Fast Singularity Preserving Collocation Method For The Second Kind Of Volterra Integral Equations

Nowadays,more and more models of mathematical problems can be transformed into problems of solving integral equations.Volterra type integral equation plays an important role in integral problems and has been widely used in engineering,physics,biology and other fields.For example,heat conduction problem,Lighthill problem,isochron model,etc.All of them can be simulated by Volterra integral equation.We find that many integral equations have weak singular kernel in the actual physical background,so this paper presents a fast configuration method for this type of integral equations.Since the multi-scale wavelet basis selected by us has the property of vanishing moment,the coefficient matrix obtained by using the collocation method is sparse.In this paper,under the premise of maintaining the optimal order of convergence,the appropriate strategy is used to truncate the coefficient matrix of the equations.Finally,we use gaussian integration to calculate this integral equation.In order to improve the accuracy of numerical calculation,we also refine the integral interval.In the forth part of the paper,we use this method to calculate three examples.We compared the numerical results with accurate solution,we can find that the wavelet base is stable.At the same time,we also find the fast singularity preserving collocation method we use have many advantages.Such as,high speed,high precision,and the approximate solution is stable,having the second order convergence.