On determination of minor coefficient in a parabolic equation of the second order

Бесплатный доступ

An inverse problem of recovering the minor time-dependent coefficient in a parabolic equation of the second order is considered. The unknown coefficient is the controlling parameter. The inverse problem lies in finding the solution of an initial-boundary value problem for this parabolic equation and this time-dependent coefficient using data of the initial-boundary value problem and point conditions of overdetermination. Cases of the Dirichlet boundary conditions and oblique derivative conditions are considered. Conditions under which the theorem of existence and solution uniqueness is applicable for the given inverse problem is described; the numerical solution method is described, and its justification is given. All the considerations are carried out in Sobolev spaces. Solution of the direct problem is based on the finite element method and the finite difference method. The proposed algorithm for the numerical solution consists of three stages: initialization of the massive that describes geometry of the area and the boundary vector; implementation of integrative calculation of the desired coefficient using the finite element method; implementation of the finite difference method. Results of numerical experiments are presented, and numerical solution of the model inverse problem is constructed in the case of Neumann boundary conditions; dependency of an error in calculation of the controlling parameter on the variation of the equation coefficients and the noise level of the overdetermination data for domains with different number of nodes that depend on an observation point is described. Results of the calculations show a good convergence of the method. In the case when introduced noise level is 10 %, the error between the desired and the obtained solution increases from 8 to 35 times, though the graph of recovered coefficient remains close to the solution graph and repeats its outlines.

Еще

Finite element method, parabolic equation, inverse problem

Короткий адрес: https://sciup.org/147232792

IDR: 147232792   |   DOI: 10.14529/mmph180404

Список литературы On determination of minor coefficient in a parabolic equation of the second order

  • Alifanov, O.M. Inverse Heat Transfer Problems / O.M. Alifanov. - Springer-Verlag, Berlin Heidelberg, 1994. - 331 p.
  • Kabanikhin, S.I. Inverse and Ill-Posed Problems: theory and applications / S.I. Kabanikhin. - De Gruyter, Berlin/Boston, 2012. - 459 p.
  • Marchuk, G.I. Mathematical Models in Environmental Problems. Studies in Mathematics and its Applications / G.I. Marchuk. - Elsevier, 1986. - Vol. 16. - 216 p.
  • Orlande, H.R.B. Inverse problems in heat transfer: New trends on solution methodologies and applications / H.R.B. Orlande // Journal of Heat Transfer. - 2012. - Vol. 134. - Issue 3. - 031011 (13 pages).
  • Ozisik, M.N. Inverse heat transfer / M.N. Ozisik, H.A.B. Orlando. - New-York: Taylor & Francis, 2000. - 352 p.
  • Prilepko, A.I. Methods for solving inverse problems in Mathematical Physics / A.I. Prilepko, D.G. Orlovsky, I.A. Vasin. - New-York: Marcel Dekker, Inc., 2000. - 744 p.
  • Прилепко, А.И. О разрешимости обратных краевых задач определения коэффициента перед младшей производной в параболическом уравнении / А.И. Прилепко, В.В. Соловьев // Дифференциальные уравнения. - 1987. - Т. 23, № 1. - С. 136-143.
  • Pyatkov, S.G. Parameter identification and control in heat transfer processes / S.G. Pyatkov, O.V. Goncharenko // Вестник ЮУрГУ. Серия «Математическое моделирование и программирование». - 2017. - Т. 10, № 2. - С. 51-62.
  • Пятков, С.Г. О некоторых классах коэффициентных обратных задач для параболических систем уравнений / С.Г. Пятков, М.Л. Самков // Математические труды. - 2012. - Т. 15, № 1. - С. 155-177.
  • Pyatkov, S.G. On some parabolic inverse problems with the pointwise overdetermination / S.G. Pyatkov, V.V. Rotko // AIP Conference Proceedings. - 2017. - Vol. 1907. - Issue 1. - 020008.
  • Dehghan, M. Method of lines solutions of the parabolic inverse problem with an overspecification at a points / M. Dehghan, F. Shakeri // Numerical Algorithms. - 2009. - Vol. 50, no 4. - P. 417-437.
  • Dehghan, M. Numerical computation of a control function in a partial differential equation / M. Dehghan // Applied mathematics and computation. - 2004. - Vol. 147. - Issue 2. - P. 397-408.
  • Dehghan, M. Determination of a control parameter in the two-dimensional diffusion equation / M. Dehghan // Applied Numerical Mathematics. - 2001. - Vol. 37. - Issue 4. - P. 489-502.
  • Dehghan, M. Fourth-order techniques for identifying a control parameter in the parabolic equations / M. Dehghan // International Journal of Engineering Science. - 2002. - Vol. 40. - Issue 4. - P. 433-447.
  • Wang, S. Numerical solutions to two inverse problems for identifying control parameters in 2-dimensional parabolic partial differential equations / S. Wang // American Society of Mechanical Engineers, Heat Transfer Division, HTD. - 1992. - Vol. 194. - P. 11-16. https://www.scopus.com/inward/record.uri?eid=2-s2.0-0027069329&partnerID=40&md5= cdb8e3110b98004d91707746c1ead322
  • Wang, W. Inverse heat problem of determining time-dependent source parameter in reproducing kernel space / W. Wang, B. Han, M. Yamamoto // Nonlinear Analysis: Real World Applications. - 2013. - Vol. 14, no. 1. - pp. 875-887.
  • Li Fu-le, Wu Zi-ku, Ye Chao-rong. A finite difference solution to a two-dimensional parabolic inverse problem / Fu-le Li, Zi-ku Wu, Chao-rong Ye // Applied Mathematical Modelling. - 2012. - Vol. 36, Issue 5. - pp. 2303-2313.
  • Tatari, M. He's variational iteration method for computing a control parameter in a semi-linear inverse parabolic equation / M. Tatari, M. Dehghan // Chaos, Solitons and Fractals. - 2007. - Vol. 33. - Issue 2. - pp. 671-677.
  • Mohebbi, A. A numerical algorithm for determination of a control parameter in two-dimensional parabolic inverse problems / A. Mohebbi // Acta Mathematicae Applicatae Sinica. English series. - 2015. - Vol. 31. - Issue 1. - pp. 213-224.
  • Vabishchevich, P.N. Numerically solving the identification problem for the lower coefficient of a parabolic equation / P.N. Vabishchevich, V.I. Vasil'ev // Математические заметки СВФУ. - 2014. - Т. 21, № 4. - С. 71-87.
  • Triebel, H. Interpolation Theory, Function Spaces, Differential Operators / H. Triebel. - Berlin, VEB Deutscher Verlag der Wissenschaften, 1978. - 528 p.
Еще
Статья научная