### Abstract

A new approach to the reconstruction of a boundary condition in an inverse problem for a nonlinear singularly perturbed reaction–diffusion–advection equation with data on the reaction front position is proposed. The problem is solved via gradient minimization of a cost functional with an initial approximation chosen by applying asymptotic analysis methods. The efficiency of the proposed approach is demonstrated by numerical experiments.

This is a preview of subscription content, access via your institution.

## REFERENCES

V. G. Danilov, V. P. Maslov, and K. A. Volosov,

*Mathematical Modeling of Heat and Mass Transfer Processes*(Kluwer, Dordrecht, 1995).V. F. Butuzov and A. B. Vasil’eva, “Singularly perturbed problems with boundary and interior layers: Theory and applications,” Adv. Chem. Phys.

**97**, 47–179 (1997).Z. Liu, Q. Liu, H.-C. Lin, C. S. Schwartz, Y.-H. Lee, and T. Wang, “Three-dimensional variational assimilation of MODIS aerosol optical depth: Implementation and application to a dust storm over East Asia,” J. Geophys. Res.: Atm.

**116**(D23), 23206 (2010).H. Egger, K. Fellner, J.-F. Pietschmann, and B. Q. Tang, “Analysis and numerical solution of coupled volume-surface reaction–diffusion systems with application to cell biology,” Appl. Math. Comput.

**336**, 351–367 (2018).N. M. Yaparova, “Method for determining particle growth dynamics in a two-component alloy,” Steel Transl.

**50**(2), 95–99 (2020).X. Wu and M. Ni, “Existence and stability of periodic contrast structure in reaction–advection–diffusion equation with discontinuous reactive and convective terms,” Commun. Nonlinear Sci. Numer. Simul.

**91**, 105457 (2020).G. Lin, Y. Zhang, X. Cheng, M. Gulliksson, P. Forssen, and T. Fornstedt, “A regularizing Kohn–Vogelius formulation for the model-free adsorption isotherm estimation problem in chromatography,” Appl. Anal.

**97**(1), 13–40 (2018).Y. Zhang, G. Lin, M. Gulliksson, P. Forssen, T. Fornstedt, and X. Cheng, “An adjoint method in inverse problems of chromatography,” Inverse Probl. Sci. Eng.

**25**(8), 1112–1137 (2017).A. I. Volpert, V. A. Volpert, and Vl. A. Volpert,

*Traveling Wave Solutions of Parabolic Systems*(Am. Math. Soc., Providence, R.I., 2000).H. Meinhardt,

*Models of Biological Pattern Formation*(Academic, London, 1982).R. FitzHugh, “Impulses and physiological states in theoretical model of nerve membrane,” Biophys. J.

**l**(1), 445–466 (1961).J. D. Murray,

*Mathematical Biology*, Vol 1:*An Introduction*(Springer, New York, 2002).H. Egger, J.-F. Pietschmann, and M. Schlottbom, “Identification of nonlinear heat conduction laws,” J. Inverse Ill-Posed Probl.

**23**(5), 429–437 (2015).A. Gholami, A. Mang, and G. Biros, “An inverse problem formulation for parameter estimation of a reaction–diffusion model of low grade gliomas,” J. Math. Biol.

**72**(1–2), 409–433 (2016).R. R. Aliev and A. V. Panfilov, “A simple two-variable model of cardiac excitation,” Chaos Solitons Fractals

**7**(3), 293–301 (1996).E. A. Generalov, N. T. Levashova, A. E. Sidorova, P. M. Chumankov, and L. V. Yakovenko, “An autowave model of the bifurcation behavior of transformed cells in response to polysaccharide,” Biophysics

**62**(5), 876–881 (2017).A. Mang, A. Gholami, C. Davatzikos, and G. Biros, “PDE-constrained optimization in medical image analysis,” Optim. Eng.

**19**(3), 765–812 (2018).S. I. Kabanikhin and M. A. Shishlenin, “Recovering a time-dependent diffusion coefficient from nonlocal data,” Numer. Anal. Appl.

**11**, 38–44 (2018).V. Mamkin, J. Kurbatova, V. Avilov, Yu. Mukhartova, A. Krupenko, D. Ivanov, N. Levashova, and A. Olchev, “Changes in net ecosystem exchange of CO

_{2}, latent and sensible heat fluxes in a recently clear-cut spruce forest in Western Russia: Results from an experimental and modeling analysis,” Environ. Res. Lett.**11**(12), 125012 (2016).N. T. Levashova, J. V. Muhartova, and A. V. Olchev, “Two approaches to describe the turbulent exchange within the atmospheric surface layer,” Math. Models Comput. Simul.

**9**(6), 697–707 (2017).N. Levashova, A. Sidorova, A. Semina, and M. Ni, “A spatio-temporal autowave model of shanghai territory development,” Sustainability

**11**(13), 3658 (2019).S. A. Zakharova, M. A. Davydova, and D. V. Lukyanenko, “A spatio-temporal autowave model of shanghai territory development,” Inverse Probl. Sci. Eng.

**29**(3), 365–377 (2020).V. M. Isakov, S. I. Kabanikhin, A. A. Shananin, M. A. Shishlenin, and S. Zhang, “Algorithm for determining the volatility function in the Black–Scholes model,” Comput. Math. Math. Phys.

**59**(10), 1753–1758 (2019).M. K. Kadalbajoo and V. Gupta, “A brief survey on numerical methods for solving singularly perturbed problems,” Appl. Math. Comput.

**217**(18), 3641–3716 (2010).J. R. Cannon and P. DuChateau, “An inverse problem for a nonlinear diffusion equation,” SIAM J. Appl. Math.

**39**(2), 272–289 (1980).P. DuChateau and W. Rundel, “Unicity in an inverse problem for an unknown reaction term in a reaction–diffusion equation,” J. Differ. Equations

**59**, 155–165 (1985).M. S. Pilant and W. Rundell, “An inverse problem for a nonlinear parabolic equation,” Commun. Partial Differ. Equations

**11**(4), 445–457 (1986).S. I. Kabanikhin, “Definitions and examples of inverse and ill-posed problems,” J. Inverse Ill-Posed Probl.

**16**(4), 317–357 (2008).S. I. Kabanikhin,

*Inverse and Ill-Posed Problems Theory and Applications*(De Gruyter, Berlin, 2011).B. Jin and W. Rundell, “A tutorial on inverse problems for anomalous diffusion processes,” Inverse Probl.

**31**, 035003 (2015).A. Belonosov and M. Shishlenin, “Regularization methods of the continuation problem for the parabolic equation,” Lect. Notes Comput. Sci.

**10187**, 220–226 (2017).B. Kaltenbacher and W. Rundell, “On the identification of a nonlinear term in a reaction–diffusion equation,” Inverse Probl.

**35**, 115007 (2019).A. Belonosov, M. Shishlenin, and D. Klyuchinskiy, “A comparative analysis of numerical methods of solving the continuation problem for 1D parabolic equation with the data given on the part of the boundary,” Adv. Comput. Math.

**45**(2), 735–755 (2019).B. Kaltenbacher and W. Rundell, “The inverse problem of reconstructing reaction–diffusion systems,” Inverse Probl.

**36**(065011) (2020).D. V. Lukyanenko, M. A. Shishlenin, and V. T. Volkov, “Solving of the coefficient inverse problems for a nonlinear singularly perturbed reaction–diffusion–advection equation with the final time data,” Commun. Nonlinear Sci. Numer. Simul.

**54**, 233–247 (2018).D. V. Lukyanenko, I. V. Prigorniy, and M. A. Shishlenin, “Some features of solving an inverse backward problem for a generalized Burgers’ equation,” J. Inverse Ill-posed Probl.

**28**(5), 641–649 (2020).D. V. Lukyanenko, A. A. Borzunov, and M. A. Shishlenin, “Solving coefficient inverse problems for nonlinear singularly perturbed equations of the reaction–diffusion–advection type with data on the position of a reaction front,” Commun. Nonlinear Sci. Numer. Simul.

**99**, 105824 (2021).D. Lukyanenko, T. Yeleskina, I. Prigorniy, T. Isaev, A. Borzunov, and M. Shishlenin, “Inverse problem of recovering the initial condition for a nonlinear equation of the reaction–diffusion–advection type by data given on the position of a reaction front with a time delay,” Mathematics

**9**(4), 342 (2021).D. V. Lukyanenko, V. B. Grigorev, V. T. Volkov, and M. A. Shishlenin, “Solving of the coefficient inverse problem for a nonlinear singularly perturbed two-dimensional reaction–diffusion equation with the location of moving front data,” Comput. Math. Appl.

**77**(5), 1245–1254 (2019).D. V. Lukyanenko, M. A. Shishlenin, and V. T. Volkov, “Asymptotic analysis of solving an inverse boundary value problem for a nonlinear singularly perturbed time-periodic reaction–diffusion–advection equation,” J. Inverse Ill-Posed Probl.

**27**(5), 745–758 (2019).A. B. Vasil’eva, V. F. Butuzov, and N. N. Nefedov, “Singularly perturbed problems with boundary and internal layers,” Proc. Steklov Inst. Math.

**268**, 258–273 (2010).A. N. Tikhonov, A. V. Goncharsky, V. V. Stepanov, and A. G. Yagola,

*Numerical Methods for the Solution of Ill-Posed Problems*(Kluwer Academic, Dordrecht, 1995).O. M. Alifanov, E. A. Artuhin, and S. V. Rumyantsev,

*Extreme Methods for the Solution of Ill-Posed Problems*(Nauka, Moscow, 1988) [in Russian].S. I. Kabanikhin and M. A. Shishlenin, “Quasi-solution in inverse coefficient problems,” J. Inverse Ill-Posed Probl.

**16**(7), 705–713 (2008).E. Hairer and G. Wanner,

*Solving Ordinary Differential Equations*, Vol. 2:*Stiff and Differential-Algebraic Problems*(Springer-Verlag, Berlin, 1996).H. H. Rosenbrock, “Some general implicit processes for the numerical solution of differential equations,” Computer J.

**5**(4), 329–330 (1963).A. Alshin, E. Alshina, N. Kalitkin, and A. Koryagina, “Rosenbrock schemes with complex coefficients for stiff and differential algebraic systems,” Comput. Math. Math. Phys.

**46**, 1320–1340 (2006).X. Wen, “High order numerical methods to a type of delta function integrals,” J. Comput. Phys.

**226**(2), 1952–1967 (2007).H. Egger, H. W. Engl, and M. V. Klibanov, “Global uniqueness and holder stability for recovering a nonlinear source term in a parabolic equation,” Inverse Probl.

**21**(1), 271–290 (2005).L. Belina and M. V. Klibanov, “A globally convergent numerical method for a coefficient inverse problem,” SIAM J. Sci. Comput.

**31**(1), 478–509 (2008).M. V. Klibanov, M. A. Fiddy, L. Beilina, N. Pantong, J. Schenk, “Picosecond scale experimental verification of a globally convergent algorithm for a coefficient inverse problem,” Inverse Probl.

**26**(4), 045003 (2010).D. Chaikovskii and Y. Zhang, “Convergence analysis for forward and inverse problems in singularly perturbed time-dependent reaction-advection-diffusion equations,” (2021). arXiv:2106.15249 [math.NA].

A. G. Yagola, A. S. Leonov, and V. N. Titatenko, “Data errors and an error estimation for ill-posed problems,” Inverse Probl. Eng.

**10**(2), 117–129 (2002).K. Y. Dorofeev, V. N. Titatenko, and A. G. Yagola, “Algorithms for constructing a posteriori errors of solutions to ill-posed problems,” Comput. Math. Math. Phys.

**43**(1), 10–23 (2003).A. S. Leonov, “A posteriori accuracy estimations of solutions to ill-posed inverse problems and extra-optimal regularizing algorithms for their solution,” Numer. Anal. Appl.

**5**(1), 68–83 (2012).

## Funding

This work was supported by the Russian Foundation for Basic Research, project no. 20-31-70016.

## Author information

### Authors and Affiliations

### Corresponding author

## Ethics declarations

The authors declare that they have no conflicts of interest.

## Additional information

Translated by I. Ruzanova

## Rights and permissions

## About this article

### Cite this article

Argun, R.L., Gorbachev, A.V., Lukyanenko, D.V. *et al.* Features of Numerical Reconstruction of a Boundary Condition in an Inverse Problem for a Reaction–Diffusion–Advection Equation with Data on the Position of a Reaction Front.
*Comput. Math. and Math. Phys.* **62**, 441–451 (2022). https://doi.org/10.1134/S0965542522030022

Received:

Revised:

Accepted:

Published:

Issue Date:

DOI: https://doi.org/10.1134/S0965542522030022

### Keywords:

- inverse problem with data on the position of a reaction front
- inverse boundary value problem
- reaction–diffusion–advection equation