A numerical solver based on Haar wavelet to find the solution of fifth-order differential equations having simple, two-point and two-point integral conditions

Abstract

This article introduces a Haar wavelet-based numerical method for solving fifth-order linear and nonlinear differential equations. This method easily handles both homogeneous and nonhomogeneous equations. It also works with variable and constant coefficients under various conditions. The method is flexible, making it easy to work with boundary, integral, and two-point integral conditions. These three different cases of given information are coupled with fifth-order linear and nonlinear differential equations, and the method proves to be effective in these cases. The outcomes of the Haar wavelet collocation technique are compared with approaches found in existing literature. The method demonstrates second-order convergence, and experimental results support this idea as well. The CPU time is used to evaluate the efficiency of the method, and the maximum absolute errors (\(L_\infty \)) are utilized to assess the accuracy level. Different examples are studied along with various given information, and the method is found to be adaptable to different types of boundary conditions and particular integral conditions.