Clenshaw-Curtis Spectral Collocation Method For Volterra Type Integral Equations

Volterra integral equations are widely used in science,engineering,physics and other fields,such as communication theory,circuit simulation,population genetics,heat transfer,etc.But for the most of the integral equations,it is difficult to obtain the analytical solutions,so numerical solution problem of integral equation is particularly critical.In recent years,it has become the focus in the reseach of scholars both at home and abroad,and has gained some mature results.It is found that the numerical solutions obtained by spectral method have high precision.On this basis,we solve three kinds of Volterra integral equations(VIEs)using the Clenshaw-Curtis spectral collocation method,constructe their discrete schemes respectively,and prove that the discrete schemes converge in the L~∞norm space.The numerical results show that this method has the advantages of high precision and fast convergence.Firstly,we solve the the VIEs of the second kind.We take the Clenshaw-Curtis points as the collocation points,assume the weight function ω(x)=1,approximate the integral term of the equation by the Clenshaw-Curtis quadrature formula,implemente Lagrange interpolation for the unknown function at collocation points,and further simplify to obtain the discrete scheme of the equation.When the number of nodes is the same,there are better error accuracy compared with Legendre spectral collocation method.Then,for the solution to VIEs with highly oscillatory kernel.The key lies in discretizing the integral terms with high oscillatory kernel.We assume the weight function W(s):=k(ω,x,s),obtaine the integral weight by calculating the weighted moments,and then use Clenshaw-Curtis quadrature formula to approximate the integral terms.We solve the VIEs with Fourier kernel,the numerical results obtaine the expected spectral accuracy,and the error decreses with the increse of frenquency ω.Finally,for the solution to Volterra integro-differential equations(VIDEs),we rewrite the VIDEs into the equivalent VIEs systems,then use Clenshaw-Curtis spectral collocation method to solve the equation systems.It with higher error accuracy than Chebyshev collocation spectal method,when the number of nodes is the same.