The Numerical Solutions of Nonlinear Time-Fractional Differential Equations by LMADM

This paper presents a numerical scheme for solving nonlinear time-fractional differential equations in the sense of Caputo. This method relies on the Laplace transform together with the modified Adomian method (LMADM), compared with the Laplace transform combined with the standard Adomian Method (LADM). Furthermore, for the comparison purpose, we applied LMADM and LADM for solving nonlinear time-fractional differential equations to identify the differences and similarities. Finally, we provided two examples regarding the nonlinear time-fractional differential equations, which showed that the convergence of the current scheme results in high accuracy and small frequency to solve this type of equations.

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STABILITY-PRESERVING HIGH-ORDER METHODS FOR MULTITERM FRACTIONAL DIFFERENTIAL EQUATIONS

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127412500733 ◽

2012 ◽

Vol 22 (04) ◽

pp. 1250073 ◽

Cited By ~ 12

Author(s):

ROBERTO GARRAPPA

Keyword(s):

Differential Equations ◽

Laplace Transform ◽

Fractional Differential Equations ◽

Numerical Experiments ◽

Numerical Approximation ◽

Numerical Solutions ◽

Good Stability ◽

Stability Properties ◽

The Laplace Transform ◽

Fractional Differential

The numerical approximation of linear multiterm fractional differential equations is investigated. Convolution quadratures based on Runge–Kutta methods together with formulas for the efficient inversion of the Laplace transform are considered to provide highly accurate numerical solutions. Implementation issues are discussed and good stability properties are shown. The effectiveness of the algorithm is analyzed by means of some numerical experiments.

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Numerical Solutions for a Kind of Time-Fractional Differential Equations

Lecture Notes in Electrical Engineering - Proceedings of the 9th International Symposium on Linear Drives for Industry Applications, Volume 1 ◽

10.1007/978-3-642-40618-8_41 ◽

2013 ◽

pp. 319-325

Author(s):

Kaili Wang ◽

Yue Long ◽

Dan Li

Keyword(s):

Differential Equations ◽

Fractional Differential Equations ◽

Numerical Solutions ◽

Fractional Differential ◽

Time Fractional Differential Equations

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Numerical Solutions of Fractional Differential Equations by Using Laplace Transformation Method and Quadrature Rule

Fractal and Fractional ◽

10.3390/fractalfract5030111 ◽

2021 ◽

Vol 5 (3) ◽

pp. 111

Author(s):

Samaneh Soradi-Zeid ◽

Mehdi Mesrizadeh ◽

Carlo Cattani

Keyword(s):

Differential Equations ◽

Laplace Transform ◽

Numerical Method ◽

Fractional Differential Equations ◽

Numerical Scheme ◽

Numerical Solutions ◽

Quadrature Rule ◽

Differential Problem ◽

Transform Method ◽

Fractional Differential

This paper introduces an efficient numerical scheme for solving a significant class of fractional differential equations. The major contributions made in this paper apply a direct approach based on a combination of time discretization and the Laplace transform method to transcribe the fractional differential problem under study into a dynamic linear equations system. The resulting problem is then solved by employing the numerical method of the quadrature rule, which is also a well-developed numerical method. The present numerical scheme, which is based on the numerical inversion of Laplace transform and equal-width quadrature rule is robust and efficient. Some numerical experiments are carried out to evaluate the performance and effectiveness of the suggested framework.

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

Global Journal of Pure and Applied Sciences ◽

10.4314/gjpas.v26i1.6 ◽

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.

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On Hyers–Ulam stability of fractional differential equations with Prabhakar derivatives

Analysis ◽

10.1515/anly-2017-0029 ◽

2018 ◽

Vol 38 (1) ◽

pp. 37-46 ◽

Cited By ~ 3

Author(s):

Mohammad Hossein Derakhshan ◽

Alireza Ansari

Keyword(s):

Differential Equations ◽

Laplace Transform ◽

Fractional Derivative ◽

Fractional Differential Equations ◽

Ulam Stability ◽

Nonlinear Fractional Differential Equations ◽

The Laplace Transform ◽

Fractional Differential ◽

Linear And Nonlinear ◽

The Stability

AbstractIn this article, we study the Hyers–Ulam stability of the linear and nonlinear fractional differential equations with the Prabhakar derivative. By using the Laplace transform, we show that the introduced fractional differential equations with the Prabhakar fractional derivative is Hyers–Ulam stable. The results generalize the stability of ordinary and fractional differential equations in the Riemann–Liouville sense.

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NUMERICAL SOLUTIONS OF NONLINEAR FRACTIONAL DIFFERENTIAL EQUATIONS VIA LAPLACE TRANSFORM

Facta Universitatis Series Mathematics and Informatics ◽

10.22190/fumi200428019c ◽

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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A New Method for Laplace Transforms of Multi-Term Fractional Differential Equations of the Caputo Type

Journal of Computational and Nonlinear Dynamics ◽

10.1115/1.4051336 ◽

2021 ◽

Author(s):

Masataka Fukunaga

Keyword(s):

Differential Equations ◽

Laplace Transform ◽

Fractional Differential Equations ◽

Initial Conditions ◽

Expansion Method ◽

Laplace Transforms ◽

Transform Method ◽

Change Of Variables ◽

The Laplace Transform ◽

Fractional Differential

Abstract The Laplace transform method is one of the powerful tools in studying the frac- tional differential equations (FDEs). In this paper, it is shown that the Heaviside expansion method for integer order differential equations is also applicable to the Laplace transforms of multi-term Caputo fractional differential equations (FDEs) of zero initial conditions if the orders of Caputo derivatives are integer multiples of a common real number. The particular solution of a linear multi-term Caputo FDE is obtained by its Laplace transform and the Heaviside expansion method. A Caputo FDE of non zero initial conditions is transformed to an Caputo FDE of zero initial conditions by an appropriate change of variables. In the latter, the terms originated from the initial conditions appear as nonhomogeneous terms. Thus, the Caputo FDE of nonzero initial conditions is obtained as the particular solutions to the equivalent Caputo FDE of zero initial conditions. The solutions of a linear multi-term Caputo FDEs of nonzero initial conditions are expressed through the two parameter Mittag-Leffler functions.

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