Euler-Maruyama Methods For Fractional Non-Linear Stochastic Integro-Differential Equations

In recent years,fractional differential equations as an important branch of differential equation have attracted people’s great attention.And they develop rapidly in many disciplines such as mathematics,fluid mechanics,fractional control system and physical engineering related to fractal dimension.However,it is difficult to obtain the analytical solution of fractional nonlinear differential equations in general.Even if some solutions of equations can be obtained,their calculation is very complicated due to the influence of special functions.Therefore,many scholars are interested in constructing numerical solutions of fractional nonlinear differential equations.It is well known that differential equations are often affected by noise perturbation when solving the problems of various systems,so it becomes more and more important to study stochastic differential equation models with noise terms.Combining the advantages of fractional differential equations and stochastic differential equations,researchers in various fields pay more attention to a novel model that is fractional stochastic differential equations with noise terms.In this thesis,we will construct and analyze Euler-Maruyama(EM)numerical methods for the Riemann-Liouville fractional nonlinear stochastic integrodifferential equations with the initial value condition.And then the strong convergence of the EM methods are proved.In addition,on the based on the study of constant order fractional stochastic integro-differential equations,more and more scholars take variableorder fractional stochastic integro-differential equations as a new research direction.Compared with the traditional constant order fractional stochastic integro-differential equation,the variable-order fractional stochastic integro-differential equation has variable order,so it is more difficult to obtain the anlytical solution of the equations.In the second part of this thesis,we extend the Euler-Maruyama method to variable-order fractional nonlinear stochastic integro-differential equations,obtain its numerical solutions,and prove the strong convergence of the EM method.The specific works of this thesis are organized as follows:In Chapter 2,we introduce the Riemann-Liouville fractional nonlinear stochastic integro-differential equations with initial value conditions,thenconstruct the corresponding Euler-Maruyama numerical method,and strictly prove the stability and strong convergence of this EM method.Finally,three numerical examples are provided to illustrate the effectiveness of the EM method and verify the theoretical results.In Chapter 3,a derivative term is added to the Riemann-Liouville fractional nonlinear stochastic integro-differential equation studied in Chapter 2.And then we construct the Euler-Maruyama method,and prove that the strong convergence order is(1-α)∧0.5.In Chapter 4,we extend Euler-Maruyama method to a class of variable-order fractional nonlinear stochastic integro-differential equations on the basis of the study of constant-order fractional nonlinear stochastic integro-differential equations,construct the corresponding numerical method,and strictly prove the strong convergence of the EM method.Finally,the proposed EM method is applied to a practical example to support the theoretical analysis.