A class of Runge–Kutta methods for nonlinear Volterra integral equations of the second kind with singular kernels

Abstract

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<α≤1 $0<\alpha\leq1$.

Keywords

• Fractional order Riccati differential equations
• Runge–Kutta methods
• Subtraction of the singularity
• Nonlinear Volterra integral equations of the second kind
• Caputo fractional derivative