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.
Highlights
Over the years, researchers have been attracted to study the scientific problems modelled in fractional differential equations due to their constant appearance in the different disciplines of mathematical sciences and engineering such as fluid mechanics, viscoelasticity, mathematical physics, mathematical biology, system identification, control theory, electrochemistry and signal processing [1,2,3,4,5,6]
Bernoulli wavlet operational matrix of fractional order integration has been derived to approximate the numerical solution of fractional differential equations in [13]
Enormous efforts and advances have been conducted to obtain the numerical solutions of fractional differential equations
Summary
Researchers have been attracted to study the scientific problems modelled in fractional differential equations due to their constant appearance in the different disciplines of mathematical sciences and engineering such as fluid mechanics, viscoelasticity, mathematical physics, mathematical biology, system identification, control theory, electrochemistry and signal processing [1,2,3,4,5,6]. Several analytical and numerical techniques have been developed to solve such kind of equations in the literature. Among these methods, Li and Sun [7] derived the generalized block pulse operational matrix to find the solution of fractional differential equations in terms of block pulse function. Doha et al [10] used the shifted Jacobi operational matrix of fractional derivatives applied together with spectral tau-method for solving fractional differential equations. Kazem et al [11] constructed an orthogonal fractional order Legendre function based on Legendre polynomials to solve fractional differential equations. Bernoulli wavlet operational matrix of fractional order integration has been derived to approximate the numerical solution of fractional differential equations in [13]. Albadarneh et al [15] adopted the fractional finite difference method for solving linear and nonlinear fractional differential equations
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