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Stability analysis for some numerical schemes of partial differential equation with extra measurements

Year 2019, Volume: 48 Issue: 5, 1324 - 1335, 08.10.2019
https://doi.org/10.15672/HJMS.2019.661

Abstract

This paper is devoted to study the stability analysis of some finite difference schemes for  an inverse problem with unknowns time-dependent coefficients subject to extra measurements. We prove that the popular forward time centered space scheme is a conditional method. But the backward time centered space and Crank Nicolson methods are suitable schemes because they are unconditional methods. We justify this advantage of the stability analysis versus the some numerical methods with an example. All the results and a numerical example are in two-dimensional setting.

References

  • [1] M. Asadzadeh, D. Rostamy and F. Zabihi, A posteriori error estimates for the solu- tions of a coupled wave system, J. Math. Sci. 175 (2), 228–248, 2011.
  • [2] M. Asadzadeh, D. Rostamy and F. Zabihi, Discontinuous Galerkin and multiscale variational schemes for a coupled damped nonlinear system of Schrödinger equations, Numer. Methods Partial Differential Equations, 29 (5), 1–34, 2013.
  • [3] C. Ashyralyyev and M. Dedeturk, A finite difference method for the inverse elliptic problem with the dirichlet condition, Contemp. Anal. Appl. Math. 1, 132–155, 2013.
  • [4] H. Azari, W. Allegretto, Y. Lin and S. Zhang, Numerical procedures for recovering a time dependent coefficient in a parabolic differential equation, Dyn. Contin. Discrete Impuls. Syst. Ser. B Appl. Algorithms, 11, 181–199, 2004.
  • [5] J. Blazek, Computational Fluid Dynamics: Principles and Applications, Elsevier, 2005.
  • [6] J.R. Cannon, Determination of an unknown coefficient in a parabolic differential equa- tion, Duke. Math. J. 30, 313–323, 1963.
  • [7] J.R. Cannon and H.M. Yin, Numerical solutions of some parabolic inverse problems, Numer. Methods Partial Differential Equations, 2, 177–191, 1990.
  • [8] M. Dehghan, Identification of a time-dependent coefficient in a partial differential equation subject to an extra measurement, Numer. Methods Partial Differential Equa- tions, 21, 611–622, 2005.
  • [9] S.K. Godunov and V.S. Ryabenkii, Difference Schemes: An Introduction to the Un- derlying Theory, Studies in Mathematics and Its Applications, 1987.
  • [10] P. Jonas and A.K. Louis, Approximate inverse for a one-dimensional inverse heat conduction problem, Inverse Problems, 16, 175–185, 2000.
  • [11] B.F. Jones, Various methods for finding unknown coefficients in a parabolic differential equations,Comm. Pure Appl. Math. 16, 33–44, 1963.
  • [12] A. Kirsch, An Introduction to the Mathematical Theory of Inverse Problems, Springer, New York, 1999.
  • [13] M. Lakestani and M. Dehghan, The use of chebyshev cardinal functions for the so- lution of a partial differential equation with an unknown time-dependent coefficient subject to an extra measurement, J. Comput. Appl. Math. 235, 669–678, 2010.
  • [14] D. Lesnic and L. Elliot, The decomposition approach to inverse heat conduction, J. Math. Anal. Appl. 232, 82–98, 1999.
  • [15] Y. Lin, Analytical and numerical solutions for a class of nonlocal nonlinear parabolic differential equations, SIAM J. Math. Anal. 25 (6), 1577–1594, 1994.
  • [16] J.A. Macbain, Inversion theory for a parameterized diffusion problem, SIAM J. Appl. Math. 18, 1386–1391, 1987.
  • [17] A.R. Mitchell and D.F. Griffiths, The finite difference method in partial differential equations, John Wiley & Sons, Chichester, 1980.
  • [18] A. Mohebbi and M. Dehghan, High-order scheme for determination of a control pa- rameter in an inverse problem from the over-specified data, Commun. Comput. Phys. 181, 1947–1954, 2010.
  • [19] D. Rostamy, The new streamline diffusion for 3D Coupled Schrodinger equations with a cross-phase modulation, ANZIAM J. 55, 51–78, 2013.
  • [20] D. Rostamy, F. Zabihi, A. Niroomand and A. Mollazeynal, New Finite Element Method for Solving a Wave Equation with a Nonlocal Conservation Condition, Trans- port Theory Statist. Phys. 42, 41–62, 2013.
  • [21] D. Rostamy and A. Abdollahi, Gegenbauer Cardinal Functions for the Inverse Source Parabolic Problem with a Time-Fractional Diffusion Equation, Transport Theory Statist. Phys. 46, 307–329, 2017.
  • [22] A. Samarskii and A. Vabishchevich, Numerical methods for solving inverse problems of mathematical physics, Walter de Gruyter, Berlin, 2007.
  • [23] J.C. Strikwerda, Finite difference schemes and partial differential equations, Wadsworth & Brooks/ Cole, California, 1989.
  • [24] A. Tikhonov and V. Arsenin, Solution methods for Ill-posed problems, Nauka, Moscow, 1986.
  • [25] A. Tveito and R. Winther, Introduction to Partial Differential Equations, Springer- Verlag, New York, Inc., 1998.
Year 2019, Volume: 48 Issue: 5, 1324 - 1335, 08.10.2019
https://doi.org/10.15672/HJMS.2019.661

Abstract

References

  • [1] M. Asadzadeh, D. Rostamy and F. Zabihi, A posteriori error estimates for the solu- tions of a coupled wave system, J. Math. Sci. 175 (2), 228–248, 2011.
  • [2] M. Asadzadeh, D. Rostamy and F. Zabihi, Discontinuous Galerkin and multiscale variational schemes for a coupled damped nonlinear system of Schrödinger equations, Numer. Methods Partial Differential Equations, 29 (5), 1–34, 2013.
  • [3] C. Ashyralyyev and M. Dedeturk, A finite difference method for the inverse elliptic problem with the dirichlet condition, Contemp. Anal. Appl. Math. 1, 132–155, 2013.
  • [4] H. Azari, W. Allegretto, Y. Lin and S. Zhang, Numerical procedures for recovering a time dependent coefficient in a parabolic differential equation, Dyn. Contin. Discrete Impuls. Syst. Ser. B Appl. Algorithms, 11, 181–199, 2004.
  • [5] J. Blazek, Computational Fluid Dynamics: Principles and Applications, Elsevier, 2005.
  • [6] J.R. Cannon, Determination of an unknown coefficient in a parabolic differential equa- tion, Duke. Math. J. 30, 313–323, 1963.
  • [7] J.R. Cannon and H.M. Yin, Numerical solutions of some parabolic inverse problems, Numer. Methods Partial Differential Equations, 2, 177–191, 1990.
  • [8] M. Dehghan, Identification of a time-dependent coefficient in a partial differential equation subject to an extra measurement, Numer. Methods Partial Differential Equa- tions, 21, 611–622, 2005.
  • [9] S.K. Godunov and V.S. Ryabenkii, Difference Schemes: An Introduction to the Un- derlying Theory, Studies in Mathematics and Its Applications, 1987.
  • [10] P. Jonas and A.K. Louis, Approximate inverse for a one-dimensional inverse heat conduction problem, Inverse Problems, 16, 175–185, 2000.
  • [11] B.F. Jones, Various methods for finding unknown coefficients in a parabolic differential equations,Comm. Pure Appl. Math. 16, 33–44, 1963.
  • [12] A. Kirsch, An Introduction to the Mathematical Theory of Inverse Problems, Springer, New York, 1999.
  • [13] M. Lakestani and M. Dehghan, The use of chebyshev cardinal functions for the so- lution of a partial differential equation with an unknown time-dependent coefficient subject to an extra measurement, J. Comput. Appl. Math. 235, 669–678, 2010.
  • [14] D. Lesnic and L. Elliot, The decomposition approach to inverse heat conduction, J. Math. Anal. Appl. 232, 82–98, 1999.
  • [15] Y. Lin, Analytical and numerical solutions for a class of nonlocal nonlinear parabolic differential equations, SIAM J. Math. Anal. 25 (6), 1577–1594, 1994.
  • [16] J.A. Macbain, Inversion theory for a parameterized diffusion problem, SIAM J. Appl. Math. 18, 1386–1391, 1987.
  • [17] A.R. Mitchell and D.F. Griffiths, The finite difference method in partial differential equations, John Wiley & Sons, Chichester, 1980.
  • [18] A. Mohebbi and M. Dehghan, High-order scheme for determination of a control pa- rameter in an inverse problem from the over-specified data, Commun. Comput. Phys. 181, 1947–1954, 2010.
  • [19] D. Rostamy, The new streamline diffusion for 3D Coupled Schrodinger equations with a cross-phase modulation, ANZIAM J. 55, 51–78, 2013.
  • [20] D. Rostamy, F. Zabihi, A. Niroomand and A. Mollazeynal, New Finite Element Method for Solving a Wave Equation with a Nonlocal Conservation Condition, Trans- port Theory Statist. Phys. 42, 41–62, 2013.
  • [21] D. Rostamy and A. Abdollahi, Gegenbauer Cardinal Functions for the Inverse Source Parabolic Problem with a Time-Fractional Diffusion Equation, Transport Theory Statist. Phys. 46, 307–329, 2017.
  • [22] A. Samarskii and A. Vabishchevich, Numerical methods for solving inverse problems of mathematical physics, Walter de Gruyter, Berlin, 2007.
  • [23] J.C. Strikwerda, Finite difference schemes and partial differential equations, Wadsworth & Brooks/ Cole, California, 1989.
  • [24] A. Tikhonov and V. Arsenin, Solution methods for Ill-posed problems, Nauka, Moscow, 1986.
  • [25] A. Tveito and R. Winther, Introduction to Partial Differential Equations, Springer- Verlag, New York, Inc., 1998.
There are 25 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Mathematics
Authors

Nazi Abdollahi This is me 0000-0002-5340-6982

Davood Rostamy This is me 0000-0001-9585-8904

Publication Date October 8, 2019
Published in Issue Year 2019 Volume: 48 Issue: 5

Cite

APA Abdollahi, N., & Rostamy, D. (2019). Stability analysis for some numerical schemes of partial differential equation with extra measurements. Hacettepe Journal of Mathematics and Statistics, 48(5), 1324-1335. https://doi.org/10.15672/HJMS.2019.661
AMA Abdollahi N, Rostamy D. Stability analysis for some numerical schemes of partial differential equation with extra measurements. Hacettepe Journal of Mathematics and Statistics. October 2019;48(5):1324-1335. doi:10.15672/HJMS.2019.661
Chicago Abdollahi, Nazi, and Davood Rostamy. “Stability Analysis for Some Numerical Schemes of Partial Differential Equation With Extra Measurements”. Hacettepe Journal of Mathematics and Statistics 48, no. 5 (October 2019): 1324-35. https://doi.org/10.15672/HJMS.2019.661.
EndNote Abdollahi N, Rostamy D (October 1, 2019) Stability analysis for some numerical schemes of partial differential equation with extra measurements. Hacettepe Journal of Mathematics and Statistics 48 5 1324–1335.
IEEE N. Abdollahi and D. Rostamy, “Stability analysis for some numerical schemes of partial differential equation with extra measurements”, Hacettepe Journal of Mathematics and Statistics, vol. 48, no. 5, pp. 1324–1335, 2019, doi: 10.15672/HJMS.2019.661.
ISNAD Abdollahi, Nazi - Rostamy, Davood. “Stability Analysis for Some Numerical Schemes of Partial Differential Equation With Extra Measurements”. Hacettepe Journal of Mathematics and Statistics 48/5 (October 2019), 1324-1335. https://doi.org/10.15672/HJMS.2019.661.
JAMA Abdollahi N, Rostamy D. Stability analysis for some numerical schemes of partial differential equation with extra measurements. Hacettepe Journal of Mathematics and Statistics. 2019;48:1324–1335.
MLA Abdollahi, Nazi and Davood Rostamy. “Stability Analysis for Some Numerical Schemes of Partial Differential Equation With Extra Measurements”. Hacettepe Journal of Mathematics and Statistics, vol. 48, no. 5, 2019, pp. 1324-35, doi:10.15672/HJMS.2019.661.
Vancouver Abdollahi N, Rostamy D. Stability analysis for some numerical schemes of partial differential equation with extra measurements. Hacettepe Journal of Mathematics and Statistics. 2019;48(5):1324-35.