HOMOTOPY PERTURBATION TRANSFORM METHOD FOR SOLVING SYSTEMS OF NONLINEAR PARTIAL FRACTIONAL DIFFERENTIAL EQUATIONS

2021 ◽  
Vol 21 (2) ◽  
pp. 355-364
Author(s):  
ABDELKADER KEHAILI ◽  
ABDELKADER BENALI ◽  
ALI HAKEM
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Fractional Differential Equations ◽  
Efficient Method ◽  
Fractional Partial Differential Equations ◽  
Transform Method ◽  
Homotopy Perturbation ◽  
Computational Work ◽  
Fractional Differential ◽  
Partial Fractional Differential Equations

In this paper, we apply an efficient method called the Homotopy perturbation transform method (HPTM) to solve systems of nonlinear fractional partial differential equations. The HPTM can easily be applied to many problems and is capable of reducing the size of computational work.

scholarly journals The Use of Sumudu Transform for Solving Certain Nonlinear Fractional Heat-Like Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-12 ◽  
Author(s):  
Abdon Atangana ◽  
Adem Kılıçman
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Variable Coefficients ◽  
Fractional Partial Differential Equations ◽  
Science And Engineering ◽  
Transform Method ◽  
Homotopy Perturbation ◽  
Sumudu Transform ◽  
Fractional Differential ◽  
Linear And Nonlinear

We make use of the properties of the Sumudu transform to solve nonlinear fractional partial differential equations describing heat-like equation with variable coefficients. The method, namely, homotopy perturbation Sumudu transform method, is the combination of the Sumudu transform and the HPM using He’s polynomials. This method is very powerful, and professional techniques for solving different kinds of linear and nonlinear fractional differential equations arising in different fields of science and engineering.

Numerical and analytical solutions of nonlinear differential equations involving fractional operators with power and Mittag-Leffler kernel

2018 ◽  
Vol 13 (1) ◽  
pp. 13 ◽  
Author(s):  
H. Yépez-Martínez ◽  
J.F. Gómez-Aguilar
Keyword(s):  
Differential Equations ◽  
Fractional Derivative ◽  
Fractional Differential Equations ◽  
Fractional Derivatives ◽  
Nonlinear Differential Equations ◽  
Fractional Operators ◽  
Transform Method ◽  
Homotopy Perturbation ◽  
Fractional Differential ◽  
Derivative Order

Analytical and numerical simulations of nonlinear fractional differential equations are obtained with the application of the homotopy perturbation transform method and the fractional Adams-Bashforth-Moulton method. Fractional derivatives with non singular Mittag-Leffler function in Liouville-Caputo sense and the fractional derivative of Liouville-Caputo type are considered. Some examples have been presented in order to compare the results obtained, classical behaviors are recovered when the derivative order is 1.

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 Homotopy Perturbation Transform method for solving the partial and the time-fractional differential equations with variable coefficients

2020 ◽  
Vol 26 (1) ◽  
pp. 35-55
Author(s):  
Abdelkader Kehaili ◽  
Ali Hakem ◽  
Abdelkader Benali
Keyword(s):  
Differential Equations ◽  
Laplace Transform ◽  
Fractional Derivative ◽  
Exact Solutions ◽  
Fractional Differential Equations ◽  
Variable Coefficients ◽  
Transform Method ◽  
Homotopy Perturbation ◽  
Fractional Differential ◽  
Time Fractional Differential Equations

In this paper, we present the exact solutions of the Parabolic-like equations and Hyperbolic-like equations with variable coefficients, by using Homotopy perturbation transform method (HPTM). Finally, we extend the results to the time-fractional differential equations. Keywords: Caputo’s fractional derivative, fractional differential equations, homotopy perturbation transform method, hyperbolic-like equation, Laplace transform, parabolic-like equation.

scholarly journals A note on Riccati-Bernoulli Sub-ODE method combined with complex transform method applied to fractional differential equations

2018 ◽  
Vol 7 (4) ◽  
pp. 279-285 ◽  
Author(s):  
Mahmoud A.E. Abdelrahman
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Dimensional Space ◽  
Nonlinear Ordinary Differential Equation ◽  
Transform Method ◽  
Fractional Equations ◽  
Dimensional Space Time ◽  
Ode Method ◽  
Fractional Differential ◽  
Partial Fractional Differential Equations

AbstractIn this paper, the fractional derivatives in the sense of modified Riemann–Liouville and the Riccati-Bernoulli Sub-ODE method are used to construct exact solutions for some nonlinear partial fractional differential equations via the nonlinear fractional Zoomeron equation and the (3 + 1) dimensional space-time fractional mKDV-ZK 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 Riccati-Bernoulli Sub-ODE method appears to be easier and more convenient by means of a symbolic computation system.

scholarly journals A new modified homotopy perturbation method for fractional partial differential equations with proportional delay

2020 ◽  
Vol 19 ◽  
pp. 58-73
Author(s):  
Ahmad. A. H. Mtawal ◽  
Sameehah. R. Alkaleeli
Keyword(s):  
Exact Solution ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Perturbation Method ◽  
Homotopy Perturbation Method ◽  
Proportional Delay ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Transform Method ◽  
Homotopy Perturbation

In this paper, we suggest and analyze a technique by combining the Shehu transform method and the homotopy perturbation method. This method is called the Shehu transform homotopy method (STHM). This method is used to solve the time-fractional partial differential equations (TFPDEs) with proportional delay. The fractional derivative is described in Caputo's sense. The solutions proposed in the series converge rapidly to the exact solution. Some examples are solved to show the STHM is easy to apply.

scholarly journals A Hybrid Natural Transform Homotopy Perturbation Method for Solving Fractional Partial Differential Equations

2016 ◽  
Vol 2016 ◽  
pp. 1-7 ◽  
Author(s):  
Shehu Maitama
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Perturbation Method ◽  
Analytical Method ◽  
Homotopy Perturbation Method ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Transform Method ◽  
Homotopy Perturbation ◽  
Linear And Nonlinear

A hybrid analytical method for solving linear and nonlinear fractional partial differential equations is presented. The proposed analytical approach is an elegant combination of the Natural Transform Method (NTM) and a well-known method, Homotopy Perturbation Method (HPM). In this analytical method, the fractional derivative is computed in Caputo sense and the nonlinear term is calculated using He’s polynomial. The proposed analytical method reduces the computational size and avoids round-off errors. Exact solution of linear and nonlinear fractional partial differential equations is successfully obtained using the analytical method.

An efficient hybrid computational technique for solving nonlinear local fractional partial differential equations arising in fractal media

2018 ◽  
Vol 7 (3) ◽  
pp. 229-235 ◽  
Author(s):  
Amit Prakash ◽  
Hardish Kaur
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Planck Equation ◽  
Computational Technique ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Transform Method ◽  
Homotopy Perturbation ◽  
Fractal Media ◽  
Sumudu Transform

AbstractIn present work, nonlinear fractional partial differential equations namely transport equation and Fokker-Planck equation involving local fractional differential operators, are investigated by means of the local fractional homotopy perturbation Sumudu transform method. The proposed method is a coupling of homotopy perturbation method with local fractional Sumudu transform and is used to describe the non-differentiable problems. Numerical simulation results are projected to show the efficiency of the proposed technique.

scholarly journals NUMERICAL SOLUTIONS OF NONLINEAR FRACTIONAL DIFFERENTIAL EQUATIONS VIA LAPLACE TRANSFORM

2021 ◽  
pp. 249
Author(s):  
Süleyman Çetinkaya ◽  
Ali Demir
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Iterative Method ◽  
Laplace Transform ◽  
Fractional Differential Equations ◽  
Numerical Solutions ◽  
Fractional Partial Differential Equations ◽  
Nonlinear Fractional Differential Equations ◽  
Time Space ◽  
Fractional Differential

In this study, solutions of time-space fractional partial differential equations(FPDEs) are obtained by utilizing the Shehu transform iterative method. The utilityof the technique is shown by getting numerical solutions to a large number of FPDEs.

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