The Numerical Method For Stachastic Linear Pantograph Equations

As a significant kind of unbounded delay differential equations,stochastic pantograph differential equations can be considered as deterministic pantograph differential equations with random noise.When the numerical method with a constant step-size is applied to the pantograph equations,the most difficult problem is the limited computer memory.In this thesis,motivated by the numerical method processing deterministic pantograph differential equations,we transform stochastic pantograph differential equations into stochastic delay differential equations with a constant delay.Then we apply four numerical methods with fixed step-size to transformed equations.Moreover,we discuss the convergence and stability properties for the first three numerical methods.Eventually,we analysize the stability properties for the last numerical method.By the technique of the limiting equations to deterministic problems,we get the conditions of mean-square stability for the linear q-methods methods and the split-step backward Euler methods.We investigate explicit and implicit Euler methods to the transformed equations.We prove that both methods preserve the strong convergence order 0.5.In addition,we present that the explicit Euler method is unstable and achieve the sufficient and necessary condition of the mean-square stability for the implicit Euler method.In general,we consider linear ?-methods to the transformed equations.Also we attest that the numerical methods are of strong convergence order 0.5.And we demonstrate the sufficient condition of the mean-square stability by utilizing matrix analysis.In the end,we consider the split-step backward Euler methods and discuss the sufficent condition of the mean-square stability.The numerical results verify the reliability of all the conclusions eventually.