Transform Framework Delivers Exact Solutions and Verifiable Benchmarks for Fractional Diffusion

A self-contained analytical and computational benchmark framework has resolved key verification challenges in anomalous transport by uniting classical integral transform representations with reproducible numerical tests for time-fractional diffusion equations. Formulated by researcher Alvaro H. Salas from the FIZMAKO Research Group at Universidad Nacional de Colombia in Manizales, the study provides a rigorous reference architecture for evolution models featuring nonlocal temporal memory across the real line. The work investigates an inhomogeneous equation governed by a Caputo fractional derivative, spatial diffusion, and a prescribed source term, addressing the mathematical demands of non-Markovian relaxation. Rather than offering isolated approximations, the research integrates closed-form special-function solutions, contour integral evaluations, continuous stability bounds, and an exact manufactured benchmark. This formulation establishes an independent standard against which emerging numerical algorithms, spectral schemes, and scientific machine-learning solvers can be verified with high precision without circular comparisons.

The mathematical foundation of the framework centers on the capacity of fractional differential operators to capture hereditary mechanisms and anomalous transport that cannot be described by integer-order equations. In standard evolution models, the instantaneous rate of change depends exclusively on the current state, discarding prior physical history. In contrast, fractional formulations incorporate history-dependent convolution kernels that retain information from earlier stages, making them suitable for subdiffusion in porous media, heterogeneous thermal relaxation, polymeric systems, and biological transport. The governing model adopts a temporal Caputo fractional derivative of order between zero and one, defined through a singular power-law memory kernel. This operator incorporates conventional initial conditions while preserving nonlocal temporal memory. As the memory parameter approaches one from below, the governing equation recovers the classical first-order time derivative and the standard Gaussian heat conduction equation.

To resolve the analytical structure of the forced fractional diffusion model, the study establishes an integral transform pipeline combining the Laplace transform in time and the Fourier transform in space. The temporal Laplace transform converts the nonlocal Caputo operator into an algebraic multiplier governed by the transform variable raised to the fractional power while retaining initial conditions directly. Simultaneously, the spatial Fourier transform diagonalizes the diffusion operator on the unbounded domain, replacing spatial derivatives with algebraic multiplication by the squared wavenumber. This combination reduces the partial differential equation to an explicit algebraic relation in transform space, with a characteristic denominator coupling temporal memory and spatial dissipation. Inversion through inverse Laplace transforms yields mild solutions governed by Mittag-Leffler relaxation functions. These functions dictate algebraic rather than exponential decay, capturing persistent subdiffusive memory and providing explicit convolution representations for both initial state propagation and source forcing.

The spatial distribution of the diffusing field is governed by a fundamental Green kernel expressed through the M-Wright special function, which replaces the Gaussian density in fractional diffusion. To evaluate and manipulate this kernel analytically, the research introduces a Mellin-Barnes contour integral representation defined along a vertical Barnes line in the complex plane. This contour formulation incorporates explicit branch-cut selections, analyticity strips, and rigorous pole-separation conditions that isolate gamma-function poles along the non-positive real axis. By shifting the integration path across the complex plane and systematically computing gamma-function residues, the author demonstrates that the Mellin-Barnes integral directly reproduces the classical Wright series expansion. This contour formulation provides a mathematical bridge between special-function theory, asymptotic front analysis, and discrete numerical evaluations. Crucially, it establishes exact analytical machinery that operates without requiring heavy computational approximations, yielding a definitive benchmark for subdiffusive spreading fronts.

Analytical validation of the derived Green kernel confirms consistency with classical diffusion theory and established fractional literature. The Laplace-Fourier inversion reproduces the canonical fundamental solution reported by Mainardi and collaborators, verifying that the concentration profile depends on an exact similarity variable scaling with position divided by the fractional power of time. Furthermore, taking the fractional order toward unity recovers the classical Gaussian heat kernel, as the Mittag-Leffler Fourier multiplier converges pointwise to the exponential multiplier and the M-Wright density simplifies to the normal distribution. The mathematical formulation also demonstrates exact total mass conservation, confirming that the spatial integral of the Green kernel remains normalized to unity. Additionally, the second spatial moment, corresponding to the physical mean-square displacement, scales proportionally to time raised to the fractional order divided by the gamma function of the order plus one, confirming subdiffusive spreading.

Alongside exact kernel derivations, the framework establishes a continuous stability estimate in the square-integrable function space to quantify solution behavior under initial perturbations and external driving forces. Working directly from the Laplace-Fourier mild solution representation, the author derives an explicit continuous bound showing that the spatial norm of the solution at any finite time remains bounded by the initial data norm and the cumulative time integral of source forcing. The proof relies on Plancherel theorem and analytical bounds for Mittag-Leffler functions along the negative real axis, where the primary relaxation factor remains bounded between zero and one while the secondary kernel is constrained by a constant. The research highlights the theoretical distinction between this continuous stability estimate and discrete numerical stability theorems. While continuous bounds confirm that the underlying equation is well-posed, they do not guarantee numerical stability for time-stepping schemes, necessitating independent computational verification.

To bridge continuous analytical formulations and practical numerical implementation, the study constructs a smooth manufactured solution designed for rigorous solver verification. The prescribed exact solution combines a Gaussian spatial profile with a power-law temporal growth term, generating a closed-form benchmark that satisfies the fractional diffusion equation continuously. By applying the Caputo derivative and spatial second derivative to this manufactured profile, the author derives the exact continuous source term required to balance the governing equation. This manufactured benchmark tests the entire computational pipeline, including the evaluation of fractional memory integrals, spatial centered-difference operators, source routines, and boundary conditions on truncated domains. For numerical implementation, the spatial domain is truncated to a symmetric interval between negative six and positive six. Because the manufactured Gaussian profile decays to roundoff levels near machine precision at the boundaries, exact Dirichlet conditions introduce no visible truncation error.

The computational benchmark implements a semi-implicit fractional Adams predictor-corrector scheme applied to the Volterra integral formulation of the governing equation. In this architecture, the spatial second derivative is approximated using a three-point centered finite-difference operator on a uniform grid. The Caputo fractional memory integral is discretized using history-dependent fractional Adams weights, separating historical evolution from the implicit advance at the current time level. An explicit Adams-Bashforth predictor first generates an intermediate solution estimate at the advanced time step, which is inserted into a semi-implicit Adams-Moulton corrector. This semi-implicit treatment handles spatial diffusion implicitly while evaluating the source term at the advanced time step, producing a tridiagonal linear system solved efficiently at each time increment. By detailing explicit formulas for every grid weight, boundary vector, and matrix coefficient, the research provides a fully reproducible computational protocol.

The computational performance of the scheme was evaluated through systematic refinement tests revealing the convergence characteristics of the numerical discretization. In an initial combined refinement experiment with a fractional order of zero point seven and diffusion coefficient of zero point one, the spatial grid and time step were refined simultaneously across three progressive levels. The final-time maximum error decreased monotonically from two point seventeen times ten to the negative third power on the coarsest grid to one point thirty-six times ten to the negative fourth power on the finest grid, matching reductions in discrete root-mean-square error. Calculating the observed convergence order yielded an empirical rate near two point zero zero, demonstrating apparent second-order accuracy. However, the study cautions against overinterpreting this combined result, explaining that simultaneous refinement entangles spatial and temporal errors, which can falsely suggest high-order temporal performance when global error is governed by spatial resolution.

To resolve individual truncation contributions, the investigation performed decoupled spatial and temporal convergence tests. In the isolated spatial experiment, the time step was fixed at one thousand intervals while the spatial mesh was refined. The resulting errors decreased cleanly with measured spatial convergence rates near two point zero zero, confirming that apparent second-order accuracy originates entirely from the centered spatial finite-difference operator. Conversely, when the spatial grid was held fixed at one thousand two hundred and eighty intervals while the time step was varied, temporal convergence rates reached approximately one point ninety-three before declining to one point thirty-two on the finest increments. This degradation occurred because temporal error decayed below the residual spatial discretization error floor. Complementary spatial consistency residuals decreased by approximately a factor of four with each grid halving, validating the theoretical second-order behavior of the discrete operator.

Parametric sensitivity evaluations confirmed that the manufactured-solution benchmark retains stability and accuracy across diverse fractional regimes, testing memory exponents of zero point two, zero point five, zero point seven, and zero point nine. Across all tested values, the numerical scheme maintained smooth error profiles without generating spurious oscillations, keeping final maximum errors consistently near one point thirty-six times ten to the negative fourth power. Graphical diagnostics established that numerical approximations and exact analytical profiles were visually indistinguishable, exhibiting symmetric error distributions that confirm proper boundary handling. The author concludes that this unified framework provides a reproducible reference foundation for verifying diverse computational methods, including spectral collocation schemes, finite-element formulations, convolution quadrature, and physics-informed neural networks. Prospective extensions include applying Laplace-Fourier transform benchmarking to multi-dimensional domains, fractional reaction-diffusion systems, and Fokker-Planck equations, ensuring that computational solvers are evaluated against rigorous mathematical ground truth.

Subject of Research: Analytical and computational benchmark framework for time-fractional diffusion equations with Caputo memory

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

Article References: Salas, A. H. (2026). Laplace–Mellin–Barnes transform solutions of time-fractional diffusion equations: Green kernels, stability estimates, and numerical benchmarks. Frontiers in Applied Mathematics and Statistics, 12, 1875542. https://doi.org/10.3389/fams.2026.1875542

Image Credits: AI Generated

DOI: 10.3389/fams.2026.1875542

Keywords: fractional diffusion, Caputo derivative, Laplace transform, Fourier transform, Green kernel, M-Wright function, Mellin-Barnes integral, Mittag-Leffler function, subdiffusion, anomalous transport, stability estimates, manufactured solution, Adams predictor-corrector, numerical benchmarks, convergence analysis, tridiagonal systems, finite-difference method