scholarly journals Optimum solutions of space fractional order diffusion equation

2018 ◽  
Vol 22 (Suppl. 1) ◽  
pp. 329-339
Author(s):  
Sajjad Ali ◽  
Kamal Shah ◽  
Yongjin Li ◽  
Muhammad Arif
Keyword(s):  
Differential Equations ◽  
Fractional Order ◽  
Fractional Derivatives ◽  
Approximate Solutions ◽  
Science And Engineering ◽  
Approximate Analytical Solutions ◽  
Special Cases ◽  
Fractional Differential ◽  
Optimal Homotopy Asymptotic Method ◽  
Excellent Accuracy

The present paper is concerned with the implementation of optimal homotopy asymptotic method to handle the approximate analytical solutions of fractional order partial differential equations. Fractional differential equations have great importance regarding distinct fields of science and engineering. Approximate solutions of space fractional order diffusion model and its various special cases are handled using the innovative proposed method. The space fractional derivatives are described in the Caputo sense. The results obtained by the proposed method are compared with various methods. The proposed method demonstrates excellent accuracy and strength over various methods.

scholarly journals Optimum Solutions of Fractional Order Zakharov–Kuznetsov Equations

Complexity ◽  
2019 ◽  
Vol 2019 ◽  
pp. 1-9 ◽  
Author(s):  
Rashid Nawaz ◽  
Laiq Zada ◽  
Abraiz Khattak ◽  
Muhammad Jibran ◽  
Adam Khan
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Fractional Order ◽  
Fractional Derivatives ◽  
Approximate Solutions ◽  
Convergent Series ◽  
Partial Differential ◽  
Higher Dimensional ◽  
Optimal Homotopy Asymptotic Method

In this paper, the Optimal Homotopy Asymptotic Method is extended to derive the approximate solutions of fractional order two-dimensional partial differential equations. The fractional order Zakharov–Kuznetsov equation is solved as a test example, while the time fractional derivatives are described in the Caputo sense. The solutions of the problem are computed in the form of rapidly convergent series with easily calculable components using Mathematica. Reliability of the proposed method is given by comparison with other methods in the literature. The obtained results showed that the method is powerful and efficient for determination of solution of higher-dimensional fractional order partial differential equations.

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 Revised Variational Iteration Method for Solving Systems of Nonlinear Fractional-Order Differential Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
C. Ünlü ◽  
H. Jafari ◽  
D. Baleanu
Keyword(s):  
Differential Equations ◽  
Fractional Order ◽  
Iteration Method ◽  
Fractional Derivatives ◽  
Variational Iteration Method ◽  
Approximate Solutions ◽  
Initial Position ◽  
Fractional Order Differential Equations ◽  
Variational Iteration ◽  
Fractional Differential

A modification of the variational iteration method (VIM) for solving systems of nonlinear fractional-order differential equations is proposed. The fractional derivatives are described in the Caputo sense. The solutions of fractional differential equations (FDE) obtained using the traditional variational iteration method give good approximations in the neighborhood of the initial position. The main advantage of the present method is that it can accelerate the convergence of the iterative approximate solutions relative to the approximate solutions obtained using the traditional variational iteration method. Illustrative examples are presented to show the validity of this modification.

scholarly journals Stability of Fractional Differential Equations with New Generalized Hattaf Fractional Derivative

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
Khalid Hattaf
Keyword(s):  
Stability Analysis ◽  
Differential Equations ◽  
Fractional Derivative ◽  
Fractional Differential Equations ◽  
Fractional Derivatives ◽  
Direct Method ◽  
Lyapunov Direct Method ◽  
Science And Engineering ◽  
Fractional Differential ◽  
The Stability

This paper aims to study the stability of fractional differential equations involving the new generalized Hattaf fractional derivative which includes the most types of fractional derivatives with nonsingular kernels. The stability analysis is obtained by means of the Lyapunov direct method. First, some fundamental results and lemmas are established in order to achieve the goal of this study. Furthermore, the results related to exponential and Mittag–Leffler stability existing in recent studies are extended and generalized. Finally, illustrative examples are presented to show the applicability of our main results in some areas of science and engineering.

scholarly journals An Integral Operational Matrix of Fractional-Order Chelyshkov Functions and Its Applications

Symmetry ◽  
2020 ◽  
Vol 12 (11) ◽  
pp. 1755
Author(s):  
M. S. Al-Sharif ◽  
A. I. Ahmed ◽  
M. S. Salim
Keyword(s):  
Differential Equations ◽  
Fractional Order ◽  
Fractional Differential Equations ◽  
Fractional Integral ◽  
Operational Matrix ◽  
Operational Matrices ◽  
Algebraic Equations ◽  
Science And Engineering ◽  
Numerical Examples ◽  
Fractional Differential

Fractional differential equations have been applied to model physical and engineering processes in many fields of science and engineering. This paper adopts the fractional-order Chelyshkov functions (FCHFs) for solving the fractional differential equations. The operational matrices of fractional integral and product for FCHFs are derived. These matrices, together with the spectral collocation method, are used to reduce the fractional differential equation into a system of algebraic equations. The error estimation of the presented method is also studied. Furthermore, numerical examples and comparison with existing results are given to demonstrate the accuracy and applicability of the presented method.

scholarly journals Variational Iteration Method for a Fractional-Order Brusselator System

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
H. Jafari ◽  
Abdelouahab Kadem ◽  
D. Baleanu
Keyword(s):  
Fractional Order ◽  
Fractional Differential Equations ◽  
Iteration Method ◽  
Fractional Derivatives ◽  
Variational Iteration Method ◽  
Variational Iteration ◽  
Promising Tool ◽  
Approximate Analytical Solutions ◽  
Fractional Differential ◽  
Linear And Nonlinear

This paper presents approximate analytical solutions for the fractional-order Brusselator system using the variational iteration method. The fractional derivatives are described in the Caputo sense. This method is based on the incorporation of the correction functional for the equation. Two examples are solved as illustrations, using symbolic computation. The numerical results show that the introduced approach is a promising tool for solving system of linear and nonlinear fractional differential equations.

scholarly journals Convergence of Variational Iteration Method for Solving Singular Partial Differential Equations of Fractional Order

2014 ◽  
Vol 2014 ◽  
pp. 1-11
Author(s):  
Asma Ali Elbeleze ◽  
Adem Kılıçman ◽  
Bachok M. Taib
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Fractional Order ◽  
Iteration Method ◽  
Series Solution ◽  
Fractional Derivatives ◽  
Variational Iteration Method ◽  
Approximate Solutions ◽  
Partial Differential ◽  
Variational Iteration

We are concerned here with singular partial differential equations of fractional order (FSPDEs). The variational iteration method (VIM) is applied to obtain approximate solutions of this type of equations. Convergence analysis of the VIM is discussed. This analysis is used to estimate the maximum absolute truncated error of the series solution. A comparison between the results of VIM solutions and exact solution is given. The fractional derivatives are described in Caputo sense.

scholarly journals New Method for Solving Linear Fractional Differential Equations

2011 ◽  
Vol 2011 ◽  
pp. 1-8 ◽  
Author(s):  
S. Z. Rida ◽  
A. A. M. Arafa
Keyword(s):  
Power Series ◽  
Differential Equations ◽  
Fractional Order ◽  
Fractional Differential Equations ◽  
Function Method ◽  
Fractional Derivatives ◽  
Linear Differential Equations ◽  
Fractional Differential ◽  
Linear Differential ◽  
Linear Fractional Differential Equations

We develop a new application of the Mittag-Leffler Function method that will extend the application of the method to linear differential equations with fractional order. A new solution is constructed in power series. The fractional derivatives are described in the Caputo sense. To illustrate the reliability of the method, some examples are provided. The results reveal that the technique introduced here is very effective and convenient for solving linear differential equations of fractional order.

scholarly journals Fractional differential equations and Volterra–Stieltjes integral equations of the second kind

2019 ◽  
Vol 38 (4) ◽  
Author(s):  
Avyt Asanov ◽  
Ricardo Almeida ◽  
Agnieszka B. Malinowska
Keyword(s):  
Differential Equation ◽  
Integral Equation ◽  
Differential Equations ◽  
Fractional Differential Equations ◽  
Equivalence Relation ◽  
Numerical Experiments ◽  
Fractional Derivatives ◽  
Approximate Solutions ◽  
Stieltjes Integral ◽  
Fractional Differential

Abstract In this paper, we construct a method to find approximate solutions to fractional differential equations involving fractional derivatives with respect to another function. The method is based on an equivalence relation between the fractional differential equation and the Volterra–Stieltjes integral equation of the second kind. The generalized midpoint rule is applied to solve numerically the integral equation and an estimation for the error is given. Results of numerical experiments demonstrate that satisfactory and reliable results could be obtained by the proposed method.

scholarly journals An Efficient Mechanism to Solve Fractional Differential Equations Using Fractional Decomposition Method

Symmetry ◽  
2021 ◽  
Vol 13 (6) ◽  
pp. 984
Author(s):  
Mahmoud S. Alrawashdeh ◽  
Seba A. Migdady ◽  
Ioannis K. Argyros
Keyword(s):  
Differential Equations ◽  
Fractional Calculus ◽  
Laplace Transform ◽  
Fractional Differential Equations ◽  
Decomposition Method ◽  
Large Scale ◽  
Approximate Solutions ◽  
Science And Engineering ◽  
Large Scale Problems ◽  
Fractional Differential

We present some new results that deal with the fractional decomposition method (FDM). This method is suitable to handle fractional calculus applications. We also explore exact and approximate solutions to fractional differential equations. The Caputo derivative is used because it allows traditional initial and boundary conditions to be included in the formulation of the problem. This is of great significance for large-scale problems. The study outlines the significant features of the FDM. The relation between the natural transform and Laplace transform is a symmetrical one. Our work can be considered as an alternative to existing techniques, and will have wide applications in science and engineering fields.

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