The Application Of Collocation Method For Sobolev Equations

Sobolev equations are an important type of mathematical physics equations,which has been widely used in the research of physical problems such as the heat conduction problems,wave notionand shear of second order fluid.The orthogonal collocation method has many advantages,the calcultions are much easier,do not involve the numerical intergration and have the higher-order convergence.Thus,the orthogonal collocation method has been widely used in many fields such as physical and chemical engineering.This paper mainly discusses the application of orthogonal collocation method for Sobolev equations.For the different forms of Sobolev equations such as constant coefficient,variable coefficient and nonlinearity,various orthogonal collocation schemes are established,including the semi-discrete orthogonal collocation scheme,full discrete orthogonal collocation scheme and characteristical collocation scheme.The numerical solutions and the solutions of discrete Galerkin method are equivalent,then the existence and uniqueness of the numerical solutions are proved.The error estimations of the optimal order are obtained by introducing the techniques of intermediate variables.Finally,the convergence results of the theoretical analysis are confirmed by different numerical examples,which show the efficiency of the collocation method.