Global series solutions of nonlinear differential equations with shocks using Walsh functions
An orthonormal basis set composed of Walsh functions is used for deriving global solutions (valid over the entire domain) to nonlinear differential equations that include discontinuities. Function gn(x) g n ( x ) of the set, a scaled Walsh function in sequency order, is comprised of n  piecewise constant values (square waves) across the domain xa?x?xb x a ? x ? x b . Only two square wave lengths are allowed in any function and a new derivation of the basis functions applies a fractal-like algorithm (infinitely self-similar) focused on the distribution of wave lengths. This distribution is determined by a recursive folding algorithm that propagates fundamental symmetries to successive values of n . Functions, including those with discontinuities, may be represented on the domain as a series in gn(x) g n ( x ) with no occurrence of a Gibbs phenomenon (ringing) across the discontinuity. A much more powerful, self-mapping characteristic of the series is closure under multiplication  the product of any two Walsh functions is also a Walsh function. This self-mapping characteristic transforms the solution of nonlinear differential equations to the solution of systems of polynomial equations if the original nonlinearities can be represented as products of the dependent variables and the convergence of the series for n?? n ? ? can be demonstrated. Fundamental operations (reciprocal, integral, derivative) on Walsh function series representations of functions with discontinuities are defined. Examples are presented for solution of the time dependent Burger?s equation and for quasi-one-dimensional nozzle flow including a shock.
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