Taylor Wavelet Method Solves Fractional Pollution Models

Researchers have introduced a new numerical technique designed to estimate unknown parameters in complex environmental models. The study, published in Boundary Value Problems, presents the Taylor wavelet collocation method, or TWCM, as a tool for solving inverse time-fractional advection-diffusion equations with variable coefficients. The model is designed to represent the spatiotemporal evolution of pollutant concentrations in heterogeneous media, including both soil and atmospheric environments.

The central problem addressed by the study is the inverse nature of the model. The proposed method allows for the estimation of the state field and an unknown time-dependent boundary function.

The Taylor wavelet collocation method works by reducing the complex inverse problem to a finite-dimensional linear algebraic system. This linear algebraic system is solved for the Taylor-wavelet coefficients.

The study also addresses noisy interior observations. To address this, the researchers incorporated Tikhonov regularization into the discrete reconstruction process. This technique enhances the stability of the solution when interior observations contain noise.

The study includes comparative analyses to evaluate the performance of the new method against established numerical techniques. The authors compared TWCM with the rationalized Haar wavelet method, radial basis function methods, and finite-difference benchmarks. The results indicate that the Taylor wavelet collocation method significantly outperforms these established approaches.

The ability to estimate unknown model quantities under these varying conditions is a critical feature of the proposed approach. The results demonstrate the method's potential to provide valuable insights for environmental monitoring and management.

The research was conducted by a team of authors including Mahmoud Abd El-Hady, Mohamed El-Gamel, Mohammad Izadi, and Yasser Kashwaa. The article is available as an open access publication. The study was published on August 28, 2026, and is currently in an early sharing stage to provide faster access to peer-reviewed, accepted research. This version is citable and carries a permanent DOI, though it is subject to further edits before the final Version of Record is released.

While the study demonstrates the method’s potential, it is important to note that the results are based on numerical simulations and comparative analyses.

Subject of Research: Numerical methods for inverse time-fractional advection-diffusion equations in environmental modeling

Article Title: Parameter estimation using Taylor wavelet collocation method for inverse time-fractional advection-diffusion equations with variable coefficients

Article References: El-Hady, M. A., El-Gamel, M., Izadi, M., & Kashwaa, Y. (2026). Parameter estimation using Taylor wavelet collocation method for inverse time-fractional advection-diffusion equations with variable coefficients. Boundary Value Problems. https://doi.org/10.1186/s13661-026-02349-y

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DOI: 10.1186/s13661-026-02349-y

Keywords: Taylor wavelet collocation method, inverse problems, time-fractional equations, advection-diffusion, environmental modeling, Tikhonov regularization, numerical analysis, pollutant concentration, Parameter, estimation, Taylor, wavelet