On mixed derivatives type high dimensional multi-term fractional partial differential equations approximate solutions

2017 ◽  
Author(s):  
Imran Talib ◽  
Fethi Bin Muhammad Belgacem ◽  
Naseer Ahmad Asif ◽  
Hammad Khalil
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Approximate Solutions ◽  
High Dimensional ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Mixed Derivatives

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 On approximate solutions of fractional order partial differential equations

2018 ◽  
Vol 22 (Suppl. 1) ◽  
pp. 287-299
Author(s):  
Muhammad Chohan ◽  
Sajjad Ali ◽  
Kamal Shah ◽  
Muhammad Arif
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Fractional Order ◽  
Approximate Solutions ◽  
Fractional Partial Differential Equations ◽  
Convergence Region ◽  
Partial Differential ◽  
Approximate Analytical Solutions ◽  
Fractional Models ◽  
Optimal Homotopy Asymptotic Method

The present paper is concerned with the implementation of optimal homotopy asymptotic method to handle the approximate analytical solutions of fractional partial differential equations. Approximate solutions of fractional models in both 1-D and 2-D cases are handled using the innovative proposed method. The consequences show excellent accuracy and strength of the planned method. Using this method, one can easily handle the convergence of approximation series solution for the fractional partial differential equations and can adjust the convergence region when required. The method is effective and explicit. Moreover, this method is flexible with respect to geometry and ease of implementation for fractional order models of physical and biological problems.

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 Exact Solution of Two-Dimensional Fractional Partial Differential Equations

2020 ◽  
Vol 4 (2) ◽  
pp. 21 ◽  
Author(s):  
Dumitru Baleanu ◽  
Hassan Kamil Jassim
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Decomposition Method ◽  
Approximate Solutions ◽  
Two Dimensional ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Transform Method ◽  
Sumudu Transform ◽  
Fractional Differential

In this study, we examine adapting and using the Sumudu decomposition method (SDM) as a way to find approximate solutions to two-dimensional fractional partial differential equations and propose a numerical algorithm for solving fractional Riccati equation. This method is a combination of the Sumudu transform method and decomposition method. The fractional derivative is described in the Caputo sense. The results obtained show that the approach is easy to implement and accurate when applied to various fractional differential equations.

scholarly journals An Accurate Approximate-Analytical Technique for Solving Time-Fractional Partial Differential Equations

Complexity ◽  
2017 ◽  
Vol 2017 ◽  
pp. 1-12 ◽  
Author(s):  
M. Bishehniasar ◽  
S. Salahshour ◽  
A. Ahmadian ◽  
F. Ismail ◽  
D. Baleanu
Keyword(s):  
Partial Differential Equations ◽  
Finite Difference ◽  
Differential Equations ◽  
Finite Difference Methods ◽  
Computational Cost ◽  
Approximate Solutions ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Analytical Technique ◽  
Analytical Scheme

The demand of many scientific areas for the usage of fractional partial differential equations (FPDEs) to explain their real-world systems has been broadly identified. The solutions may portray dynamical behaviors of various particles such as chemicals and cells. The desire of obtaining approximate solutions to treat these equations aims to overcome the mathematical complexity of modeling the relevant phenomena in nature. This research proposes a promising approximate-analytical scheme that is an accurate technique for solving a variety of noninteger partial differential equations (PDEs). The proposed strategy is based on approximating the derivative of fractional-order and reducing the problem to the corresponding partial differential equation (PDE). Afterwards, the approximating PDE is solved by using a separation-variables technique. The method can be simply applied to nonhomogeneous problems and is proficient to diminish the span of computational cost as well as achieving an approximate-analytical solution that is in excellent concurrence with the exact solution of the original problem. In addition and to demonstrate the efficiency of the method, it compares with two finite difference methods including a nonstandard finite difference (NSFD) method and standard finite difference (SFD) technique, which are popular in the literature for solving engineering problems.

scholarly journals Local Fractional Laplace Variational Iteration Method for Solving Linear Partial Differential Equations with Local Fractional Derivative

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Ai-Min Yang ◽  
Jie Li ◽  
H. M. Srivastava ◽  
Gong-Nan Xie ◽  
Xiao-Jun Yang
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Fractional Derivative ◽  
Iteration Method ◽  
Variational Iteration Method ◽  
Approximate Solutions ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Variational Iteration ◽  
Linear Partial Differential Equations

The local fractional Laplace variational iteration method was applied to solve the linear local fractional partial differential equations. The local fractional Laplace variational iteration method is coupled by the local fractional variational iteration method and Laplace transform. The nondifferentiable approximate solutions are obtained and their graphs are also shown.

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