Abstract
This paper aims to obtain an approximate solution for fractional order Riccati differential equations (FRDEs). FRDEs are equivalent to nonlinear Volterra integral equations of the second kind. In order to solve nonlinear Volterra integral equations of the second kind, a class of Runge–Kutta methods has been applied. Runge–Kutta methods have been implemented to solve nonsingular integral equations. In this work Volterra integral equations are singular. The singularity by a suitable subtraction technique will be weakened; then, this method will be applied to gain an approximate solution. Fractional derivatives are defined in the Caputo form of order 0<alphaleq1.
Highlights
A generalization of the classical Newtonian calculus is called fractional calculus and appears in many natural phenomena such as physical, chemical, sociological, biological, and economical processes
First, fractional order Riccati differential equations (FRDEs) will be converted into nonlinear Volterra integral equations of the second kind and we will look for the solution by a class of Runge–Kutta methods
The rest of this paper is organized as follows: we present a brief review of a class of Runge–Kutta methods for nonlinear Volterra integral equations of the second kind
Summary
A generalization of the classical Newtonian calculus is called fractional calculus and appears in many natural phenomena such as physical, chemical, sociological, biological, and economical processes. First, FRDEs will be converted into nonlinear Volterra integral equations of the second kind and we will look for the solution by a class of Runge–Kutta methods. In [33], tau approximation method has been applied to solve weakly singular Volterra-Hammerstein integral equations. An efficient approach based on combining the radial basis functions and discrete collocation method has been implemented to solve nonlinear Volterra integral equations of the second kind in [39]. The rest of this paper is organized as follows: we present a brief review of a class of Runge–Kutta methods for nonlinear Volterra integral equations of the second kind. All calculations have been done using Maple on a computer with Intel Core i5-2430M CPU at 2.400 GHz, 4.00 GB of RAM and 64-bit operating system (Windows 7)
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