Applied and Computational Mathematics Exact and explicit approximate solutions to the multi-order fractional Burgers-Poisson and fractional Burgers-Poisson equations Ikechukwu Nwamba. (original) (raw)

Solution of the Generalized Burgers Equation Using Homotopy Perturbation Method with General Fractional Derivative

Symmetry

This research paper introduces the generalized Burgers equation, a mathematical model defined using the general fractional derivative, the most recent operator in fractional calculus. The general fractional derivative can be reduced into three well-known operators, providing a more tractable form of the equation. We apply the homotopy perturbation method (HPM), a powerful analytical technique, to obtain the solution of the generalized Burgers equation. The results are illustrated using a practical example, and we present an analysis of the three reduced operators. In addition, a graphical analysis is provided to visualize the behavior of the solution. This study sheds light on the application of the homotopy perturbation method and the general fractional derivative in solving the generalized Burgers equation, contributing to the field of nonlinear differential equations.

Homotopy perturbation method for fractional-order Burgers-Poisson equation

2010

In this paper, the fractional-order Burgers-Poisson equation is introduced by replacing the first-order time derivative by fractional derivative of order α. Both exact and approximate explicit solutions are obtained by employing homotopy perturbation method. The numerical results reveal that the proposed method is very effective and simple for handling fractional-order differential equations.

A Comparative Study of effective techniques for solving a new model of (1+n) dimensional fractional Burgers’ equation

Delta Journal of Science

The present work offers a new model of (n+1)-dimensional fractional Burgers' equation ((n+1)D-FBE) and presents a comparative numerical study of three efficient semi analytical techniques for solving the ((n+1)D-FBEs). These techniques include the Laplace Adomian decomposition method (LADM), the Laplace variational iteration method (LVIM) and the reduced differential transform method (RDTM). The suggested approaches consider the use of the suitable initial conditions and find the solutions without any discretization or limiting traditions. Furthermore, their solutions are in the form of quickly convergent series with easily calculable terms. Numerical studies of four numerical applications are provided to certify the effectiveness and reliability of the suggested approaches, also to compare their computational effectiveness with each other and with other supplementary methods in the available literature. In addition to explore the properties of the solutions when changing the fractional derivative parameter. Numerical results demonstrate the effectiveness and accuracy of the suggested methods.

Fractional Homotopy Perturbation Transform Method for Solving the Time-Fractional KdV , K ( 2 , 2 ) and Burgers Equations

International Journal of Open Problems in Computer Science and Mathematics, 2015

In this paper, the fractional homotopy perturbation transform method (FHPTM) is employed to obtain approximate analytical solutions of the time-fractional KdV, K(2,2) and Burgers equations. The FHPTM can easily be applied to many problems and is capable of reducing the size of computational work. The fractional derivative is described in the Caputo sense. The results show that the FHPTM is an appropriate method for solving nonlinear fractional derivative equation.

Analysis of Homotopy Perturbation Method for Solving Fractional Order Differential Equations

Mathematics

The analysis of Homotopy Perturbation Method (HPM) for the solution of fractional partial differential equations (FPDEs) is presented. A unified convergence theorem is given. In order to validate the theory, the solution of fractional-order Burger-Poisson (FBP) equation is obtained. Furthermore, this work presents the method to find the solution of FPDEs, while the same partial differential equation (PDE) with ordinary derivative i.e., for α = 1 , is not defined in the given domain. Moreover, HPM is applied to a complicated obstacle boundary value problem (BVP) of fractional order.

A Novel Homotopy Perturbation Method with Applications to Nonlinear Fractional Order KdV and Burger Equation with Exponential-Decay Kernel

Journal of Function Spaces

In this paper, we introduce the Yang transform homotopy perturbation method (YTHPM), which is a novel method. We provide formulae for the Yang transform of Caputo-Fabrizio fractional order derivatives. We derive an algorithm for the solution of Caputo-Fabrizio (CF) fractional order partial differential equation in series form and show its convergence to the exact solution. To demonstrate the novel approach, we include some examples with detailed solutions. We use tables and graphs to compare the exact and approximate solutions.

Analysis of nonlinear fractional partial differential equations with the homotopy analysis method

Communications in Nonlinear Science and Numerical Simulation, 2009

In this article, the homotopy analysis method is applied to solve nonlinear fractional partial differential equations. On the basis of the homotopy analysis method, a scheme is developed to obtain the approximate solution of the fractional KdV, K(2, 2), Burgers, BBM-Burgers, cubic Boussinesq, coupled KdV, and Boussinesq-like B(m, n) equations with initial conditions, which are introduced by replacing some integer-order time derivatives by fractional derivatives. The homotopy analysis method for partial differential equations of integer-order is directly extended to derive explicit and numerical solutions of the fractional partial differential equations. The solutions of the studied models are calculated in the form of convergent series with easily computable components. The results of applying this procedure to the studied cases show the high accuracy and efficiency of the new technique.

A Comparative Study of the Fractional-Order System of Burgers Equations

Symmetry

This paper is related to the fractional view analysis of coupled Burgers equations, using innovative analytical techniques. The fractional analysis of the proposed problems has been done in terms of the Caputo-operator sense. In the current methodologies, first, we applied the Elzaki transform to the targeted problem. The Adomian decomposition method and homotopy perturbation method are then implemented to obtain the series form solution. After applying the inverse transform, the desire analytical solution is achieved. The suggested procedures are verified through specific examples of the fractional Burgers couple systems. The current methods are found to be effective methods having a close resemblance with the actual solutions. The proposed techniques have less computational cost and a higher rate of convergence. The proposed techniques are, therefore, beneficial to solve other systems of fractional-order problems.

Exact and numerical solutions of higher-order fractional partial differential equations: A new analytical method and some applications

Pramana

In this paper, the solution methodology of higher-order linear fractional partial deferential equations (FPDEs) as mentioned in eqs (1) and (2) below in Caputo definition relies on a new analytical method which is called the Laplace-residual power series method (L-RPSM). The main idea of our proposed technique is to convert the original FPDE in Laplace space, and then apply the residual power series method (RPSM) by using the concept of limit to obtain the solution. Some interesting and important numerical test applications are given and discussed to illustrate the procedure of our method, and also to confirm that this method is simple, understandable and very fast for obtaining the exact and approximate solutions (ASs) of FPDEs compared with other methods such as RPSM, variational iteration method (VIM), homotopy perturbation method (HPM) and Adomian decomposition method (ADM). The main advantage of the proposed method is its simplicity in computing the coefficients of terms of series solution by using only the concept of limit at infinity and not as the other well-known analytical method such as, RPSM that need to obtain the fractional derivative (FD) each time to determine the unknown coefficients in series solutions, and VIM, ADM, or HPM that need the integration operators which is difficult in fractional case.

The approximate solutions of time-fractional Burger’s and coupled time-fractional Burger’s equations

DOAJ (DOAJ: Directory of Open Access Journals), 2019

In this paper, we extend the fractional Sumudu decomposition method (FSDM) to solve nonlinear fractional partial differential equations. The time fractional Burger's equations and coupled time fractional Burger's equations with initial conditions are chosen to illustrate our method. As a result, we successfully obtain some available approximate solutions of them. The results reveal that the proposed method is very effective and simple for obtaining approximate solutions of nonlinear fractional partial differential equations. The fractional derivatives are considered in the Caputo sense.