Numerical simulations for initial value inversion problem in a two-dimensional degenerate parabolic equation

Zui-Cha Deng; Fan-Li Liu; Liu Yang; Zui-Cha Deng; Fan-Li Liu; Liu Yang

doi:10.3934/math.2021187

AIMS Mathematics

2021, Volume 6, Issue 4: 3080-3104. doi: 10.3934/math.2021187

Previous Article Next Article

Research article

Numerical simulations for initial value inversion problem in a two-dimensional degenerate parabolic equation

1.
School of Mathematics and Physics, Lanzhou Jiaotong University, Lanzhou, Gansu, 730070, People's Republic of China
2.
Computer Science and Technology Experimental Teaching Center, Lanzhou Jiaotong University, Lanzhou, Gansu, 730070, People's Republic of China

Received: 05 November 2020 Accepted: 31 December 2020 Published: 12 January 2021
MSC : 35R30, 49J20

In this paper, we study the inverse problem of identifying the initial value of a two-dimensional degenerate parabolic equation, which often appears in the fields of engineering, physics, and computer image processing. Firstly, the difference scheme of forward problem is established by using the finite volume method. Then stability and convergence of the difference equations are proved rigorously. Finally, the Landweber iteration and conjugate gradient method are used to solve the inverse problem, and some typical numerical examples are shown to verify the validity of our iterative algorithm. Numerical results show that the algorithm is stable and efficient.
- initial value inversion problem,
- degenerate parabolic equation,
- difference scheme,
- stability,
- numerical simulation
Citation: Zui-Cha Deng, Fan-Li Liu, Liu Yang. Numerical simulations for initial value inversion problem in a two-dimensional degenerate parabolic equation[J]. AIMS Mathematics, 2021, 6(4): 3080-3104. doi: 10.3934/math.2021187

Related Papers:

Abstract

In this paper, we study the inverse problem of identifying the initial value of a two-dimensional degenerate parabolic equation, which often appears in the fields of engineering, physics, and computer image processing. Firstly, the difference scheme of forward problem is established by using the finite volume method. Then stability and convergence of the difference equations are proved rigorously. Finally, the Landweber iteration and conjugate gradient method are used to solve the inverse problem, and some typical numerical examples are shown to verify the validity of our iterative algorithm. Numerical results show that the algorithm is stable and efficient.

References

[1]	G. Albuja, A.I. Ávila, A family of new globally convergent linearization schemes for solving Richards' equation, Appl. Numer. Math., 159 (2021), 281-296.
[2]	K. Beauchard, P. Cannarsa, M. Yamamoto, Inverse source problem and null controllability for multidimensional parabolic operators of Grushin type, Inverse Problems, 30 (2014), 025006. doi: 10.1088/0266-5611/30/2/025006
[3]	M. Berardi, F. Difonzo, F. Notarnicola, M. Vurro, A transversal method of lines for the numerical modeling of vertical infiltration into the vadose zone, Appl. Numer. Math., 135 (2019) 264-275.
[4]	M. Berardi, F. Difonzo, L. Lopez, A mixed MoL-TMoL for the numerical solution of the 2D Richards' equation in layered soils, Comput. Math. Appl., 79 (2020), 1990-2001. doi: 10.1016/j.camwa.2019.07.026
[5]	N. Brandhorst, D. Erdal, I. Neuweiler, Soil moisture prediction with the ensemble Kalman filter: Handling uncertainty of soil hydraulic parameters, Adv. Water Res., 110 (2017), 360-370. doi: 10.1016/j.advwatres.2017.10.022
[6]	P. Cannarsa, J. Tort, M. Yamamoto, Determination of source terms in a degenerate parabolic equation, Inverse Problems, 26 (2010), 105003. doi: 10.1088/0266-5611/26/10/105003
[7]	P. Cannarsa, P. Martinez, J. Vancostenoble, Carleman estimates for a class of degenerate parabolic operators, SIAM J Control Optim, 47 (2008), 1-19. doi: 10.1137/04062062X
[8]	P. Cannarsa, P. Martinez, J. Vancostenoble, Null controllability of degenerate heat equations, Adv. Differ. Equ., 10 (2005), 153-190.
[9]	J. R. Cannon, The One-Dimensional Heat Equation, Addison-Wesley, 1984.
[10]	J. R. Cannon, Y. Lin, S. Xu, Numerical procedure for the determination of an unknown coefficient in semilinear parabolic partial differential equations, Inverse Problems, 10 (1994), 227-243.
[11]	J. Cheng, J. J. Liu, A quasi Tikhonov regularization for a two-dimensional backward heat problem by a fundamental solution, Inverse Problems, 24 (2008), 065012. doi: 10.1088/0266-5611/24/6/065012
[12]	M. Dehghan, Identification of a time-dependent coefficient in a partial differential equation subject to an extra measurement, Numer. Meth. Part. Diff. Equ., 21 (2005), 611-622. doi: 10.1002/num.20055
[13]	M. Dehghan, Determination of a control function in three-dimensional parabolic equations, Math. Comput. Simul., 61 (2003), 89-100. doi: 10.1016/S0378-4754(01)00434-7
[14]	M. Dehghan, M. Tatari, Determination of a control parameter in a one-dimensional parabolicequation using the method of radial basis functions, Math. Comput. Model., 44 (2006), 1160-1168. doi: 10.1016/j.mcm.2006.04.003
[15]	M. Dehghan, An inverse problems of finding a source parameter in a semilinear parabolic equation, Appl. Math. Model., 25 (2001), 743-754. doi: 10.1016/S0307-904X(01)00010-5
[16]	Z. C. Deng, K. Qian, X. B. Rao, L. Yang, G. W. Luo, An inverse problem of identifying the source coefficient in a degenerate heat equation, Inverse Probl. Sci. Eng., 23 (2015), 498-517. doi: 10.1080/17415977.2014.922079
[17]	F. L. Dimet, V. Shutyaev, J. Wang, M. Mu, The problem of data assimilation for soil water movement, ESAIM: Control, Optimisation and Calculus of Variations, 10 (2004), 331-345. doi: 10.1051/cocv:2004009
[18]	A. Kirsch, An introduction to the mathematical theory of inverse problem, Springer, New York, 1999.
[19]	H. W. Engl, M. Hanke, A. Neubauer, Regularization of inverse problems, Dordrecht: Kluwer Academic Publishers, 1996.
[20]	V. Isakov, Inverse Problems for Partial Differential Equations, Springer, New York, 1998.
[21]	J. F. Lu, Z. Guan, Numerical Solution of Partial Differential Equations, Tsinghua University Press, Beijing, 2004.
[22]	P. Martinez, J. Vancostenoble, Carleman estimates for one-dimensional degenerate heat equations, J. Evol. Equ., 6 (2006), 325-362. doi: 10.1007/s00028-006-0214-6
[23]	O. A. Oleinik, E. V. Radkevic, Second order differential equations with non-negative characteristic form, Rhode Island and Plenum Press, New York: American Mathematical Society, 1973.
[24]	M. Hanke, Conjugate Gradient Type Methods for Ill-Posed Problems, Harlow, Longman Scientific and Technical, Essex, 1995.
[25]	I. S. Pop, Regularization Methods in the Numerical Analysis of Some Degenerate Parabolic Equations, IWR, University of Heidelberg, 1998.
[26]	X. B. Rao, Y. X. Wang, K. Qian, Z. C. Deng, L. Yang, Numerical simulation for an inverse source problem in a degenerate parabolic equation, Appl. Math. Model., 39 (2015), 7537-7553. doi: 10.1016/j.apm.2015.03.016
[27]	R. B. Ricardo, Numerical Methods and Analysis for Degenerate Parabolic Equations and Reaction-Diffusion Systems, 2008.
[28]	Z. Z. Sun, Numerical Solution of Partial Differential Equations, Science Press, Beijing, 2005.
[29]	J. Tort, J. Vancostenoble, Determination of the insolation function in the nonlinear Sellers climate model, Ann. I. H. Poincare-AN, 29 (2012), 683-713. doi: 10.1016/j.anihpc.2012.03.003
[30]	D. K. Wang, Y. Q. Hou, J. Y. Peng, Partial Differential Equation Method for Image Processing, Science Press, Beijing, 2008.
[31]	L. Yang, Z. C. Deng, J. N. Yu, G. W. Luo, Optimization method for the inverse problem of reconstructing the source term in a parabolic equation, Math. Comput. Simul., 80 (2009), 314-326. doi: 10.1016/j.matcom.2009.06.031
[32]	L. Yang, Z. C. Deng, An inverse backward problem for degenerate parabolic equations, Numer. Meth. Part. Differ. Equ., 33 (2017), 1900-1923. doi: 10.1002/num.22165
[33]	L. Yang, Y. Liu, Z. C. Deng, Multi-parameters identification problem for a degenerate parabolic equation, J. Comput. Appl. Math., 366 (2020), 112422. doi: 10.1016/j.cam.2019.112422

Reader Comments

Your name:*

Email:*
© 2021 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)