Numerical Analysis Of Stochastic Fractional Integro-Differential Equations And Stochastic Fractional Integro-differential Equations With Weak Singular Kernel

Integro differential equation is an important branch of modern mathematics.It is an effective tool to solve various practical problems.It is widely used in geometry,mechanics,life science and other fields.With the development of fractional calculus and some random factors that can not be ignored in real life,stochastic fractional Integro differential equations appear in population dynamics,signal processing and other statistical fields.At present,many scholars have studied the suitability of the solutions of these equations,and because it is difficult to obtain the exact solutions of the model in practice,the numerical methods and related properties of solving the equations are also under study.The weak singularity of fractional derivative and the low regularity of random noise bring inevitable difficulties to the specific analysis.Especially,when the integral core is singular,it is more challenging.In this paper,the semi implicit Euler schemes for stochastic fractional Integro differential equations and stochastic fractional Integro differential equations with weak singular kernels are numerically analyzed.In the first part,the convergence of semi implicit Euler methods for nonlinear stochastic fractional Integro differential equations is studied.It is proved that the semi implicit Euler methods have strong first-order convergence;In addition,on the premise that the exact solution satisfies the mean square stability,we study the mean square stability of semi implicit Euler solution of nonlinear stochastic fractional order integral differential equation.Finally,the convergence of the numerical solution is verified by numerical examples.In the second part,firstly,we use the semi implicit Euler method to solve the equation,and prove the convergence of the method under the local Lipschitz condition;Furthermore,we obtain the convergence order of the method under the special global Lipschitz condition,and the semi implicit Euler method is more stable than the Euler method.Finally,a numerical example is used to verify the convergence of the numerical solution.