A Backward Problem for the Homogeneous Diffusion Equation with Coupling Operator and Random Noise
Main Article Content
Abstract
In this work, we consider the problem of recovering the heat distribution for a homogeneous diffusion equation with white noise. As commonly acknowledged, the problem is severely ill-posed according to Hadamard's definition. Consequently, we propose the Fourier truncation method to regularize this problem. With different assumptions on the exact solution, the estimation of the expectation of the error between the regularized solution and the exact solution was obtained. Finally, we provided an example to illustrate our theoretically obtained results.
Keywords:
Homogeneous diffusion equation, truncation method, regularized solution, heat distribution, white noise.
References
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S. Dipierro, E. Valdinoci, Description of An Ecological Niche for A Mixed Local/Nonlocal Dispersal: an Evolution Equation and a New Neumann Condition Arising from the Superposition of Brownian and Levy Processes, Physica A: Stat. Mech. Appl, Vol. 575, 2021, pp. 126052.
Z. Y. Fang, B. Shang, C. Zhang; Regularity Theory for Mixed Local and Nonlocal Parabolic p-Laplace Equations, J Geom Anal, Vol. 32, No. 1, pp. 22.
D. Mugnai, P. E. Lippi, On Mixed Local and Nonlocal Operators with (α,β) Neuumann Conditions, Renr Circ Mat Palermo Ser 2, Vol. 71, No. 3, 2022, pp.1035-1048.
R. E. Showalter, The Final Value Problem for Evolution Equations, J. Math. Anal. Appl., Vol. 47, 1974,
pp. 563-572.
G. Clark, C. Oppenheimer, Quasireversibility Methods for Non-Well-Posed Problem, Electronic Journal of Differential Equations, Vol. 1994, No. 8, 1994, pp. 1-9.
M. Denche, K. Bessila, A Modified Quasi-boundary Value Method for Ill-posed Problems, J. Math. Anal. Appl, Vol. 301, 2005, pp. 419-426.
G. H. Zheng, Q. G. Zhang; Determining the Initial Distribution in Space - Fractional Diffusion By A Negative Exponential Regularization Method, Inverse Prol. Sci. End., Vol. 25, No. 7, 2017, pp. 965-977.
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B. Kaltenbacher, W. Rundell, Inverse Problems for Fractional Partial Differential Equations, American Mathematical Society, 2023.
Y. V. Egorov, M. A. Shubin, Foundations of the Classical Theory of Partial Differential Equations, Springer Berlin, 2013.
G. F. Roach, I. G. Stratis, A. N. Yannacopoulos, Mathematical Analysis of Deterministic and Stochastic Problems in Complex Media Electromagnetics, Princeton University Press, 2012.
L. Cavalier, Inverse Problems in Statistics. Inverse Problems and High - Dimensional Estimation, Lecture notes in Statistics 203, Springer, 2009.