RECONSTRUCTING THE HISTORICAL DISTRIBUTION OF DIFFUSION EQUATION WITH POLY-FRACTIONAL OPERATOR IN MULTI-DIMENSIONAL SPACE
Author : Tran Thi Khieu
Abstract :The content of the report discusses a research study focusing on the reconstruction of the past distribution for diffusion equations in multidimensional space, involving a poly-fractional operator. The mathematical expression for the operator is displayed as: Pα((−∆)γ )u := Xm i=1 αi(−∆)γiu, where m ∈ N, 0 < γ1 ≤ · · · ≤ γm ≤ 1 and αi > 0 for i = 1, . . . , m. This equation is centered on the page and uses LaTeX-style mathematical notation, including symbols like Pα, ∆ (Laplacian), and γ. The text explains that the problem arises from the superposition of multiple stochastic processes with different scales, such as classical random walks and L´evy flights. It mentions that the problem is extremely ill-posed in the sense of Hadamard. The authors describe using an appropriate filtering method to construct a regularized solution that converges to the original solution in the L 2-sense. Additionally, imposing a Gevrey-type a-priori condition allows for obtaining a H¨older convergence rate. The abstract concludes by noting that three numerical examples using MATLAB were performed to demonstrate the ill-posedness of the problem and the effectiveness of the regularization method. There are no images, graphs, or additional figures in the attachment—just plain text. The font appears to be a standard serif typeface, such as Times New Roman, commonly used in academic publications. The page has standard margins, and there are no visible headers, footers, or page numbers in the visibleportion of the image. The text is dense, with no bullet points or subheadings, maintaining a formal and technical tone throughout.
Keywords :diffusion equations, poly-fractional operator, stochastic processes, random walks, Lévý flights, Hadamard ill-posedness, regularization, Gevrey condition, H¨older convergence, MATLAB, numerical examples.
Conference Name :International Conference on Modern Industrial Mathematics (ICMIM-25)
Conference Place Ho Chi Minh City, Vietnam
Conference Date 28th Mar 2025
