A local discontinuous Galerkin method for the Caputo–Hadamard Burgers equation: Analysis and parameter estimation

This paper develops and analyzes a fully discrete numerical scheme for the time-fractional Burgers equation with a Caputo–Hadamard derivative, which incorporates a logarithmic kernel particularly suitable for modeling ultraslow diffusion processes. The proposed scheme combines a nonuniform L1 approximation on exponentially graded meshes for temporal discretization with a local discontinuous Galerkin method for spatial discretization. By employing a discrete fractional Grönwall inequality, we prove unconditional stability and optimal L2 error estimates of order min{2 − α, rα} in time and k + 1 in space, where α ∈ (0, 1) is the fractional order, r is the mesh grading parameter, and k is the polynomial degree. Numerical experiments validate the theoretical convergence rates. Furthermore, we integrate the forward solver into a Broyden–Fletcher–Goldfarb–Shanno optimization framework to simultaneously estimate the fractional order α and the viscosity coefficient μ from noisy observations. A comparative study with the classical Caputo model demonstrates that the Caputo–Hadamard formulation yields significantly more accurate parameter calibration for processes exhibiting ultraslow diffusion, emphasizing the critical importance of selecting a fractional operator consistent with the underlying physical mechanism.