Numerical identification of the time-dependent coefficient in the heat equation with fractional Laplacian
We consider the inverse problem of identifying a time-dependent source coefficient in a one-dimensional heat equation governed by the spectral Dirichlet fractional Laplacian from a weighted integral measurement. We present the continuous semigroup formulation and reduce the inverse problem to a Volterra equation under a nondegeneracy condition. For the numerical approximation, the spectral fractional operator is discretised by the fractional matrix power $A_h=L_h^s$ of the standard Dirichlet difference Laplacian and combined with a Crank--Nicolson scheme. We prove unconditional stability and an $O(τ^2+h^2)$ convergence estimate under a second-order spatial consistency assumption. A scalar reconstruction formula for the unknown coefficient is derived together with a discrete identifiability condition, and conditional convergence of the coupled state--coefficient reconstruction is established. For noisy integral measurements, regularised differentiation is used to approximate the required derivative data. Numerical experiments illustrate stable reconstruction for \(1\%\)--\(5\%\) relative noise and are consistent with the derived conditional error estimate.