Piecewise-linear approximation of solution of linear differential equations by Walsh functions

An algorithm for solving linear differential equations(DEs) by Walsh functions (WFs) is proposed. In this algorithm, approximate solutions are determined in a form of piecewise-linear approximation (PWLA) by means of fast algorithms of inverse Walsh transforms. For this purpose, derivatives of the solutions are expanded into Walsh series with unknown coefficients. In other words, the solutions are expressed by termwise integrals of Walsh series in terms of time variable. In this approach, the accuracy of the solutions is improved and hence the number of computations is reduced greatly, compared with that of the conventional stairstep approximations for the same order of the approximations of the solutions.