一个与非线性反应项相关的时间分数阶平流方程的反问题

IF 1.1 4区 工程技术 Q3 ENGINEERING, MULTIDISCIPLINARY
Hoang-Hung Vo, Triet Le Minh, Phong Luu Hong, Canh Vo Van
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引用次数: 4

摘要

分数阶导数是当代数学研究中的一个重要概念,不仅因为它比经典阶导数在数学上更具有普遍性,而且在理解许多物理现象方面也有实际应用。特别是,分数阶导数与长幂律粒子跳变有关,可以理解为瞬态异常亚扩散模型(见Sabzikar F, Meerschaert M, Chen J.回火分数阶微积分)。计算物理学报,2015;29 (3):14 - 28;刘建军,张建军,张建军,等。物理学报。2002;55:48-54;李建军,李建军,李建军,等。物理世界。2005;18:19-22;张勇,Meerschaert MM, Packman AI。连接河床沉积物跨尺度运输。地球物理学报,2012;39(20):20404。gl053476 doi: 10.1029/2012)。基于Scher H给出的模型,Montroll EW。非晶态固体中的异常跃迁时间色散。郑光华,魏涛。求解时间分数阶逆扩散问题的谱正则化方法。物理学报,1995;12(6):2455-2477。应用数学计算,2011;18:1972 - 1990,研究了二维半无限域中非线性平流方程的反问题,从最终位置x = 1处提供的观测数据恢复了平流方程的初始分布。这个问题在Hadamard意义上是严重病态的。因此,我们提出一种正则化方法来构造该问题的近似解。由此,在精确解的先验界假设下,得到了正则解的收敛速率。最后,通过数值实验验证了所提正则化方法的有效性。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
An inverse problem for a time-fractional advection equation associated with a nonlinear reaction term
Fractional derivative is an important notion in the study of the contemporary mathematics not only because it is more mathematically general than the classical derivative but also it really has applications to understand many physical phenomena. In particular, fractional derivatives are related to long power-law particle jumps, which can be understood as transient anomalous sub-diffusion model (see Sabzikar F, Meerschaert M, Chen J. Tempered fractional calculus. J Comput Phys. 2015;293:14–28; Sokolov IM, Klafter J, Blumen A. Fractional kinetics. Phys Today. 2002;55:48–54; Sokolov IM, Klafter J. Anomalous diffusion spreads its wings. Phys World. 2005;18:19–22; Zhang Y, Meerschaert MM, Packman AI. Linking fluvial bed sediment transport across scales. Geophys Res Lett. 2012;39(20):20404. doi:10.1029/2012GL053476). Based on the models given in Scher H, Montroll EW. Anomalous transit-time dispersion in amorphous solids. Phys Rev B. 1975;12(6):2455–2477 and Zheng GH, Wei T. Spectral regularization method for solving a time-fractional inverse diffusion problem. Appl Math Comput. 2011;218:1972–1990, we study an inverse problem for the advection equation with a nonlinear reaction term in a two-dimensional semi-infinite domain for which we recover the initial distribution from the observation data provided at the final location x = 1. This problem is severely ill-posed in the sense of Hadamard. Thus, we propose a regularization method to construct an approximate solution for the problem. From that, convergence rate of the regularized solution is obtained under some a priori bound assumptions on the exact solution. Eventually, a numerical experiment is given to show the effectiveness of the proposed regularization methods.
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来源期刊
Inverse Problems in Science and Engineering
Inverse Problems in Science and Engineering 工程技术-工程:综合
自引率
0.00%
发文量
0
审稿时长
6 months
期刊介绍: Inverse Problems in Science and Engineering provides an international forum for the discussion of conceptual ideas and methods for the practical solution of applied inverse problems. The Journal aims to address the needs of practising engineers, mathematicians and researchers and to serve as a focal point for the quick communication of ideas. Papers must provide several non-trivial examples of practical applications. Multidisciplinary applied papers are particularly welcome. Topics include: -Shape design: determination of shape, size and location of domains (shape identification or optimization in acoustics, aerodynamics, electromagnets, etc; detection of voids and cracks). -Material properties: determination of physical properties of media. -Boundary values/initial values: identification of the proper boundary conditions and/or initial conditions (tomographic problems involving X-rays, ultrasonics, optics, thermal sources etc; determination of thermal, stress/strain, electromagnetic, fluid flow etc. boundary conditions on inaccessible boundaries; determination of initial chemical composition, etc.). -Forces and sources: determination of the unknown external forces or inputs acting on a domain (structural dynamic modification and reconstruction) and internal concentrated and distributed sources/sinks (sources of heat, noise, electromagnetic radiation, etc.). -Governing equations: inference of analytic forms of partial and/or integral equations governing the variation of measured field quantities.
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