Analysis Of Numerical Solution Of Volterra Integral Equation Collocation Method

Integral equations are an important branch of mathematics and Volterra integral equation(VIE)plays an important role in integral equations.The study of VIE relates to many fields,such as physics,biology and chemistry.The common heat conduction model,Lighthill model and isochronous pendulum problem are modeled by VIE.However,for general VIE,the numerical solution is difficult to be obtained,so solving the numerical solutions of VIE had raised much attention.In recent years,smooth transformation and collocation method have been applied to VIE by many scholars,and some results have been achieved.In this paper,the smooth transformation and collocation method are studied to solve the third kind linear VIE and the cordial Volterra integral equation(CVIE).The regularity,existence and uniqueness of exact solutions and convergence of numerical solutions,etc.are studied.Firstly,we apply the smoothing transformation to solve the third kind linear VIE whose the exact solution smoothness is low.Some of the conditions satisfied by the smooth transformation function are given.The compactness of the equation integral operator of the smooth transformation is discussed.According to the different conditions of ? in the equation,the general form of smooth transformation function is given and the regularity of the exact solution of the transformed equation is discussed.Secondly,we apply collocation method to the transformed equation and discuss the solvability and study the convergence of the numerical solution of the collocation equation.By the inverse transformation of the smooth transform function,we discuss the convergence order of the numerical solution of the pre-transformation equation and the transformed equation.Finally,on the basis of discussion about the third kind linear VIE,we discuss a kind of multi-solution CVIE.We add an extra condition to the original equation and study the existence and uniqueness of the exact solution of the equation.We apply the collocation method on the modified mesh to the equation before and after the transformation,and discuss the solvability of the collocation equation.