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

Abstract

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.

Basis Functions for a Transient Analysis of Linear Commensurate Fractional-Order Systems

“…Numerical algorithms of the Laplace inversion can also be used for a transient analysis [20,21]. An excellent discussion of the selected algorithms is given in [22,23].…”

Numerical Solutions for Multi-Term Fractional Order Differential Equations with Fractional Taylor Operational Matrix of Fractional Integration

“…There are various numerical methods have been developed for solving FDEs in literature such as predictor-corrector method [25], Laplace transforms [26], Taylor collocation method [27], variational iteration method and homotopy perturbation method [8] (Chapter 6), Adomian decomposition method [28], Tau method [29], inverse Laplace transform [30], Haar wavelet collocation method [31], generalized block pulse operational matrix [32], shifted Legendre-tau method [33], fractional multi-step differential transformed method [34], q-homotopy analysis transform method [35], conformable Laplace transform [36], fractional B-splines collocation method [37], finite difference method [38], homotopy analysis method [39] and so on.…”

Numerical approximation of tempered fractional Sturm‐Liouville problem with application in fractional diffusion equation

“…Fractional calculus provides suitable mathematical modeling for nonlocal processes like anomalous diffusion behavior of living particles, the transportation of electrons in amorphous semiconductors in an electrical field etc., 16 so to understand the behavior of these motions, one needs to solve TFDE numerically as to find analytic solutions of such problems are sometimes not possible. Recently, many authors provided different approaches to find the numerical solution of fractional differential equations and fractional integro‐differential equations including TFDEs which can be found in References 20‐32. To establish a proper numerical schemes for non‐ocal problems such as TFDE, one needs to have its proper spectral theory.…”