A collocation boundary value method for linear Volterra integral equations
This paper is devoted to the classical linear Volterra integral equation of the second kind \[ u(t)+\int\limits_{0}^{t}K(t,s)ds=f(t), \; t\in I=[0,T], \tag{1} \] with the continuous kernel \(K(t,s)\) defined on \(D=\{(s,t):\;0\leq s\leq t\leq T\}\). This equation has a unique continuous solution for any \(f(t)\in C(I)\). High order numerical schemes for (1) are devised by using special multistep collocation methods, which depend on numerical approximations of the solution in the next several steps. Stability analysis illustrates that these methods enjoy wide absolutely stable regions. With the help of efficient evaluation for highly oscillatory integrals, these methods are applied to solving Volterra integral equations with highly oscillatory kernels. Both theoretical and numerical results show they share the property that the higher the oscillation, the better the accuracy of the approximations.
scientific article; zbMATH DE number 6849347
scientific article