Solving Initial Value Problem by Matching Asymptotic Expansions

Abstract

An asymptotic expansion in powers of small dimensionless dispersion coefficient is sought to solve an initial value problem posed for an advection diffusion equation modeling orientation of pulp fibers in a steady fully turbulent flow. The regular expansion is shown to be nonuniform in a small neighborhood of $\phi=0$. Although the highest order derivative with respect to the orientation angle $\phi$ is multiplied by the small parameter, application of matched asymptotic expansions to obtain the inner solution in a small neighborhood of $\phi=0$ matchable with the regular expansion turned out to be unsuccessful. The multiple scales do not lead to the solution either. The problem is solved by matching two asymptotic expansions, one solving the initial value problem in a small neighborhood of the initial point, while another one solves the equation at large distances from the initial point. Thus, this is an example of using the method of matched asymptotic expansions to satisfy the given initial condition by the long-distance approximation of the solution to a nonsingular partial differential equation.