Abstract
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.
Highlights
The need and demand for making more accurate and precise calculations in many industrial and technological fields have caused fractional calculus to become very popular in recent years
It is known that many physical processes in nature can be modeled by using fractional calculus, such as control systems [1,2], material modeling and mechanics [3,4], chaotic systems [5,6], power and energy systems [7,8] and medicine [9,10]
Due to the increasing growth in fractional order models in the various area of science, the analysis and the development of numerical methods for fractional differential equations (FDEs) and partial differential equations with time or space fractional derivatives have become an attractive area of research [11]
Summary
The need and demand for making more accurate and precise calculations in many industrial and technological fields have caused fractional calculus to become very popular in recent years. By the application of the inverse Laplace transform, a solution is presented along a smooth curve, which is evaluated by the quadrature rule. It was shown that this problem has a unique solution, using Laplace transform methods under some strong conditions (in particular, the linearity of the differential equations). Uddin et al [20] extended this work to approximate the solution of fractional order differential equations by an integral representation in the complex plane. McLean and Thomée [21] applied the Laplace transform and a quadrature rule to solve a fractional-order evolution equation in a Banach space framework.
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