The sequential spectral method for integro -differential equations

Using the Galerkin method to solve nonlinear integro-differential equations of elliptic or parabolic type one needs to solve the resulting nonlinear systems of algebraic or ordinary differential equations. To solve these equations with Newtons method or a variant thereof can be very difficult and one needs a good initial guess for the methods to converge. Also there might be multiple solutions and it is virtually impossible to track all of them. In addition it is hard to study the parameter dependence of solutions. We developed a remedy for these problems by developing the sequential spectral method which avoids solving a nonlinear system altogether. In the sequential spectral method a scalar nonlinear algebraic or ordinary differential equation is solved at the initial stage and then the solution of the original problem is obtained through iterations, we never have to solve a nonlinear system at any stage of the method. The sequential spectral method converges linearly for steady state problems and superlinearly in the case of evolution. With the sequential spectral method we can obtain solutions to any desired accuracy with much less effort than with the Galerkin method. We can also increase the spectral degree of accuracy while the method is running. In addition one can easily detect the existence of multiple solutions by observing only a single equation and one can track those solutions. The behavior of the solution and the dependence on parameters can be estimated and one can also determine the blow up time for the corresponding parameter values by studying only a single equation. We further show that the sequential spectral method can be applied to a system of nonlinear elliptic partial differential equations.