Time-fractional interface problems, found in heat transfer with discontinuous conductivities and fluid flows with surface tension forces, are challenging due to irregular interfaces and the history-dependent nature of fractional derivatives. This paper presents two numerical methods for simulating time-fractional double interface problems. The first method uses the Haar wavelet collocation technique, while the second relies on a meshless approach with radial basis functions. The fractional derivatives are replaced with the Caputo sense, the resulting first-order time derivatives are handled using the finite difference method, and the spatial operator is approximated using the two proposed methods. Gauss elimination is used to solve linear problems. Quasi-Newton linearization method is used for nonlinear problems. Both methods accommodate constant and variable coefficients, handling discontinuities and singularities in both solutions and coefficients. To evaluate the effectiveness of the proposed methods, numerical experiments are carried out. The accuracy of each method is quantified using the L∞ error norm, and a comparative analysis highlights the validity and advantages of the approaches. Moreover, the proposed schemes are rigorously analyzed to establish their stability, and the existence and uniqueness of the solutions.
Bilal et al. (Sat,) studied this question.