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.