数学与统计学院知识库

本文分两部分。第一部分研究了用伴随问题的方法来求解时间分数阶扩散方程中反演空间依赖源项的问题。由正问题解的级数表达式，我们得到了解的更高的正则性估计。然后我们用Tikhonov正则化方法求解反源问题，并且采用共轭梯度算法求解Tikhonov泛函的极小元。一维和二维的数值算例验证了该算法的有效性。

第二部分主要考虑了多维情形下的时间分数阶扩散波方程用带有噪音的边界数据反演空间源项的问题。由正问题解的表达式，我们得到了在更高的条件下正问题解的正则性。同时，通过Tichmarsh convolution定理和齐次化原理我们证明了反源问题的唯一性。进一步我们用最佳摄动算法求解反源问题。一维的数值算例说明了算法的有效性。

~This paper divide two parts. In the first part, we consider an inverse space-dependent source problem for a time-fractional diffusion equation  by an adjoint problem approach. Based on the series expression of the solution for the direct problem, we improve the regularity of the weak solution for the direct problem under strong conditions. And we provide the existence and uniqueness for the adjoint problem. Further, we use the Tikhonov regularization method to solve the inverse source problem and provide a conjugate gradient algorithm to find an approximation to the minimizer of the Tikhonov regularization functional. Numerical examples in one-dimensional and two-dimensional cases are provided to show the effectiveness of the proposed method.
The second part is devoted to identify a space-dependent source term in a multi-dimensional time fractional diffusion-wave equation from a part of noisy boundary data. Based on the series expression of solution for the direct problem, we improve the regularity of the weak solution for the direct problem under strong conditions. And we obtain the uniqueness of inverse space-dependent source term problem by the Titchmarsh convolution theorem and the Duhamel principle. Further, we use an optimal perturbation algorithm to find an approximation to the source term. Numerical examples in one-dimensional case are provided to show the effectiveness of the proposed  method.