scholarly journals The Variational Iteration Transform Method for Solving the Time-Fractional Fornberg–Whitham Equation and Comparison with Decomposition Transform Method

Mathematics ◽  
2021 ◽  
Vol 9 (2) ◽  
pp. 141
Author(s):  
Nehad Ali Shah ◽  
Ioannis Dassios ◽  
Essam R. El-Zahar ◽  
Jae Dong Chung ◽  
Somaye Taherifar
Keyword(s):  
Decomposition Method ◽  
Approximate Analytical Solution ◽  
Exact Result ◽  
Solution Procedure ◽  
Transform Method ◽  
Variational Iteration ◽  
Non Linear ◽  
Linear Problems ◽  
Adomian’S Polynomials ◽  
Whitham Equation

In this article, modified techniques, namely the variational iteration transform and Shehu decomposition method, are implemented to achieve an approximate analytical solution for the time-fractional Fornberg–Whitham equation. A comparison is made between the results of the variational iteration transform method and the Shehu decomposition method. The solution procedure reveals that the variational iteration transform method and Shehu decomposition method is effective, reliable and straightforward. The variational iteration transform methods solve non-linear problems without using Adomian’s polynomials and He’s polynomials, which is a clear advantage over the decomposition technique. The solutions achieved are compared with the corresponding exact result to show the efficiency and accuracy of the existing methods in solving a wide variety of linear and non-linear problems arising in various science areas.

scholarly journals Analytical solution for non-linear local fractional Bratu-type equation in a fractal space

2020 ◽  
Vol 24 (6 Part B) ◽  
pp. 3941-3947
Author(s):  
Shao-Wen Yao ◽  
Wen-Jie Li ◽  
Kang-Le Wang
Keyword(s):  
Analytical Solution ◽  
Fractional Derivative ◽  
Variational Formulation ◽  
Decomposition Method ◽  
Approximate Analytical Solution ◽  
Type Equation ◽  
Transform Method ◽  
Inverse Transform ◽  
Non Linear ◽  
Fractal Space

In this paper, the non-linear local fractional Bratu-type equation is described by the local fractional derivative in a fractal space, and its variational formulation is successfully established according to semi-inverse transform method. Finally, we find the approximate analytical solution of the local fractional Bratu-type equation by using Adomina decomposition method.

scholarly journals Approximate analytical solutions of non-linear local fractional heat equations

2019 ◽  
Vol 23 (Suppl. 3) ◽  
pp. 837-841 ◽  
Author(s):  
Shuxian Deng
Keyword(s):  
Analytical Solutions ◽  
Decomposition Method ◽  
Adomian Decomposition Method ◽  
Heat Equations ◽  
Transform Method ◽  
Fractional Complex Transform ◽  
Fractional Heat Equation ◽  
Approximate Analytical Solutions ◽  
Adomian Decomposition ◽  
Non Linear

Consider the non-linear local fractional heat equation. The fractional complex transform method and the Adomian decomposition method are used to solve the equation. The approximate analytical solutions are obtained.

scholarly journals Multistage optimal homotopy asymptotic method for the K(2,2) equation arising in solitary waves theory

2021 ◽  
Vol 25 (Spec. issue 2) ◽  
pp. 199-205
Author(s):  
Shah Ali ◽  
Imtiaz Ahmad ◽  
Hanaa Abu-Zinadah ◽  
Khedher Mohamed ◽  
Hijaz Ahmad
Keyword(s):  
Asymptotic Method ◽  
Large Time ◽  
Approximate Analytical Solution ◽  
Variational Iteration Method ◽  
Time Span ◽  
Numerical Comparison ◽  
Non Linear ◽  
Modern Physics ◽  
Optimal Homotopy Asymptotic Method ◽  
Linear Problems

The paper is concern to the approximate analytical solution of K(2,2) using the multistage homotopy asymptotic method which are used in modern physics and engineering. The suggested algorithm is an accurate, effective, and simple to-utilize semi-analytic tool for non-linear problems, and in this manner the current investigation highlights the efficiency and accuracy of the method for the solution of non-linear PDE for large time span. Numerical comparison with the variational iteration method and with homotopy asymptotic method shows the efficacy and accuracy of the proposed method.

scholarly journals New Modified Variational Iteration Laplace Transform Method Compares Laplace Adomian Decomposition Method for Solution Time-Partial Fractional Differential Equations

2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Mohamed Z. Mohamed ◽  
Tarig M. Elzaki ◽  
Mohamed S. Algolam ◽  
Eltaib M. Abd Elmohmoud ◽  
Amjad E. Hamza
Keyword(s):  
Differential Equations ◽  
Decomposition Method ◽  
Numerical Solutions ◽  
Adomian Decomposition Method ◽  
Variational Iteration Method ◽  
Transform Method ◽  
Variational Iteration ◽  
Modified Method ◽  
Adomian Decomposition ◽  
Fractional Differential

The objective of this paper is to compute the new modified method of variational iteration and the Laplace Adomian decomposition method for the solution of nonlinear fractional partial differential equations. We execute a comparatively newfangled analytical mechanism that is denoted by the modified Laplace variational iteration method (MLVIM) and Laplace Adomian decomposition method (LADM). The effect of the numerical results indicates that the double approximation is handy to execute and reliable when applied. It is shown that numerical solutions are gained in the form of approximately series which are facilely computable.

scholarly journals Application of DGJ method for solving non-linear local fractional heat equations

2019 ◽  
Vol 23 (3 Part A) ◽  
pp. 1571-1576 ◽  
Author(s):  
Shu-Xian Deng ◽  
Xin-Xin Ge
Keyword(s):  
Heat Equation ◽  
Analytical Solutions ◽  
Decomposition Method ◽  
Heat Equations ◽  
Initial Value ◽  
Transform Method ◽  
Fractional Complex Transform ◽  
Fractional Heat Equation ◽  
Approximate Analytical Solutions ◽  
Non Linear

In this paper, the initial value problem for a new non-linear local fractional heat equation is considered. The fractional complex transform method and the DGJ decomposition method are used to solve the problem, and the approximate analytical solutions are also obtained.

scholarly journals Analytical solutions of nonlinear Klein-Gordon equations using Multistep Modified Reduced Differential Transform Method

2019 ◽  
Vol 23 (Suppl. 1) ◽  
pp. 317-326 ◽  
Author(s):  
Hussin Che ◽  
Ahmad Ismail ◽  
Adem Kilicman ◽  
Amirah Azmi
Keyword(s):  
Approximate Analytical Solution ◽  
Approximate Solutions ◽  
Differential Transform Method ◽  
Transform Method ◽  
Adomian Polynomials ◽  
Differential Transform ◽  
Non Linear ◽  
Klein Gordon ◽  
Time Region ◽  
Reduced Differential Transform Method

This paper explores the approximate analytical solution of non-linear Klein-Gordon equations (NKGE) by using multistep modified reduced differential transform method (MMRDTM). Through this proposed strategy, the non-linear term is substituted by associating Adomian polynomials obtained by utilization of a multistep approach. The NKGE solutions can be obtained with a reduced number of computed terms. In addition, the approximate solutions converge rapidly in a wide time region. Three examples are provided to illustrate the effectiveness of the proposed method to obtain solutions for the NKGE. Graphical results are shown to represent the behavior of the solution so as to demonstrate the validity and accuracy of the MMRDTM.

Analytic study for fractional coupled Burger’s equations via Sumudu transform method

2018 ◽  
Vol 7 (4) ◽  
pp. 323-332 ◽  
Author(s):  
Amit Prakash ◽  
Vijay Verma ◽  
Devendra Kumar ◽  
Jagdev Singh
Keyword(s):  
Decomposition Method ◽  
Homotopy Perturbation Method ◽  
Approximate Analytical Solution ◽  
Transform Method ◽  
Numerical Examples ◽  
Homotopy Perturbation ◽  
Differential Transform ◽  
Sumudu Transform ◽  
Generalized Differential ◽  
Burger's Equations

AbstractIn this work, we aim to apply a reliable analytic algorithm based on homotopy perturbation Sumudu transform method (HPSTM) to examine the nonlinear time-fractional coupled Burger’s equations. The approximate analytical solution and some numerical examples show the accuracy and efficiency of the proposed method, which is simple and accurate in comparison to the Adomain decomposition method (ADM), homotopy perturbation method (HPM) and generalized differential transform method (GDTM).

scholarly journals Analytical Solutions of Fractional-Order Diffusion Equations by Natural Transform Decomposition Method

Entropy ◽  
2019 ◽  
Vol 21 (6) ◽  
pp. 557 ◽  
Author(s):  
Rasool Shah ◽  
Hassan Khan ◽  
Saima Mustafa ◽  
Poom Kumam ◽  
Muhammad Arif
Keyword(s):  
Diffusion Equation ◽  
Fractional Order ◽  
Decomposition Method ◽  
Analytical Techniques ◽  
High Rate ◽  
Diffusion Equations ◽  
Numerical Examples ◽  
Analytical Technique ◽  
Non Linear ◽  
Linear Problems

In the present article, fractional-order diffusion equations are solved using the Natural transform decomposition method. The series form solutions are obtained for fractional-order diffusion equations using the proposed method. Some numerical examples are presented to understand the procedure of the Natural transform decomposition method. The Natural transform decomposition method has shown the least volume of calculations and a high rate of convergence compared to other analytical techniques, the proposed method can also be easily applied to other non-linear problems. Therefore, the Natural transform decomposition method is considered to be one of the best analytical technique, to solve fractional-order linear and non-linear partial deferential equations, particularly fractional-order diffusion equation.

scholarly journals A New Improved Adomian Decomposition Method for Solving Emden{Fowler Type Equations of Higher{Order

2020 ◽  
pp. 9-17
Author(s):  
Zainab Ali Abdu AL-Rabahi ◽  
Yahya Qaid Hasan
Keyword(s):  
Exact Solution ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Decomposition Method ◽  
Adomian Decomposition Method ◽  
Higher Order ◽  
Adomian Decomposition ◽  
Non Linear ◽  
Linear Problems ◽  
Suggested Modification

In this paper, we present a suggested modification for Adomain decomposition method to solve Emden{Fowler Types Equations of higher-order ordinary differential equations. The proposed method can be applied to linear and non-linear problems. By using some illustrative examples, we tested the reliability and effectiveness of the proposed method and we found that the obtained results approximate the exact solution. Thus, we can conclude that this proposed method is efficient and reliable .

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