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The aim of this paper is to obtain the numerical solution of singular ordinary differential equations using the Haar-wavelet approach. The proposed method is mathematically simple and provides highly accurate solutions. In this method, we derive the Haar operational matrix using Haar function. Haar operational matrix is a basic tool and applied in system analysis to evaluate the numerical solution of differential equations. The convergence of the proposed method is discussed through its error analysis. To illustrate the efficiency of this method, solutions of four singular differential equations are obtained.

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In this paper, we propose an extended numerical algorithm for the numerical solution of the Benjamin–Bona–Mahony–Burgers equation. This algorithm involves the application of wavelet theory. First, we use the Quasilinearization technique of linearization and apply the 3-scale Haar wavelet approach for truncation error. This algorithm is constructed from two wavelet functions that make it robust and highly accurate. A multi-resolution is used to generate the Haar basis function. We consider three cases of a mathematical problem for the accuracy of the presented algorithm. The obtained results show good agreement with analytical solutions and have better accuracy.

This article proposes a modified numerical algorithm based on a finite-difference scheme and the 3-Scale Haar approach for numerical simulation of the fractional KdV-Burgers differential equation. The multi-resolution utilize to develop the wavelet basis functions. The convergence of the proposed algorithm convinces its error analysis. The Haar solution shows decent concurrence with the analytical solutions and other existing techniques present in the literature. The Haar solutions reveal that the proposed procedure is profoundly viable and helpful for the fractional partial differential equation.