Mean Square Convergence Of Numerical Methods For Two Classes Of Stochastic Fractional Delay Integro-differential Equations

Fractional delay integro-differential equations(FDIDEs)have many practical applications in scientific fields such as Population kinetic,Electrical engineering,Mechanics and Medicine.It belongs to memory equation models that describe systems with memory and genetic characteristics.But this model does not consider the effects of stochastic factors,this paper considers an equation that is closer to the real situation:Stochastic fractional delay integro-differential equations(SFDIDEs).Since the theoretical solution of such equations is very difficult,this article will study the numerical methods to solve such equations and give it mean square convergence analysis of numerical methods.During the collection of literature,the relevant literature has not yet been found,so researches on such equations are meaningful.This article mainly studies two classes of issues on SFDIDEs:the first classes of problem is semilinear SFDIDEs;the other problem is a nonlinear SFDIDEs.For the first classes of problem,this article considers non-autonomous situation.When the drift and diffusion terms in the equation satisfy the global lipschitz condi-tion,linear growth conditions and H¨older continuous,the unique sufficient condition of the equation solution is given by using the Picard iterative method.Due to the difficulty of theoretical solution,an exponential Euler method is constructed and the order of mean square convergence is proved to be 1/2.The theoretical results are verified by numerical experiments.For the second classes of problem,this article considers the authority of au-tonomy.Under the assumptions that the drift and diffusion terms in the equation satisfy the global Lipschitz condition and linear growth condition,the existence and uniqueness of the solution is proved by using contraction mapping principle.Due to the difficulty of theoretical solution,a numerical method(Euler-Maruyama method)is constructed and the order of mean square convergence is proved to beα-1/2,α∈(1/2,1].The numerical experiments verified the effectiveness of the theo-retical results.