A high-order Haar wavelet approach to solve differential equations of fifth-order with simple, two-point and two-point integral conditions

This study introduces a high-order Haar wavelet collocation method (HHWCM) as an enhanced version of the classical Haar wavelet collocation method (HWCM) for solving fifth-order ordinary differential equations (FoDEs) subject to simple, two-point, and integral boundary conditions. By incorporating a quasi-linearization strategy, the proposed method avoids Jacobian computations and achieves higher accuracy with faster convergence. The stability and convergence of the approach are rigorously analyzed. Numerical experiments on both linear and nonlinear FoDEs demonstrate that HHWCM significantly outperforms HWCM and other existing numerical methods in terms of precision, computational efficiency, and robustness across diverse problem settings.