What does this research mean for the field?

Explicit analytical solutions and unique classical solutions can be rigorously established for direct and inverse source problems in time-fractional diffusion equations governed by the Hilfer fractional derivative, although the inverse problems remain ill-posed in the Hadamard sense. Novelty: ClaimNovelty.METHODOLOGICAL. Consensus alignment: ConsensusAlignment.NEUTRAL.

What question did this study set out to answer?

To establish analytical solutions and prove existence, uniqueness, and stability properties for a direct problem and two inverse source problems in a Hilfer time-fractional diffusion equation.

August 16, 2026MathematicsOpen Access

Analytical Solutions for Direct and Inverse Source Problems in a Time-Fractional Diffusion Equation

View Full Paper

Ask AI

Bookmark

Authors

GAGhaziyah AlsahliJouf University NANura AlotaibiJouf University SBSid Ahmed Ould BeinaneJouf University

Discussion

Member takes

Implication

Mathematical analysis reveals exact series solutions for direct and inverse time-fractional diffusion equations, indicating robust reconstruction frameworks for anomalous transport processes.

Key Points

To establish analytical solutions and prove existence, uniqueness, and stability properties for a direct problem and two inverse source problems in a Hilfer time-fractional diffusion equation.
Derived explicit analytical series solutions using eigenfunction expansions and Laplace transform techniques in terms of Mittag-Leffler functions.
Reformulated the time-dependent inverse source problem into a Volterra integral equation and proved unique solvability using the Banach fixed point theorem.
Conducted numerical experiments to validate theoretical reconstructions under generalized Hilfer-type initial and homogeneous Neumann boundary conditions.
Established rigorous existence and uniqueness of classical solutions for the direct problem and both space- and time-dependent inverse source problems.
Demonstrated that both inverse source problems are ill-posed in the Hadamard sense, showing instability under data perturbations.
Recovered classical formulations for Riemann–Liouville and Caputo fractional derivatives as limiting cases of the generalized Hilfer framework.

Cite This Study

Alsahli et al. (2026) studied this question.

synapsesocial.com/papers/6a8179cbf2fb91fc834ad362 https://doi.org/10.3390/math14162948

View Full Paper

Ask AI

Bookmark

Implication

Mathematical analysis reveals exact series solutions for direct and inverse time-fractional diffusion equations, indicating robust reconstruction frameworks for anomalous transport processes.

Analytical Solutions for Direct and Inverse Source Problems in a Time-Fractional Diffusion Equation

Discussion

Member takes

Implication

Key Points

Cite This Study

Analytical Solutions for Direct and Inverse Source Problems in a Time-Fractional Diffusion Equation

Discussion

Member takes

Implication

Key Points

Cite This Study

Also Consider

Also Consider