ISSN 1997-7670 (Print)
ISSN 2541-8785 (Online)

List of issues > Series «Mathematics». 2015. Vol. 11

Application of Numerical Methods for the Volterra Equations of the First Kind that Appear in an Inverse Boundary-Value Problem of Heat Conduction

S. V. Solodusha

In applied problems related to the study of non-stationary thermal processes, often arise a situation where it is impossible to carry out direct measurements of thedesired physical quantity and its characteristics are restored on the results of indirect measurements. In this case the only way to finding the required values is associated with the solution of the inverse heat conduction problem with the initial data, known only to the part of the boundary. Such problems appear not only in the study of thermal processes, but also in the study of diffusion processes and studying the properties of materials related to the thermal characteristics. This article is devoted to the approximate solution of the Volterra equations of the first kind received as a result of the integral Laplace transform to solve the heat equation. The work consists of an introduction and three sections. In the first two sections the specificity of Volterra kernels of the corresponding integral equations and peculiarity of computing kernels over the machine arithmetic operations on real numbers with floating point are considered. In tests typically systematic accumulation of errors are illustrated. The third section presents the results of numerical algorithms based on product integration method and middle rectangles quadrature. The conditions under which used algorithms are stable and converge to the exact solution in the case of fixed digit grid in the computer representation of numbers are allocated. Series of test calculations are carried out in order to test the efficacy of difference methods.

inverse boundary-value problem of heat conduction, Volterra integral equations of the first kind, numerical methods

1. Yaparova N.M. Numerical methods for solving a boundary value inverse heatconduction problem. Inverse Problems in Science and Engineering, 2014, vol. 22,no 5, pp. 832-847.

2. Alifanov O.M., Budnik S.A., Nenarokomov A.V., Netelev A.V. Identification ofMathematical Models of Heat Transfer in Decomposing Materials [Identifikatsiya matematicheskikh modeley teploperenosa v razlagayushchikhsya materialakh].Teplovye protsessy v tekhnike [Thermal processes in engineering], 2011, no 8, pp.338-347.

3. Gamov P.A., Drozin A.D., Dudorov M.V., Roshchin V.E. The Growth Model ofNanocrystals in the Amorphous Alloy [Model’ rosta nanokristallov v amorfnomsplave]. Metally [Metals], 2012, no 6, pp. 101-106.

4. Korotkii A.I., Kovtunov D.A. Reconstruction of the boundary conditions inthe inverse problem of thermal convection of a viscous fluid [Rekonstruktsiyagranichnykh rezhimov v obratnoy zadache teplovoy konvektsii vysokovyazkoyzhidkosti]. Proceedings of the Steklov Institute of Mathematics, 2006, vol. 255, no2, pp. S81-S92.

5. Balabanov P.V., Ponomarev S.V., Trofimov A.V. Mathematical Modeling of HeatTransfer During of Chemosorption [Matematicheskoe modelirovanie teploperenosav protsesse khemocorbtsii]. Vestnik TGTU [Bulletin of the Tambov State TechnicalUniversity], 2008, vol. 14, no 2, pp. 334-341.

6. Beilina L., Klibanov M.V. Approximate Global Convergence and Adaptivity forCoefficient Inverse Problems. New York, Springer, 2012.

7. Kabanikhin S.I. Inverse and Ill-Posed Problems. Theory and Applications.Germany, De Gruyter, 2011.

8. Brunner H., van der Houwen P.J. The Numerical Solution of Volterra Equations.North-Holland, Amsterdam, 1986.

9. Brunner H. Collocation methods for Volterra integral and related functionaldifferential equations. New York, Cambridge Univ. Press, 2004.

10. Verlan’ A.F., Sizikov V.S. Integral equations: methods, algorithms, programs[Integralnye uravneniya: metody, algoritmy, programmy]. Kiev, Nauk. dumka,1986.

11. Apartsyn A.S. Nonclassical linear Volterra equations of the first kind. Boston, VSPUtrecht, 2003.

12. Solodusha S.V., Yaparova N.M. Numerical solution of the Volterra equationsof the first kind that appear in an inverse boundary-value problem of heatconduction. To appear in Numerical Analysis and Applications, available at:http://arxiv.org/abs/1407.1678 (accessed 1 November 2014).

13. Kalitkin N.N. Numerical methods [Chislennye metody]. M., Nauka, 1978.

14. Mokry I.V., Khamisov O.V., Tsapakh A.S. The Basic Mechanisms of theEmergence of Computational Errors in Computer Calculations [Osnovnyemekhanizmy vosniknoveniya vychislitelnoy oshibki pri kompyuternykh raschetakh].Materialy IV Vseross. konf. "Problemy optimizatsii i economicheskie prilozheniya"[Proc. IVth All-Russian Conference "Problems of Optimization and EconomicApplications" ], Omsk, Nasledie, 2009, pp. 185.

15. Linz P. Product integration method for Volterra integral equations of the first kind.BIT, 1971, vol. 11, pp. 413-421.

16. Geng F.Z., Cui M.G. Analytical Approximation to Solutions of SingularlyPerturbed Boundary Value Problems. Bulletin of the Malaysian MathematicalSciences Society, 2010, vol. 33, no 2, pp. 221-232.

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