Abstract
In the present paper, time fractional partial differential equation is considered, where the fractional derivative is defined in the Caputo sense. Laplace transform method has been applied to obtain an exact solution. The authors solved certain homogeneous and nonhomogeneous time fractional heat equations using integral transform. Transform method is a powerful tool for solving fractional singular Integro - differential equations and PDEs. The result reveals that the transform method is very convenient and effective.
Highlights
Laplace transform can be used to solve certain types of singular integral equations
In fractional PDE problems, we showed that Laplace transforms are - useful when the boundary conditions are time dependent
The main purpose of this work is to develop a method for finding formal solution of certain Fredholm fractional singular integral equations of second kind, analytic solution of the time fractional heat equation and system of partial fractional differential equations, which is a generalization to certain types of problems in the literature .We hope that it will benefit many researchers in the disciplines of applied mathematics, mathematical physics and engineering
Summary
Laplace transform can be used to solve certain types of singular integral equations. The mathematical formulation of physical phenomena often involves Cauchy type, or more severe, singular integral equations. There are many applications in many important fields, like fracture mechanics, elastic contact problems, the theory of porous filtering contain integral and integro - differential equation with singular kernel. The following class of Fredholm singular integral equation of second kind of the following type φ(x) = f (x) + λ. Solution: Laplace-transform of the above integral equation, leads to the following φ(x) cosh x 1 − λ2
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