A Wavelet-Based Symbolic Method for Time–Space Fractional Advection Equations with Caputo Derivatives

Resumen

This paper presents a wavelet-based numerical method for solving time–space fractional advection equations involving Caputo derivatives. The governing equation is given by
\[
d_1 \frac{\partial^{\beta} W}{\partial z^{\beta}} + d_2 \frac{\partial^{\gamma} W}{\partial u^{\gamma}} = h(z, u),
\]
where \( 0 < \beta, \gamma \leq 1 \) denote the fractional orders in the Caputo sense, and \( h(z,u) \) is a known source function. The proposed scheme uses a collocation approach based on Euler wavelets—compactly supported bases constructed from shifted and scaled Euler polynomials. This structure enables exact symbolic evaluation of fractional derivatives and facilitates the accurate enforcement of boundary conditions.

The numerical framework builds the solution through coefficient matrices and vector terms derived from a symbolic system, ensuring consistency with the governing equation at carefully selected collocation points. A central result shows that, when the exact solution is polynomial and symbolic computation is used, the method reproduces the solution exactly at all collocation nodes.

Numerical experiments support the theoretical findings, demonstrating high accuracy and computational efficiency, particularly for smooth solutions where rapid convergence is observed. Compared to existing approaches, the method offers enhanced precision and broader applicability, especially for problems involving coupled space–time nonlocality. This work expands the use of Euler wavelets in the context of fractional partial differential equations and provides a mathematically rigorous framework suitable for future extensions to nonlinear and multidimensional problems.