Numerical Schemes For Multi-Term Fractional Stochastic Integro-Differential Equations

In recent years,fractional stochastic differential equations have become one of the hot studies in mathematical field at home and abroad.On the one hand,stochastic differential equations play an irreplaceable role in capturing noise disturbances that cannot be ignored in the environment.On the other hand,fractional differential equations are an old and fresh field of study,although the development has been very slow for a long time due to the lack of modeling background support,but in recent years,people have paid attention to the fractional differential equations can describe the abnormal relaxation process in complex systems,so the fractional differential equations have also been more and more concerned by scholars.As the study of fractional equations deepens,it has been noted that when using fractional equations to simulate dynamic problems,the order of fractional orders is a time-dependent variable and non-quantitative.However,the solutions to such equations are difficult to obtain,so we work on numerical solutions to such equations.Therefore,this thesis studies a class of nonlinear multi-term fractional stochastic integro-differential equations(MFSIDE)with weak singular kernel and its variable fractional equation forms,and mainly studies MFSIDE's EulerMaruyama(EM)numerical scheme.The specific works of this thesis are organized as follows:In the second chapter,first uses the Fubini theorem in the stochastic integral differential equation to equivalently transform the multi-fractional integral differential equation MFSIDE into the stochastic Volterra integral equation(SVIE).The EM numerical scheme of the integral equation is then given,and the existence,uniqueness and stability of the MFSIDE solution are derived in detail using the EM numerical scheme.In the third chapter,first the improved EM scheme of MFSIDE is constructed,and then the strong convergence of the improved EM scheme in the sense of the mean square is proved.And it is found that the order of strong convergence is ?m-?m-1,where ?i is the order of fractional derivative satisfying 0