This manuscript deals with an inverse fractional-diffusing problem, the time-fractional heat conduction equation, which is a physical model of a problem, where one needs to identify the temperature distribution of a semi-conductor, but one transient temperature data is unreachable to measurement. Mathematically, it is designed as a time-fractional diffusion problem in a semi-infinite region, with polluted data measured at x = 1, where the solution is wanted for 0 ≤ x < 1. In view of Hadamard, the problem extremely suffers from an intrinsic ill-posedness, i.e., the true solution of the problem is computationally impossible to measure since any measurement or numerical computation is polluted by inevitable errors. In order to capture the solution, a regularization scheme based on the Meyer wavelet is therefore applied to treat the underlying problem in the presence of polluted data. The regularized solution is restored by the Meyer wavelet projection on elements of the Meyer multiresolution analysis (MRA). Furthermore, the concepts of convergence rate and stability of the proposed scheme are investigated and some new order-optimal stable estimates of the so-called Hölder-Logarithmic type are rigorously derived by carrying out both an a priori and a posteriori choice approaches in Sobolev scales. It turns out that both approaches yield the same convergence rate, except for some different constants. Finally, the computational performance of the proposed method effectively verifies the applicability and validity of our strategy. Meanwhile, the thrust of the present paper is compared with other sophisticated methods in the literature.

该手稿涉及一个逆分数扩散问题，即时间分数热传导方程，这是一个问题的物理模型，其中需要确定半导体的温度分布，但无法获得一个瞬态温度数据。测量。在数学上，它被设计为半无限区域中的时间分数扩散问题，污染数据在x = 1处测得，其中需要0≤x的解<1.从Hadamard的角度来看，该问题极易遭受固有的不适定性，即该问题的真正解决方案在计算上无法测量，因为任何测量或数值计算都不可避免地会受到误差的污染。为了捕获解决方案，因此基于Meyer小波的正则化方案因此被应用于在存在污染数据的情况下处理潜在问题。通过对Meyer多分辨率分析（MRA）的元素进行Meyer小波投影，可以恢复正则化的解决方案。此外，研究了所提方案的收敛速度和稳定性的概念，并通过在Sobolev量表中进行先验和后验选择方法，严格得出了一些新的所谓Hölder-对数类型的阶最优稳定估计。 。事实证明，除了一些不同的常数之外，两种方法都产生相同的收敛速度。最后，所提方法的计算性能有效地验证了该策略的适用性和有效性。同时，将本文的主旨与文献中的其他复杂方法进行了比较。