On the Approximation of Fractal-Fractional Differential Equations Using Numerical Inverse Laplace Transform Methods

Abstract

Laplace transform is one of the powerful tools for solving differential equations in engineering and other science subjects. Using the Laplace transform for solving differential equations, however, sometimes leads to solutions in the Laplace domain that are not readily invertible to the real domain by analytical means. Thus, we need numerical inversion methods to convert the obtained solution from Laplace domain to a real domain. In this paper, we propose a numerical scheme based on Laplace transform and numerical inverse Laplace transform for the approximate solution of fractal-fractional differential equations with order α, β. Our proposed numerical scheme is based on three main steps. First, we convert the given fractal-fractional differential equation to fractional-differential equation in Riemann-Liouville sense, and then into Caputo sense. Secondly, we transform the fractional differential equation in Caputo sense to an equivalent equation in Laplace space. Then the solution of the transformed equation is obtained in Laplace domain. Finally, the solution is converted into the real domain using numerical inversion of Laplace transform. Three inversion methods are evaluated in this paper, and their convergence is also discussed. Three test problems are used to validate the inversion methods. We demonstrate our results with the help of tables and figures. The obtained results show that Euler's and Talbot's methods performed better than Stehfest's method.

Analysis of Volterra Integrodifferential Equations with the Fractal-Fractional Differential Operator

“…In the third test, we consider problem 3 defned by equation (40) with the exact solution given in equation (49). Te problem is solved using fve NILT schemes.…”

“…Fractal-fractional diferential equations are very important topics nowadays. Some important works on the study of diferential and integrodiferential equations involving fractal-fractional operators can be found in [37][38][39][40][41][42]. In this work, a new class of integrodiferential equations with the Caputo fractal-fractional derivative operator is introduced.…”

Existence, uniqueness, and galerkin shifted Legendre's approximation of time delays integrodifferential models by adapting the Hilfer fractional attitude

“…It is a relatively new field of study that has gained significant attention from researchers because of its wide-ranging applications in different sectors of science and engineering [ [1] , [2] , [3] , [4] , [5] ]. This has resulted in the development of new mathematical tools and techniques that have been used to solve complex problems in physics, engineering, finance, and other fields [ [6] , [7] , [8] , [9] , [10] ]. Studying fractional calculus has yielded the maturation of numerous numerical strategies for solving fractional models, including Riemann, Caputo, and Grunwald-Letnikov's approaches [ [11] , [12] , [13] ].…”

On Orthogonality of Elementary Operators in Normed Spaces

“…It turned out in [43] that their results contained another proof of Ando's results in which every log-hyponormal operator is paranormal. New groups of operators similar to class A operators and paranormal operators were also introduced in [44], [45], [46], [47], [48], [49] and [50]. The author in [51] gave a group of finite operators of the form S + G whereby S ∈ L(Z) and G is compact whereby it was proved that w o (δ S,P ) = c o δ(δ S,P ), where w o (δ S,P ), c o δ(δ S,P ) denote respectively the numerical range of δ S,P and the convex hull of δ(δ S,P ) (the spectrum of δ S,P ) for certain operators S, P ∈ L(Z), δ S,P is the ant operator on L(Z) defined by δ S,P = SZ − ZP Z ∈ L(Z).…”