A Numerical Method for Solving a Class of Nonlinear Second Order Fractional Volterra Integro-Differntial Type of Singularly Perturbed Problems
In this paper, we study a class of fractional nonlinear second order Volterra integro-differential type of singularly perturbed problems with fractional order. We divide the problem into two subproblems. The first subproblems is the reduced problem when = 0. The second subproblems is fractional Volterra integro-differential problem. We use the finite difference method to solve the first problem and the reproducing kernel method to solve the second problem. In addition, we use the pade'
... the pade' approximation. The results show that the proposed analytical method can achieve excellent results in predicting the solutions of such problems. Theoretical results are presented. Numerical results are presented to show the efficiency of the proposed method. Keywords: singularly perturbed volterra integro-differntial; caputo fractional derivative; nonlinear boundary value problem Introduction Volterra integral equations are considered as a type of integral equations. In 1913, Volterra published the first book that talked about Volterra integral equations. In 1884, Volterra began working on integral equations, but his important study was in 1896. However, the name Volterra integral equation was first called by Lalesco in 1908. Volterra integral equations have many applications in science and engineering such as elasticity, semi-conductors, scattering theory, seismology, heat conduction, metallurgy, fluid flow, chemical reactions, population dynamics, and spread of epidemics  . Volterra integral equations have growlingly been recognized as useful tools for problems in science and engineering. In , they proposed and examined a spectral Jacobi-collocation approximation for fractional order integro-differntial equations. Ray et al.  , used the Legendre wavelet method to find the solutions for a system of nonlinear Volterra integro-differntial equations. In , they used Lagurre polynomials and the collocation method to solve the pantograph-type Volterra integro-differntial equations under some initial conditions. Yang et al.  , discussed the blow-up of Volterra integro-differntial equations with a dissipative linear term to show the differences of the solutions. In , they solved a non-linear system of higher order Volterra integro-differntial equations using the Single Term Walsh Series (STWS) method. Also in , they solved the fractional Fredholem-Volterra integro-differntial equations by the fractional-order functions based on the Bernoulli polynomials. We also indicate the interested reader to          for more research works on Volterra integro-differntial equations. In 1904, A German physicist called Ludwig Prandtl was revolutionized fluid dynamics. He noted that the influence of friction is experienced only very near an object moving through a fluid. In  , he presented the idea of the boundary layer and its significance for drag and streamlining. In his