One of the major objectives in the field of inverse problems is to construct a space-dependent term of an unknown source in a stable manner. Many different fields of science have used this source term, especially when the extra condition is accompanied by noise. We focus on a one-dimensional situation in a fractional diffusion problem to recover the source term that is unknown. In order to accomplish this, the major problem was transformed into an equation of operator form in a way that allowed the unique solvability of this equation to be established. The Ritz-Galerkin method is then used to implement a numerical solution to the inverse problem. In conjunction with the Galerkin method, shifted Bernoulli wavelets (BWs) are used as basis functions to reduce the main problem to an algebraic equation. It is essential to include some kind of regularization method within the numerical algorithm to obtain a stable solution to the resulting linear system. We concluded by giving numerical examples that demonstrate the proposed algorithm’s validity and efficiency in the presence of noise.

反问题领域的主要目标之一是以稳定的方式构建未知源的空间相关项。许多不同的科学领域都使用过这个源项，尤其是当额外条件伴随噪音时。我们专注于分数扩散问题中的一维情况，以恢复未知的源项。为了实现这一点，主要问题被转化为算子形式的方程，其方式允许建立该方程的唯一可解性。然后使用 Ritz-Galerkin 方法实现反问题的数值解。结合 Galerkin 方法，将移位的伯努利小波 (BW) 用作基函数，以将主要问题简化为代数方程。必须在数值算法中包含某种正则化方法以获得所得线性系统的稳定解。最后，我们通过给出数值示例来证明所提出的算法在存在噪声的情况下的有效性和效率。