Abstract
With fractional differential equations (FDEs) rising in popularity and methods for solving them still being developed, approximations to solutions of fractional initial value problems (IVPs) have great applications in related fields. This paper proves an extension of Picard’s Iterative Existence and Uniqueness Theorem to Caputo fractional ordinary differential equations, when the nonhomogeneous term satisfies the usual Lipschitz’s condition. As an application of our method, we have provided several numerical examples.
Highlights
Fractional differential equations (FDEs) are seeing a rapid rise in utility including applications in engineering, physics, economics, and chemistry [1,2,3,4,5,6,7]
Our goal is to develop an iterative method which converges to the unique solution of the initial value problems (IVPs) (1)
In order to study blow up results for fractional reaction diffusion equation, one easy approach is to show that the corresponding ordinary Caputo FDE blows up at some time t
Summary
Fractional differential equations (FDEs) are seeing a rapid rise in utility including applications in engineering, physics, economics, and chemistry [1,2,3,4,5,6,7]. Analytic or numerical computation of the solution of fractional dynamic equations has been challenging This is mainly because some of the basic properties enjoyed by integer derivatives such as the product rule and separation of variables are not available. In [9], Picard’s method has been developed to solve fractional initial value problems (IVPs) when the forcing term satisfies a Lipschitz condition where the Lipschitz’s constant is time dependent and these conditions are not global. We develop Picard’s method for Caputo FDEs with initial conditions, when the nonlinear term satisfies a time independent (global) Lipschitz’s condition.
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