- in particular for the Mathieu equation (best example), Lamé and ellipsoidal wave equations, oblate and prolate spheroidal wave equations, and equations with anharmonic potentials. Convection-diffusion equations Methods of matched asymptotic expansions have been developed to find approximate solutions to the Smoluchowski convection-diffusion equation, which is a singularly perturbed second-order differential equation. The problem has been studied particularly in the context of colloid particles in linear flow fields, where the variable is given by the pair distribution function around a test particle. In the limit of low Péclet number, the convection-diffusion equation also presents a singularity at infinite distance (where normally the far-field boundary condition should be placed) due to the flow field being linear in the interparticle separation. This problem can be circumvented with a spatial Fourier transform as shown by Jan Dhont. A different approach to solving this problem was developed by Alessio Zaccone and coworkers and consists in placing the boundary condition right at the boundary layer distance, upon assuming (in a first-order approximation) a constant value of the pair distribution function in the outer layer due to convection being dominant there. This leads to an approximate theory for the encounter rate of two interacting colloid particles in a linear flow field in good agreement with the full numerical solution. When the Péclet number is significantly larger than one, the singularity at infinite separation no longer occurs and the method of matched asymptotics can be applied to construct the full solution for the pair distribution function across the entire domain. See also Asymptotic analysis Multiple-scale analysis Activation energy asymptotics References Differential equations Asymptotic analysis