Adaptive Solution of Partial Differential Equations in Multiwavelet Bases

We construct multiresolution representations of derivative and exponential operators with linear boundary conditions in multiwavelet bases and use them to develop a simple, adaptive scheme for the solution of nonlinear, time-dependent partial differential equations. The emphasis on hierarchical representations of functions on intervals helps to address issues of both high-order approximation and efficient application of integral operators, and the lack of regularity of multiwavelets does not preclude their use in representing differential operators. Comparisons with finite difference, finite element, and spectral element methods are presented, as are numerical examples with the heat equation and Burgersʹ equation.