Analysis of Boundary-Initial Value Problems for Fractional Equations Involving Sequential Caputo Derivatives

A boundary-initial value problem for a fractional partial differential equation with sequential Caputo derivatives is studied. Exact analytical solutions are derived via the Fourier sine spectral method, with time-dependent coefficients expressed in closed form through the bivariate Mittag-Leffler function. Existence, uniqueness, and uniform convergence of the classical solution in Hölder spaces are established under the conditions α + β > 1 and α > β . A fully discrete L 1 finite-element scheme on a graded temporal mesh is validated against a manufactured solution, confirming first-order convergence and the superiority of graded over uniform grids near the initial singularity.

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