Spectral Galerkin Method For Volterra Integro-Differential Equations

This paper mainly discusses the numerical methods for Volterra integrodifferential equations of the second kind with smooth kernel function.The integral item yields the memorial property of the problem.How to resolve problems of this kind more efficiently and rapidly plays a fundamental role in designing the corresponding algorithms.In recent years,lots of scholars prefer to use the spectral method,which has the property of spectral convergence accuracy,to deal with equations of this kind.In this paper we use spectral Galerkin method in dealing with Volterra integro-differential equation and analyze its convergence property.First and foremost,we introduce the spectral Jacobi-Galerkin method.Then we prove the exponential convergence property of this method in the sense of L_w~2 and L~∞theoretically.Next we introduce the pseudo-spectral Jacobi-Galerkin method.The idea of this method is that we could use the discrete inner product with weight function to simulate the inner product with weight function.The exponential convergence property of this method in the sense of L_w~2 and L~∞is also obtained theoretically.On the other hand,the numerical results show that these methods not only deal with problems rapidly,but also attain the spectral convergence accuracy indeed.