Solution of differential and integral equations with Walsh functions

Any well-behaved periodic waveform can be expressed as a series of Walsh functions. If the series is truncated at the end of any group of terms of a given order, the partial sum will be a stairstep approximation to the waveform. The height of each step will be the average value of the waveform over the same interval. If a zero-memory nonlinear transformation is applied to a Walsh series, the output series can be derived by simple algebraic processes. The coefficients of the input series will change, but there will be no new terms not in the original groups. Nonlinear differential and integral equations can be solved as a Walsh series, since the series for the derivatives can always be integrated by simple table lookup. The differential equation is solved for the highest derivative first and the result is then integrated the required number of times to give the solution.