Legendre wavelet collocation method for singularly‐perturbed problems using operational matrix and Laplace transformation

Abstract In this paper, a collocation method based on the Legendre wavelets has been developed for the numerical solution of singularly perturbed problems (SPPs). The integral operators of Legendre scaling functions have been derived using two methods: the operational matrix method and the Laplace transformation method. The operational matrix method gives the approximated value of the integral operator of the Legendre scaling functions, whereas the Laplace transformation gives the exact solution for the integral operators. The Legendre wavelet collocation method together with its properties has been utilized to convert the SPP into the system of algebraic equations with the unknown coefficients. Moreover, the ‐error estimate in the function approximation with Legendre scaling functions has been derived, and it has been verified by taking a test function. The stability analysis of the proposed method has been discussed in detail. In the last, illustrative linear and non‐linear examples have been considered to demonstrate the applicability of the proposed method and the results have been compared with the other existing numerical methods in the recent literature.