scholarly journals Numerical Inverse Laplace Transform for Solving a Class of Fractional Differential Equations

Symmetry ◽  
2019 ◽  
Vol 11 (4) ◽  
pp. 530 ◽  
Author(s):  
Dimple Rani ◽  
Vinod Mishra ◽  
Carlo Cattani
Keyword(s):  
Differential Equations ◽  
Laplace Transform ◽  
Fractional Differential Equations ◽  
Operational Matrix ◽  
Adomian Decomposition Method ◽  
Inverse Laplace Transform ◽  
Convergence Criterion ◽  
Numerical Inverse Laplace Transform ◽  
Adomian Decomposition ◽  
Fractional Differential

This paper discusses the applications of numerical inversion of the Laplace transform method based on the Bernstein operational matrix to find the solution to a class of fractional differential equations. By the use of Laplace transform, fractional differential equations are firstly converted to system of algebraic equations then the numerical inverse of a Laplace transform is adopted to find the unknown function in the equation by expanding it in a Bernstein series. The advantages and computational implications of the proposed technique are discussed and verified in some numerical examples by comparing the results with some existing methods. We have also combined our technique to the standard Laplace Adomian decomposition method for solving nonlinear fractional order differential equations. The method is given with error estimation and convergence criterion that exclude the validity of our method.

scholarly journals Solving partial fractional differential equations by using the Laguerre wavelet-Adomian method

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Nasser Aghazadeh ◽  
Amir Mohammadi ◽  
Ghader Ahmadnezhad ◽  
Shahram Rezapour
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Fractional Integration ◽  
Operational Matrix ◽  
Adomian Decomposition Method ◽  
Nonlinear Method ◽  
Adomian Decomposition ◽  
Laguerre Wavelets ◽  
Fractional Differential ◽  
Partial Fractional Differential Equations

AbstractBy using a nonlinear method, we try to solve partial fractional differential equations. In this way, we construct the Laguerre wavelets operational matrix of fractional integration. The method is proposed by utilizing Laguerre wavelets in conjunction with the Adomian decomposition method. We present the procedure of implementation and convergence analysis for the method. This method is tested on fractional Fisher’s equation and the singular fractional Emden–Fowler equation. We compare the results produced by the present method with some well-known results.

scholarly journals Computational method based on Bernstein operational matrices for multi-order fractional differential equations

Filomat ◽  
2014 ◽  
Vol 28 (3) ◽  
pp. 591-601 ◽  
Author(s):  
Davood Rostamy ◽  
Hossein Jafari ◽  
Mohsen Alipour ◽  
Chaudry Khalique
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Operational Matrix ◽  
Adomian Decomposition Method ◽  
Computational Method ◽  
Operational Matrices ◽  
Algebraic Equations ◽  
Transform Method ◽  
Adomian Decomposition ◽  
Fractional Differential

In this paper, the Bernstein operational matrices are used to obtain solutions of multi-order fractional differential equations. In this regard we present a theorem which can reduce the nonlinear fractional differential equations to a system of algebraic equations. The fractional derivative considered here is in the Caputo sense. Finally, we give several examples by using the proposed method. These results are then compared with the results obtained by using Adomian decomposition method, differential transform method and the generalized block pulse operational matrix method. We conclude that our results compare well with the results of other methods and the efficiency and accuracy of the proposed method is very good.

scholarly journals Sinc Based Inverse Laplace Transforms, Mittag-Leffler Functions and Their Approximation for Fractional Calculus

2021 ◽  
Vol 5 (2) ◽  
pp. 43
Author(s):  
Gerd Baumann
Keyword(s):  
Differential Equations ◽  
Fractional Calculus ◽  
Laplace Transform ◽  
Fractional Differential Equations ◽  
Numerical Approximation ◽  
Laplace Transforms ◽  
Inverse Laplace Transform ◽  
Exact Methods ◽  
Sinc Methods ◽  
Fractional Differential

We shall discuss three methods of inverse Laplace transforms. A Sinc-Thiele approximation, a pure Sinc, and a Sinc-Gaussian based method. The two last Sinc related methods are exact methods of inverse Laplace transforms which allow us a numerical approximation using Sinc methods. The inverse Laplace transform converges exponentially and does not use Bromwich contours for computations. We apply the three methods to Mittag-Leffler functions incorporating one, two, and three parameters. The three parameter Mittag-Leffler function represents Prabhakar’s function. The exact Sinc methods are used to solve fractional differential equations of constant and variable differentiation order.

Sign in / Sign up

Export Citation Format

Share Document