Laplace–Mellin–Barnes transform solutions of time-fractional diffusion equations: Green kernels, stability estimates, and numerical benchmarks

Introduction We study a time-fractional diffusion equation with Caputo memory on the real line and develop a self-contained analytical-computational benchmark framework. Methods The solution is obtained using the Laplace transform in time and the Fourier transform in space, leading to Mittag–Leffler relaxation factors. The Green kernel is expressed through the M-Wright function and a Mellin–Barnes contour integral with explicit assumptions, contour conventions, and branch choices. A continuous L 2 -type stability estimate is derived, and a smooth manufactured solution is used to verify a semi-implicit fractional time-stepping method on a truncated computational interval. Results The analytical validation recovers the established M-Wright kernel, the Gaussian limit, mass conservation, and the subdiffusive second moment. The numerical tests report final-time errors, residual diagnostics, separated spatial and temporal convergence behavior, and sensitivity with respect to the fractional order. Discussion The framework organizes classical transform and special-function representations together with explicit assumptions, stability bounds, and a reproducible manufactured-solution benchmark. It can serve as a reference problem for finite-difference, spectral, collocation, physics-informed neural-network, and hybrid solvers when those methods are implemented and assessed independently.