Mathematics, Vol. 11, Pages 3101: Solving Inverse Problem of Distributed-Order Time-Fractional Diffusion Equations Using Boundary Observations and L2 Regularization

Mathematics, Vol. 11, Pages 3101: Solving Inverse Problem of Distributed-Order Time-Fractional Diffusion Equations Using Boundary Observations and L2 Regularization

Mathematics doi: 10.3390/math11143101

Authors: Lele Yuan Kewei Liang Huidi Wang

This article investigates the inverse problem of estimating the weight function using boundary observations in a distributed-order time-fractional diffusion equation. We propose a method based on L2 regularization to convert the inverse problem into a regularized minimization problem, and we solve it using the conjugate gradient algorithm. The minimization functional only needs the weight to have L2 regularity. We prove the weak closedness of the inverse operator, which ensures the existence, stability, and convergence of the regularized solution for the weight in L2(0,1). We propose a weak source condition for the weight in C[0,1] and, based on this, we prove the convergence rate for the regularized solution. In the conjugate gradient algorithm, we derive the gradient of the objective functional through the adjoint technique. The effectiveness of the proposed method and the convergence rate are demonstrated by two numerical examples in two dimensions.