On a Boundary–Initial Value Problem for a Multi-Term Sequential Caputo Fractional Equation

Fractional differential equations with sequential derivatives have become increasingly important because they can capture memory effects and complex behaviours that classical derivatives fail to describe. These tools are widely used today to model real processes in areas such as biology, chemistry, and physics, where systems often evolve in ways that depend not only on their current state but also on their history. Motivated by these applications, we investigate a new multi-term fractional differential equation involving sequential Caputo derivatives, motivated by recent research on fractional models with multiple memory effects. Unlike classical single-term models, the presence of sequential derivatives introduces additional analytical challenges, particularly in constructing explicit solution formulas.

Using techniques inspired by earlier works on sequential fractional operators, we derive the exact analytic solution of the problem in terms of the bivariate Mittag-Leffler function. The obtained representation provides a clear description of how the interaction between the two fractional orders influences the system's temporal dynamics. Additionally, several useful properties of the bivariate Mittag-Leffler function are formulated to support the construction of the solution.

The results demonstrate that multi-term sequential fractional equations admit explicit closed-form solutions under suitable boundary and initial conditions. These findings contribute to the theoretical development of sequential fractional models and offer a practical framework for analysing complex diffusion-type processes with layered memory effects.