An application of rationalized Haar functions to solution of linear differential equations. (English) Zbl 0613.65072

Ohkita, Masaaki; Kobayashi, Yasuhiro

IEEE Trans. Circuits Syst. 33, 853-862 (1986).

The rationalized Haar function RHar(r,t) \((r=0,1,...;0\leq t\leq T)\) is a step function taking only \(+1,-1\) or 0. The definition is as follows: \(RHar(0,t)=1\). For \(r\geq 1\), first decompose r uniquely into \(2^ i+j- 1\); \(i=0,1,...;j=1,2,...,2^ i\), and put \(RHar(r,t)=1\) in \(J_ 1\leq t<J_{1/2}\), -1 in \(J_{1/2}\leq t<J_ 0\) and 0 otherwise, where \(J_ u=(j-u)T/2^ i\), for \(u=1\), 1/2 or 0. First the authors discuss some fundamental properties of RHar functions, including the expansion of two products of RHar’s into series of RHar functions.
In the process for solving approximately a linear differential equations, they first rewrite the original equation into series of RHar functions, and solving a system of linear equations, they obtain the solution in form of the integral of series of RHar’s, i.e. a piecewise linear function. The authors emphasize the effectiveness of the present method at a regular singularity, and give several numerical examples for typical special functions.

Reviewer: S.Hitotumatu

MSC:

65L05	Numerical methods for initial value problems
34A30	Linear ordinary differential equations and systems, general
34B30	Special ordinary differential equations (Mathieu, Hill, Bessel, etc.)
33C45	Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)

Keywords:

series expansions; rationalized Haar function; regular singularity; numerical examples; special functions