performance of the new approach.

scholarly journals Analytical Solutions of (2+Time Fractional Order) Dimensional Physical Models, Using Modified Decomposition Method

2019 ◽  
Vol 10 (1) ◽  
pp. 122 ◽  
Author(s):  
Hassan Khan ◽  
Umar Farooq ◽  
Rasool Shah ◽  
Dumitru Baleanu ◽  
Poom Kumam ◽  
...  
Keyword(s):  
Fractional Order ◽  
Decomposition Method ◽  
Adomian Decomposition Method ◽  
Analytical Techniques ◽  
Form Solution ◽  
Physical Models ◽  
Fractional Partial Differential Equations ◽  
Analytical Technique ◽  
Adomian Decomposition ◽  
Closed Contact

In this article, a new analytical technique based on an innovative transformation is used to solve (2+time fractional-order) dimensional physical models. The proposed method is the hybrid methodology of Shehu transformation along with Adomian decomposition method. The series form solution is obtained by using the suggested method which provides the desired rate of convergence. Some numerical examples are solved by using the proposed method. The solutions of the targeted problems are represented by graphs which have confirmed closed contact between the exact and obtained solutions of the problems. Based on the novelty and straightforward implementation of the method, it is considered to be one of the best analytical techniques to solve linear and non-linear fractional partial differential equations.

scholarly journals A New Computational Approach for Solving Fractional Order Telegraph Equations

2021 ◽  
Vol 13 (3) ◽  
pp. 715-732
Author(s):  
A. Devi ◽  
M. Jakhar
Keyword(s):  
Fractional Order ◽  
Decomposition Method ◽  
Series Solution ◽  
Adomian Decomposition Method ◽  
Approximate Solutions ◽  
Computational Approach ◽  
Order Problem ◽  
Adomian Decomposition ◽  
Telegraph Equations ◽  
Derivatives Of

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.

scholarly journals Fractional Telegraph Equation and Its Solution by Natural Transform Decomposition Method

Symmetry ◽  
2019 ◽  
Vol 11 (3) ◽  
pp. 334 ◽  
Author(s):  
Hassan Eltayeb ◽  
Yahya Abdalla ◽  
Imed Bachar ◽  
Mohamed Khabir
Keyword(s):  
Decomposition Method ◽  
Decomposition Methods ◽  
Telegraph Equation ◽  
Adomian Decomposition ◽  
Fractional Telegraph Equation ◽  
Telegraph Equations ◽  
Linear And Nonlinear

In this work, the natural transform decomposition method (NTDM) is applied to solve the linear and nonlinear fractional telegraph equations. This method is a combined form of the natural transform and the Adomian decomposition methods. In addition, we prove the convergence of our method. Finally, three examples have been employed to illustrate the preciseness and effectiveness of the proposed method.

scholarly journals Numerical Analysis of the Fractional-Order Telegraph Equations

2021 ◽  
Vol 2021 ◽  
pp. 1-14
Author(s):  
Omar Fouad Azhar ◽  
Muhammad Naeem ◽  
Fatemah Mofarreh ◽  
Jeevan Kafle
Keyword(s):  
Numerical Analysis ◽  
Communication Networks ◽  
Fractional Order ◽  
High Frequency ◽  
Decomposition Method ◽  
Natural Transformation ◽  
Vital Role ◽  
Telegraph Equations ◽  
Microwave Communication ◽  
Significant Attention

This paper studied the fractional-order telegraph equations via the natural transform decomposition method with nonsingular kernel derivatives. The fractional result considered in the Caputo-Fabrizio derivative is Caputo sense. Currently, the communication system plays a vital role in a global society. High-frequency telecommunications continuously receive significant attention in the industry due to a slew of radiofrequency and microwave communication networks. These technologies use transmission media to move information-carrying signals from one location to another. We used natural transformation on fractional telegraph equations followed by inverse natural transformation to achieve the solution of the equation. To validate the technique, we have considered a few problems and compared them with the exact solutions.

scholarly journals Vibration Equation of Fractional Order Describing Viscoelasticity and Viscous Inertia

Open Physics ◽  
2019 ◽  
Vol 17 (1) ◽  
pp. 850-856 ◽  
Author(s):  
Jun-Sheng Duan ◽  
Yun-Yun Xu
Keyword(s):  
Fractional Derivative ◽  
Fractional Order ◽  
Fractional Derivatives ◽  
Harmonic Excitation ◽  
Steady State Response ◽  
Vibration System ◽  
Derivative Operator ◽  
Vibration Equation ◽  
State Response ◽  
Frequency Relation

Abstract The steady state response of a fractional order vibration system subject to harmonic excitation was studied by using the fractional derivative operator ${}_{-\infty} D_t^\beta,$where the order β is a real number satisfying 0 ≤ β ≤ 2. We derived that the fractional derivative contributes to the viscoelasticity if 0 < β < 1, while it contributes to the viscous inertia if 1 < β < 2. Thus the fractional derivative can represent the “spring-pot” element and also the “inerterpot” element proposed in the present article. The viscosity contribution coefficient, elasticity contribution coefficient, inertia contribution coefficient, amplitude-frequency relation, phase-frequency relation, and influence of the order are discussed in detail. The results show that fractional derivatives are applicable for characterizing the viscoelasticity and viscous inertia of materials.

scholarly journals Approximate analytical fractional view of convection–diffusion equations

Open Physics ◽  
2020 ◽  
Vol 18 (1) ◽  
pp. 897-905
Author(s):  
Hassan Khan ◽  
Saima Mustafa ◽  
Izaz Ali ◽  
Poom Kumam ◽  
Dumitru Baleanu ◽  
...  
Keyword(s):  
Fractional Order ◽  
Variational Iteration Method ◽  
Absolute Error ◽  
Form Solution ◽  
Diffusion Equations ◽  
Nonlinear Convection ◽  
Convection Diffusion ◽  
Suggested Technique ◽  
First Time ◽  
Modified Variational Iteration Method

Abstract In this article, a modified variational iteration method along with Laplace transformation is used for obtaining the solution of fractional-order nonlinear convection–diffusion equations (CDEs). The proposed technique is applied for the first time to solve fractional-order nonlinear CDEs and attain a series-form solution with the quick rate of convergence. Tabular and graphical representations are presented to confirm the reliability of the suggested technique. The solutions are calculated for fractional as well as for integer orders of the problems. The solution graphs of the solutions at various fractional derivatives are plotted. The accuracy is measured in terms of absolute error. The higher degree of accuracy is observed from the table and figures. It is further investigated that fractional solutions have the convergence behavior toward the solution at integer order. The applicability of the present technique is verified by illustrative examples. The simple and effective procedure of the current technique supports its implementation to solve other nonlinear fractional problems in different areas of applied science.

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