Analysis of Biharmonic and Harmonic Models by the Methods of Iterative Extensions

The article describes the results of recent years on the analysis of biharmonic and harmonic models by the methods of iterative extensions. In mechanics, hydrodynamics and heat engineering, various stationary physical systems are modeled using boundary value problems for inhomogeneous Sophie Germain and Poisson equations. Deflection of plates, flows during fluid flows are described using the biharmonic model, i.e. boundary value problem for the inhomogeneous Sophie Germain equation. Deflection of membranes, stationary temperature distributions near the plates are described using the harmonic model, i.e. boundary value problem for the inhomogeneous Poisson equation. With the help of the developed methods of iterative extensions, efficient algorithms for solving the problems under consideration are obtained.

Keywords

biharmonic and harmonic models; methods of iterative extensions.

References

1. Aubin J.-P. Approximation of Elliptic Boundary-Value Problems. New York, Wiley-Interscience, 1972.
2. Bank R.E., Rose D.J. Marching Algorithms for Elliptic Boundary Value Problems. I: the Constant Coefficient Case. SIAM Journal on Numerical Analysis, 1977, vol. 14, no. 5, pp. 792-829.
3. Marchuk G.I., Kuznetsov Yu.A., Matsokin A.M. Fictitious Domain and Domain Decomposion Methods. Russian Journal Numerical Analysis and Mathematical Modelling, 1986, vol. 1, no. 1, pp. 3-35.
4. Swarztrauber P.N. A Direct Method for Discrete Solution of Separable Elliptic Equations. SIAM Journal on Numerical Analysis, 1974, vol. 11, no. 6, pp. 1136-1150.
5. Swarztrauber P.N. The Method of Cyclic Reduction, Fourier Analysis and FACR Algorithms for the Discrete Solution of Poisson's Equations on a Rectangle. SIAM Review, 1977, vol. 19, no. 3, pp. 490-501.
6. Manteuffel T. An Incomlete Factorization Technigue for Positive Definite Linear Systems. Mathematics of Computation, 1980, vol. 38, no. 1, pp. 114-123.
7. Matsokin A.M., Nepomnyaschikh S.V. The Fictitious-Domain Method and Explicit Continuation Operators. Computational Mathematics and Mathematical Physics, 1993, vol. 33, no. 1, pp. 52-68.
8. Mukanova B. Numerical Reconstruction of Unknown Boundary Data in the Cauchy Problem for Laplace's Equation. Inverse Problems in Science and Engineering, 2012, vol. 21, no. 8, pp. 1255-1267. DOI: 10.1080/17415977.2012.744405
9. Оganesyan L.А., Rukhovets L.A. Variation-Difference Methods for solving Elliptic Equations [Variatsionno-raznostnye metody resheniya ellipticheskikh uravnenii]. Еrevan АN АrmSSR, 1979. (in Russian)
10. Sorokin S.B. Analytical Solution of Generalized Spectral Problem in the Method of Recalculating Boundary Conditions for a Biharmonic Equation. Siberian Journal Numerical Mathematics, 2013, vol. 16, no. 3, pp. 267-274. DOI: 10.1134/S1995423913030063
11. Sorokin S.B. An Efficient Direct Method for the Numerical Solution to the Cauchy Problem for the Laplace Equation. Numerical Analysis and Applications, 2019, vol. 12, no. 12, pp. 87-103. (in Russian) DOI: 10.1134/S1995423919010075
12. Ushakov А.L. About Modeelling of Deformations of Plates. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2015, vol. 8, no. 2, pp. 138-142. (in Russian) DOI: 10.14529/mmp150213
13. Ushakov A.L. Investigation of a Mixed Boundary Value Problem for the Poisson Equation. International Russian Automation Conference, Sochi, Russian Federation, 2020, article ID: 9208198, 6 p. DOI: 10.1109/RusAutoCon49822.2020.9208198
14. Ushakov A.L. Numerical Anallysis of the Mixed Boundary Value Problem for the Sophie Germain Equation. Journal of Computational and Engineering Mathematics, 2021, vol. 8, no. 1, pp. 46-59. DOI: 10.14529/jcem210104
15. Ushakov А.L. Аnalysis of the Mixed Boundary Value Problem for the Poisson's Equation. Bulletin of the South Ural State University. Series: Mathematics. Mechanics. Physics, 2021, vol. 13, no. 1, pp. 29-40. (in Russian) DOI: 10.14529/mmph210104
16. Ushakov A.L. Research of the Boundary Value Problem for the Sophie Germain Equationinin in a Cyber-Physical System. Studies in Systems, Decision and Control, 2021, vol. 338, pp. 51-63. DOI: 10.1007/978-3-030-66077-2-5
17. Ushakov A.L. Аnalysis of the Boundary Value Problem for the Poisson Equation. Bulletin of the South Ural State University. Series: Mathematics. Mechanics. Physics, 2022, vol. 14, no. 1, pp. 64-76. DOI: 10.14529/mmph220107
18. Ushakov A.L. Anallysis of the Problem for the Biharmonic Equation. Journal of Computational and Engineering Mathematics, 2022, vol. 9, no. 1, pp. 43-58. DOI: 10.14529/jcem220105