Exponentially weighted Legendre–Gauss Tau methods for linear second-order differential equations

We develop a method to solve a class of second-order ordinary differential equations with highly oscillatory solutions. The method consists in combining three different techniques: Legendre–Gauss spectral Tau method, exponential fitting, and coefficient perturbation methods. With our approach, the resulting approximate solutions are expressed in terms of an exponentially weighted Legendre polynomial basis {eωnxLn(x);n≥0}{eωnxLn(x);n≥0}, where ωnωn are appropriately chosen complex numbers. The accuracy and efficiency of the method are discussed and illustrated numerically.