A Fourier Series Technique for Approximate Solutions of Modified Anomalous Time-Fractional Sub-Diffusion Equations - Samad Kheybari; Farzaneh Alizadeh; Mohammad Taghi Darvishi; Kamyar Hosseini (Fractal and Fractional)
This article presents an innovative approximating technique for addressing modified anomalous time-fractional sub-diffusion equations (MAFSDEs) of the Caputo type. These equations generalize classical diffusion equations, which involve fractional derivatives with respect to time, capturing the non-local and history-dependent behavior typical in sub-diffusion processes. In such a model, the particle transports slower than in a standard diffusion, often due to obstacles or memory effects in the medium. The core of the proposed technique involves transforming the original problem into a family of independent fractional-order ordinary differential equations (FODEs). This transformation is achieved using the Fourier expansion method. Each of these resulting FODEs is defined under initial value conditions which are derived from the initial condition of the original problem. To solve them, for each resulting FODE, some secondary initial value problems are introduced. By solving these secondary initial value problems, some particular solutions are obtained and then we combine them linearly in an optimal manner. This combination is essential to estimate the solution of the original problem. To evaluate the accuracy and effectiveness of the proposed scheme, we conduct a various test problem. For each problem, we analyze the observed convergence order indicators and compare them with those from other methods. Our comparison demonstrates that the proposed technique provides enhanced precision and reliability in respect with the current numerical approaches in the literature.