Wavelet-Based Numerical Methods for Solving Nonlinear Differential Equations

Abstract

Nonlinear differential equations play a crucial role in modeling rather complex phenomena in physics, engineering and biology, although their analytical solution is not always available because of nonlinearities. The work here describes a wavelet-based numerical method to solve nonlinear differential equations, with excellent efficiency, by collocation and orthogonal wavelet bases. The suggested approach turns the governing differential equation into a system of nonlinear algebraic equations by extending the solution in a sequence of wavelet basis functions with compact support and multi resolution characteristics. This expression allows a good representation of localized features and minimizes the amount of computation. The algebraic system that is found is solved by means of iterative schemes, which are stable and converge quickly. An error and convergence analysis is conducted in more detail in order to assess the accuracy of the method. The effectiveness of the suggested method is examined by numerous benchmark nonlinear problems (stiff and highly nonlinear equations). The numerical performance of the wavelet-based method proves that the wavelet-based technique has a better accuracy and lower computational cost than the traditional numerical methods like finite difference and Runge-Kutta methods. The results alone indicate that wavelet-based numerical approaches are a powerful and effective alternative to solving nonlinear differential equations with great probability of expansion to high-dimensional and complex systems.