scholarly journals Two Reliable Methods for The Solution of Fractional Coupled Burgers’ Equation Arising as a Model of Polydispersive Sedimentation

2019 ◽  
Vol 4 (2) ◽  
pp. 523-534 ◽  
Author(s):  
Ali Kurt ◽  
Mehmet Şenol ◽  
Orkun Tasbozan ◽  
Mehar Chand
Keyword(s):  
Water Waves ◽  
Fractional Derivatives ◽  
Burgers Equation ◽  
Approximate Solutions ◽  
Iteration Algorithm ◽  
Shallow Water Waves ◽  
Fractional Differential ◽  
Reliable Methods ◽  
Coupled Burgers Equation ◽  
Partial Fractional Differential Equations

AbstractIn this article, we attain new analytical solution sets for nonlinear time-fractional coupled Burgers’ equations which arise in polydispersive sedimentation in shallow water waves using exp-function method. Then we apply a semi-analytical method namely perturbation-iteration algorithm (PIA) to obtain some approximate solutions. These results are compared with obtained exact solutions by tables and surface plots. The fractional derivatives are evaluated in the conformable sense. The findings reveal that both methods are very effective and dependable for solving partial fractional differential equations.

scholarly journals An Efficient Method for Systems of Variable Coefficient Coupled Burgers’ Equation with Time-Fractional Derivative

2013 ◽  
Vol 2013 ◽  
pp. 1-7
Author(s):  
Hossein Aminikhah ◽  
Nasrin Malekzadeh
Keyword(s):  
Fractional Derivative ◽  
Variable Coefficient ◽  
Fractional Derivatives ◽  
Homotopy Perturbation Method ◽  
Burgers Equation ◽  
Power Series Expansion ◽  
Approximate Solutions ◽  
Convergent Series ◽  
Time Fractional Derivative ◽  
Coupled Burgers Equation

A new homotopy perturbation method (NHPM) is applied to system of variable coefficient coupled Burgers' equation with time-fractional derivative. The fractional derivatives are described in the Caputo fractional derivative sense. The concept of new algorithm is introduced briefly, and NHPM is examined for two systems of nonlinear Burgers' equation. In this approach, the solution is considered as a power series expansion that converges rapidly to the nonlinear problem. The new approximate analytical procedure depends on two iteratives. The modified algorithm provides approximate solutions in the form of convergent series with easily computable components. Results indicate that the introduced method is promising for solving other types of systems of nonlinear fractional-order partial differential equations.

A note on exp-function method combined with complex transform method applied to fractional differential equations

2015 ◽  
Vol 4 (3) ◽  
pp. 201-208 ◽  
Author(s):  
Ozkan Guner ◽  
Ahmet Bekir ◽  
Halis Bilgil
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Fractional Differential Equations ◽  
Function Method ◽  
Fractional Derivatives ◽  
Nonlinear Ordinary Differential Equation ◽  
Transform Method ◽  
Fractional Equations ◽  
Fractional Differential ◽  
Partial Fractional Differential Equations

AbstractIn this article, the fractional derivatives in the sense of modified Riemann–Liouville and the exp-function method are used to construct exact solutions for some nonlinear partial fractional differential equations via the nonlinear fractional Liouville equation and nonlinear fractional Zoomeron equation. These nonlinear fractional equations can be turned into another nonlinear ordinary differential equation by complex transform method. This method is efficient and powerful in solving wide classes of nonlinear fractional order equations. The exp-function method appears to be easier and more convenient by means of a symbolic computation system.

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 Spectrally accurate approximate solutions and convergence analysis of fractional Burgers’ equation

2020 ◽  
Vol 9 (3) ◽  
pp. 633-644
Author(s):  
A. K. Mittal
Keyword(s):  
Numerical Solutions ◽  
Burgers Equation ◽  
Numerical Technique ◽  
Time Derivative ◽  
Pseudospectral Method ◽  
Approximate Solutions ◽  
Algebraic Equations ◽  
Time And Space ◽  
Time Space ◽  
Coupled Burgers Equation

Abstract In this paper, a new numerical technique implements on the time-space pseudospectral method to approximate the numerical solutions of nonlinear time- and space-fractional coupled Burgers’ equation. This technique is based on orthogonal Chebyshev polynomial function and discretizes using Chebyshev–Gauss–Lobbato (CGL) points. Caputo–Riemann–Liouville fractional derivative formula is used to illustrate the fractional derivatives matrix at CGL points. Using the derivatives matrices, the given problem is reduced to a system of nonlinear algebraic equations. These equations can be solved using Newton–Raphson method. Two model examples of time- and space-fractional coupled Burgers’ equation are tested for a set of fractional space and time derivative order. The figures and tables show the significant features, effectiveness, and good accuracy of the proposed method.

scholarly journals Optimal Perturbation Iteration Method for Solving Fractional Model of Damped Burgers’ Equation

Symmetry ◽  
2020 ◽  
Vol 12 (6) ◽  
pp. 958 ◽  
Author(s):  
Sinan Deniz ◽  
Ali Konuralp ◽  
Mnauel De la Sen
Keyword(s):  
Laplace Transform ◽  
Analytical Solutions ◽  
Differential Form ◽  
Fractional Derivatives ◽  
Burgers Equation ◽  
Optimal Perturbation ◽  
Fractional Differential ◽  
Fractional Type ◽  
Iteration Technique ◽  
Fractional Differential Form

The newly constructed optimal perturbation iteration procedure with Laplace transform is applied to obtain the new approximate semi-analytical solutions of the fractional type of damped Burgers’ equation. The classical damped Burgers’ equation is remodeled to fractional differential form via the Atangana–Baleanu fractional derivatives described with the help of the Mittag–Leffler function. To display the efficiency of the proposed optimal perturbation iteration technique, an extended example is deeply analyzed.

scholarly journals Approximate analytic solutions of multi-dimensional fractional heat-like models with variable coefficients

2019 ◽  
Vol 23 (6 Part B) ◽  
pp. 3725-3729
Author(s):  
Jianshe Sun
Keyword(s):  
Fractional Derivatives ◽  
Fractional Power ◽  
Approximate Solutions ◽  
Analytic Solutions ◽  
Variable Coefficients ◽  
Mathematical Tool ◽  
Power Series Method ◽  
Fractional Differential ◽  
Fractional Power Series ◽  
Powerful Mathematical Tool

In this work, the fractional power series method is applied to solve the 2-D and 3-D fractional heat-like models with variable coefficients. The fractional derivatives are described in the Liouville-Caputo sense. The analytical approximate solutions and exact solutions for the 2-D and 3-D fractional heat-like models with variable coefficients are obtained. It is shown that the proposed method provides a very effective, convenient and powerful mathematical tool for solving fractional differential equations in mathematical physics.

Matrix Approach to Discretization of Ordinary and Partial Differential Equations of Arbitrary Real Order: The Matlab Toolbox

2009 ◽  
Author(s):  
Igor Podlubny ◽  
Tomas Skovranek ◽  
Blas M. Vinagre Jara
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Fractional Derivatives ◽  
Matrix Approach ◽  
Matlab Toolbox ◽  
Partial Differential ◽  
The Matrix ◽  
Fractional Differential ◽  
Derivatives Of ◽  
Partial Fractional Differential Equations

The method developed recently by Podlubny et al. (I. Podlubny, Fractional Calculus and Applied Analysis, vol. 3, no. 4, 2000, pp. 359–386; I. Podlubny et al., Journal of Computational Physics, vol. 228, no. 8, 1 May 2009, pp. 3137–3153) makes it possible to immediately obtain the discretization of ordinary and partial differential equations by replacing the derivatives with their discrete analogs in the form of triangular strip matrices. This article presents a Matlab toolbox that implements the matrix approach and allows easy and convenient discretization of ordinary and partial differential equations of arbitrary real order. The basic use of the functions implementing the matrix approach to discretization of derivatives of arbitrary real order (so-called fractional derivatives, or fractional-order derivatives), and to solution of ordinary and partial fractional differential equations, is illustrated by examples with explanations.

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