scholarly journals Exact and Approximate Solutions of Fractional Partial Differential Equations Using Aboodh Transform Iterative Method.

2020 ◽  
Author(s):  
Gbenga OJO
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Iterative Method ◽  
Approximate Solutions ◽  
Fractional Partial Differential Equations ◽  
Partial Differential

Analytical Approach to Time-Fractional Partial Differential Equations in Fluid Mechanics

2011 ◽  
Vol 347-353 ◽  
pp. 463-466
Author(s):  
Xue Hui Chen ◽  
Liang Wei ◽  
Lian Cun Zheng ◽  
Xin Xin Zhang
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Fluid Mechanics ◽  
Approximate Solutions ◽  
Differential Transform Method ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Transform Method ◽  
Differential Transform ◽  
Generalized Differential

The generalized differential transform method is implemented for solving time-fractional partial differential equations in fluid mechanics. This method is based on the two-dimensional differential transform method (DTM) and generalized Taylor’s formula. Results obtained by using the scheme presented here agree well with the numerical results presented elsewhere. The results reveal the method is feasible and convenient for handling approximate solutions of time-fractional partial differential equations.

scholarly journals Numerical Solution of the Fractional Partial Differential Equations by the Two-Dimensional Fractional-Order Legendre Functions

2013 ◽  
Vol 2013 ◽  
pp. 1-13 ◽  
Author(s):  
Fukang Yin ◽  
Junqiang Song ◽  
Yongwen Wu ◽  
Lilun Zhang
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Fractional Order ◽  
Approximate Solutions ◽  
Operational Matrices ◽  
Legendre Functions ◽  
Two Dimensional ◽  
Algebraic Equations ◽  
Fractional Partial Differential Equations ◽  
Partial Differential

A numerical method is presented to obtain the approximate solutions of the fractional partial differential equations (FPDEs). The basic idea of this method is to achieve the approximate solutions in a generalized expansion form of two-dimensional fractional-order Legendre functions (2D-FLFs). The operational matrices of integration and derivative for 2D-FLFs are first derived. Then, by these matrices, a system of algebraic equations is obtained from FPDEs. Hence, by solving this system, the unknown 2D-FLFs coefficients can be computed. Three examples are discussed to demonstrate the validity and applicability of the proposed method.

Solving Fractional Partial Differential Equations with Variable Coefficients by the Reconstruction of Variational Iteration Method

2015 ◽  
Vol 70 (5) ◽  
pp. 375-382 ◽  
Author(s):  
Esmail Hesameddini ◽  
Azam Rahimi
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Closed Form Solution ◽  
Variational Iteration Method ◽  
Approximate Solutions ◽  
Variable Coefficients ◽  
Form Solution ◽  
Successive Approximations ◽  
Fractional Partial Differential Equations ◽  
Partial Differential

AbstractIn this article, we propose a new approach for solving fractional partial differential equations with variable coefficients, which is very effective and can also be applied to other types of differential equations. The main advantage of the method lies in its flexibility for obtaining the approximate solutions of time fractional and space fractional equations. The fractional derivatives are described based on the Caputo sense. Our method contains an iterative formula that can provide rapidly convergent successive approximations of the exact solution if such a closed form solution exists. Several examples are given, and the numerical results are shown to demonstrate the efficiency of the newly proposed method.

scholarly journals An efficient new iterative method for finding exact solutions of nonlinear time-fractional partial differential equations

2011 ◽  
Vol 16 (4) ◽  
pp. 403-414 ◽  
Author(s):  
Hüseyin Koçak ◽  
Ahmet Yıldırım
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Iterative Method ◽  
Exact Solutions ◽  
Fractional Derivatives ◽  
Convergent Series ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
New Iterative Method

In this paper, a new iterative method (NIM) is used to obtain the exact solutions of some nonlinear time-fractional partial differential equations. The fractional derivatives are described in the Caputo sense. The method provides a convergent series with easily computable components in comparison with other existing methods.

scholarly journals New Iterative Method of Solving Nonlinear Equations in Fluid Mechanics

2021 ◽  
Vol 26 (3) ◽  
pp. 163-176
Author(s):  
M. Paliivets ◽  
E. Andreev ◽  
A. Bakshtanin ◽  
D. Benin ◽  
V. Snezhko
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Iterative Method ◽  
Fluid Mechanics ◽  
Nonlinear Equations ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Adomian Decomposition ◽  
Linear And Nonlinear ◽  
New Iterative Method

Abstract This paper presents the results of applying a new iterative method to linear and nonlinear fractional partial differential equations in fluid mechanics. A numerical analysis was performed to find an exact solution of the fractional wave equation and fractional Burgers’ equation, as well as an approximate solution of fractional KdV equation and fractional Boussinesq equation. Fractional derivatives of the order α are described using Caputo's definition with 0 < α ≤ 1 or 1 < α ≤ 2. A comparative analysis of the results obtained using a new iterative method with those obtained by the Adomian decomposition method showed the first method to be more efficient and simple, providing accurate results in fewer computational operations. Given its flexibility and ability to solve nonlinear equations, the iterative method can be used to solve more complex linear and nonlinear fractional partial differential equations.

scholarly journals Application of Sinc-Galerkin Method for Solving Space-Fractional Boundary Value Problems

2015 ◽  
Vol 2015 ◽  
pp. 1-10 ◽  
Author(s):  
Sertan Alkan ◽  
Aydin Secer
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Galerkin Method ◽  
Fractional Derivatives ◽  
Approximate Solutions ◽  
Variable Coefficients ◽  
Fractional Partial Differential Equations ◽  
Efficient Tool ◽  
Partial Differential ◽  
Fractional Boundary Value Problems

We employ the sinc-Galerkin method to obtain approximate solutions of space-fractional order partial differential equations (FPDEs) with variable coefficients. The fractional derivatives are used in the Caputo sense. The method is applied to three different problems and the obtained solutions are compared with the exact solutions of the problems. These comparisons demonstrate that the sinc-Galerkin method is a very efficient tool in solving space-fractional partial differential equations.

scholarly journals Solution of Fractional Partial Differential Equations in Fluid Mechanics by Extension of Some Iterative Method

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
A. A. Hemeda
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Iterative Method ◽  
Fluid Mechanics ◽  
Kdv Equation ◽  
Adomian Decomposition Method ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Adomian Decomposition ◽  
Linear And Nonlinear

An extension of the so-called new iterative method (NIM) has been used to handle linear and nonlinear fractional partial differential equations. The main property of the method lies in its flexibility and ability to solve nonlinear equations accurately and conveniently. Therefore, a general framework of the NIM is presented for analytical treatment of fractional partial differential equations in fluid mechanics. The fractional derivatives are described in the Caputo sense. Numerical illustrations that include the fractional wave equation, fractional Burgers equation, fractional KdV equation, fractional Klein-Gordon equation, and fractional Boussinesq-like equation are investigated to show the pertinent features of the technique. Comparison of the results obtained by the NIM with those obtained by both Adomian decomposition method (ADM) and the variational iteration method (VIM) reveals that the NIM is very effective and convenient. The basic idea described in this paper is expected to be further employed to solve other similar linear and nonlinear problems in fractional calculus.

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