Wavelet-Based Collocation Techniques for Fractional Equations with Multiple Terms and Boundaries Restrictions for Optimized Approach

Abstract

This paper discusses the limitations of initial value problems in relation to the stability, uniqueness, and accuracy of boundary value problems involving fractional polynomial differential equations.  The objective is to establish a robust framework for addressing complex boundary value problems through wavelet-based numerical techniques.  The wavelet collocation method uses Taylor wavelets to figure out mixed partial derivatives while also taking into account the boundary conditions.  The method tries to find solutions to problems with boundary values that are related to the Benjamin Ona Mahony equation.  We look at the single solutions for two-dimensional and three-dimensional corner domains, as well as smooth domains with localized forcing terms that come from polynomial exponential Laplacian expressions.  The study demonstrates that, unlike initial value problems, the geometry of the domain and the magnitude of the eigenvalues influence the stability of boundary value problems and Green's functions.  The wavelet collocation method provides a highly accurate numerical solution for challenging domains in physics and engineering by effectively capturing pronounced peaks and managing the boundary layers.  This is not the same as how things are in problems with initial values.