Abstract
We consider the determination of an unknown spatial-dependent coefficient for the pseudoparabolic system with Dirichlet boundary condition from nonlocal inversion input data. Based on the positive property of the solution to the direct problem, the uniqueness and the Lipschitz conditional stability of this nonlinear inverse problem are addressed in Hölder space, by the principle of contraction mapping for specially chosen weight function describing the nonlocal inversion input. Then, an iterative algorithm is established in terms of the fixed point equation for the unknown potential to construct the approximate solution to this inverse problem, with rigorous convergence analysis and the error estimate on the iterative sequence, which also leads to the optimal approximation error for choosing the stopping rule of the iteration process. The proposed iterative algorithm is compared with classical optimization scheme with regularizing term, showing the advantages of our scheme. Some numerical results are presented to validate our proposed scheme.
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References
Prilepko, A.I., Kostin, A.B.: Inverse problems of determining the coefficient in a parabolic equation. I. Sibirsk Mat. Zh. 33(3), 146–155 (1992)
Kamynin, V.L., Kostin, A.B.: Two inverse problems of finding a coefficient in a parabolic equation. Diff. Eqns. 46(3), 372–383 (2010)
Isakov, V.: Inverse Problems for Partial Differential Equations, Vol. 127 of Appl. Math. Sci. Spring, New York (1998)
Prilepko, A.I., Orlovsky, D.G., Vasin, I.A.: Methods for Solving Inverse Problems in Mathematical Physics, Vol. 231 of Monographs and Textbooks in Pure and Applied Mathematics. Marcel Dekker Inc., New York (2000)
Chen, Q., Liu, J.: Solving an inverse parabolic problem by optimization from final measurement data. J. Comput. Appl. Math. 193(1), 183–203 (2006)
Barenblatt, G.I., Entov, V.M., Ryzhik, V.M.: Theory of Fluid Flows Through Natural Rocks. Kluwer Academic Publishers, Dordrecht (1989)
Hassanizadeh, S.M., Gray, W.G.: Thermodynamic basis of capillary pressure in porous media. Water Resour. Res. 29(10), 3389–3405 (1993)
Cuesta, C., Duijn, C.V., Hulshof, J.: Infiltration in porous media with dynamic capillary pressure: Travelling waves. Eur. J. Appl. Math. 11 (4), 381–398 (2000)
Chen, P.J., Gurtin, M.E.: On a theory of heat conduction involving two temperatures. Z. Angew. Math. Phys. 19(4), 614–627 (1968)
Ting, T.W.: Certain non-steady flows of second-order fluids. Arch. Ration. Mech. Anal. 14(1), 1–26 (1963)
Barenblatt, G.I., Bertsch, M., Passo, R.D., Ughi, M.: A degenerate pseudoparabolic regularization of a nonlinear forward-backward heat equation arising in the theory of heat and mass exchange in stably stratified turbulent shear flow. SIAM J. Math. Anal. 24(6), 1414–1439 (1993)
Novick-Cohen, A., Pego, R.L.: Stable patterns in a viscous diffusion equation. Trans. Amer. Math. Soc. 324(1), 331–351 (1991)
Showalter, R.E., Ting, T.W.: Pseudoparabolic partial differential equations. SIAM J. Math. Anal. 1(1), 1–26 (1970)
Ptashnyk, M.: Pseudoparabolic equations with convection. IMA J. Appl. Math. 72(6), 912–922 (2007)
Amirov, S.: On solvability of the first initial-boundary value problem for quasilinear pseudoparabolic equations. Appl. Math. Comput. 246(1), 112–120 (2014)
Rundell, W., Colton, D.L.: Determination of an unknown nonhomogeneous term in a linear partial differential equation from overspecified boundary data. Appl. Anal. 10(3), 231–242 (1980)
Mamayusupov, M.S.: The problem of determining coefficients of a pseudoparabolic equation. In: Studies in Integro-differential Equations, vol. 16, pp 290–297. Ilim, Frunze (1983)
Pyatkov, S.G., Shergin, S.N.: On some mathematical models of filtration type. Bulletin of the South Ural State University Series-Mathematical Modelling Programming and Computer Software 8(2), 105–116 (2015)
Lyubanova, A.S., Tani, A.: An inverse problem for pseudoparabolic equation of filtration: the existence, uniqueness and regularity. Appl. Anal. 90 (10), 1557–1571 (2011)
Lyubanova, A.S., Velisevich, A.V.: Inverse problems for the stationary and pseudoparabolic equations of diffusion. Appl. Anal. 98(11), 1997–2010 (2019)
Fu, J.-L., Liu, J.: Recovery of a potential coefficient in a pseudoparabolic system from nonlocal observation. Appl. Numer. Math. 184, 121–136 (2023)
Rundell, W.: The determination of a parabolic equation from initial and final data. Proc. Amer. Math. Soc. 99(4), 637–642 (1987)
Choulli, M., Yamamoto, M.: Generic well-posedness of an inverse parabolic problem–the Hölder-space approach. Inverse Prob. 12(3), 195–205 (1996)
Kaltenbacher, B., Neubauer, A., Scherzer, O.: Iterative Regularization Methods for Nonlinear Ill-Posed Problems. Walter de Gruyter, New York (2008)
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This work was supported by NSFC (No.11971104).
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Communicated by: Bangti Jin
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Fu, JL., Liu, J. On the determination of the spatial-dependent potential coefficient in a linear pseudoparabolic equation. Adv Comput Math 49, 28 (2023). https://doi.org/10.1007/s10444-023-10023-5
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DOI: https://doi.org/10.1007/s10444-023-10023-5
Keywords
- Pseudoparabolic equation
- Inverse problem
- Uniqueness
- Conditional stability
- Iteration
- Error estimate
- Numerics