Fast Computation of Highly Oscillatory ODE Problems: Applications in High Frequency Communication Circuits

Abstract

Two types of algorithms are presented to approximate highly oscillatory and non-oscillatory first order ordinary differential equations. In the first approach, radial basis function interpolation is used to approximate the function f(t,x), then quadrature method is used to evaluate the integral part of the equation. The method is implementable to non-oscillatory first order initial value problems. The second approach is more generic and can approximate highly oscillatory and non-oscillatory initial value problems. Accordingly, the first order initial value problem with oscillatory forcing term is transformed into an integral with oscillatory Fourier kernel. The transformed oscillatory integral is then evaluated numerically by the Levin collocation method. Finally, non-linear form of the initial value problems with oscillatory forcing term is converted into a linear form using Bernoulli’s transformation. The resulting linear oscillatory problem is then computed by the new approaches. To justify accuracy of the algorithms, few numerical examples are added from the literature.