A set of multi-dimensional orthogonal basis functions and its application to solve integral equations

Abstract

A new set of multi-dimensional orthogonal basis functions and some of their properties are introduced. These functions are extension of triangular functions (TFs) in n dimensions. Expansion of multi-variable functions with respect to these functions is presented. Also, the relation of these new functions to the block-pulse functions (BPFs) in n dimensions
will be investigated.

Many applied problems are often discussed in n dimensions,
consequently the multi-dimensional moment method using the current orthogonal basis functions will be used to solve two-variable integral equations. The obtained results are compared with those of the multi-dimensional moment method using BPFs. These comparisons show efficiency and accuracy of the new orthogonal basis functions applied to solve multi-dimensional integral equations. Finally, a study of the representational error will be made to estimate the mean integral squared error for the TF approximation of a function f(s,t) of Lebesgue measure.