A reliable algorithm based on Haar wavelet and Runge-Kutta method for the numerical solutions of (1+1)- and (1+2)-dimensional nonlinear parabolic partial differential equations

This study addresses the approximate solution of (1+1)- and (1+2)-dimensional nonlinear parabolic partial differential equations with initial and Dirichlet boundary conditions. The proposed hybrid scheme integrates the Haar wavelet with the fourth-order Runge-Kutta (RK-4) method. Spatial discretization via Haar wavelets reduces the partial differential equations to a system of time-dependent ordinary differential equations, which are then advanced using RK-4. The key advantages of this method are its ability to treat nonlinear partial differential equations directly, without linearization and solve extended domain problems without domain transformation. Rigorous stability and convergence analysis establish the reliability of the approach. Numerical experiments demonstrate its accuracy and efficiency, showing strong agreement with exact solutions and improved performance compared to existing stat-of-the-art methods.