Numerical Methods For Two Classes Of Volterra Integro-differential Equations With Fractional Order

The emergence of fractional calculus arises new questions in fundamental physics, which provides great interest for the mathematicians and physicists in the theory of fractional calculus. The fractional calculus has been considered to be the valuable tool, which can be described as dynamical behavior of real life phenomena more accurately. For instance, the nonlinear oscillation of earthquake can be well modeled with fractional derivatives. We can find the numerous applications of fractional differential equations in control theory. Usually, it is difficult to obtain analytic solutions for those kinds of equations. So solving the numerical solution becomes very important and has practical application value.In recent 40 years, many numerical methods have been proposed by scholars for solving fractional integro-differential equations, such as Adomain decomposition method, finite difference method, collocation method, wavelet method and so on. However, the theoretical system is not perfect. Thus, this paper studies the numerical algorithm for two classes of Volterra integro-differential equations with fractional order, including nonlinear Volterra integro-differential equations and Volterra’s population growth model with fractional order.As a preliminary knowledge, the first part introduces the basic definition of fractional order and the theory of reproducing kernel.The third part of this paper explains the model of the nonlinear Volterra integro-differential equations at first. Then it is presented for solving the nonlinear Volterra integro-differential equations with fractional order. Finally, we verify the superiority of the method by concrete numerical experiments.The fourth part of this paper mainly studies the numerical algorithm for the Volterra’s population growth model with fractional order. The modified reproducing kernel method to be used to solve this kinds of equations and builds an integrated theoretical system. Finally, the results of numerical experiments show that the presented algorithm is very effective and easy to operate.