On certain nonlinear elliptic equations numerical solving for thermal applications

Аuthors

Cherepanov V. V.

Moscow Aviation Institute (National Research University), 4, Volokolamskoe shosse, Moscow, А-80, GSP-3, 125993, Russia

e-mail: vvcherepanov@yandex.ru

Abstract

The work discusses some important properties of nonlinear differential equations with linear and quasilinear differential operators of elliptic type, as well as their generalizations. The most famous example of such operators is the Laplace – Beltrami operator, which is included in the key equations of the theory of functions of complex variables, the theory of measure potential and electromagnetic wave theory, plasma physics and the theory of heat and mass transfer, as well as other important applications In a narrow sense, this work solves the Dirichlet problem for a second-order elliptic type equation in Euclidean space that has a significant nonlinearity in the source. Two local iterative methods are considered that converge to solve a nonlinear problem under the condition that the source is limited. One of these methods is known but is applicable only in the case of exponential nonlinearity of the source. Another, original method has a significantly larger area of application and is based on some estimates arising from fundamental theorems.

Keywords:

second-order quasilinear elliptic operators, fundamental properties, plasma, heat and mass transfer, nonlinear Poisson equation, principle of contraction mappings, iterative solution

References

1. Gilbarg D., Trudinger N.S. Elliptic partial differential equations of second order. 2nd Ed. Berlin, Heidelberg, New York, Tokyo: Springer–Verlag, 1983, 447 p.

2. Wermer J. Potential theory. Berlin, New York, 1974, 127 p.

3. Reed M., Simon B. Methods of modern mathematical physics. Vol. 1 Functional analysis. Berlin, New York, London: Academic Press, 1972, 341 p.

4. Polianin A.D., Zaitsev V.F., Tjurov A.I. Metodi resheniya nelineinikh uravneniy matematicheskoy fiziki [Methods for solving nonlinear equations of mathematical physics]. Moscow: Fizmatlit, 2005, 256 p. (In Russ.).

5. Cherepanov V.V. O modelirovanii teplovikh vozmuschenii, vnosimikh v razrejennuju plasmu nepodvijnimi kanonitheskimi telami [On the modeling of thermal disturbances introduced into a rarefied plasma by motionless canonical bodies]. Thermal processes in engineering, 2023, vol. 15, no. 10, pp. 448–455.

6. Alekseev B.V., Kotelnikov V.A., Cherepanov V.V. Electrostatitheskiy zond v mnogo komponentnoy plasme [Electrostatic probe in multicomponent plasma]. High Temperature, 1984, vol. 22, no. 2, pp. 395–396. (In Russ.).

7. Hockney R.G., Eastwood J.W. Computer simulation using particles. Bristol, Philadelphia: IOP (Institute of Physics) Publishing, 1988, 568 p.

8. John F. Plane waves and spherical means applied to a partial differential equations. New York: Interscience, 1955, 180 p.

9. Douglis A., Nirenberg L. Interior estimates for elliptic systems of partial differential equations. Communications on Pure and Applied Mathematics, 1955, vol. 8, pp. 503–538.

10. Giraud G. Generalizations des problemes sur les operations du type elliptique. (In France). Bulletin de la Societe Mathematique de France, 1932, vol. 56, pp. 248–352.

11. Sobolev S.L. Ob odnoy teoreme functionalnogo analiza [On a theorem of functional analysis]. Mathematical collection, 1938, vol. 4, pp. 471–497. (In Russ.).

12. Friedrichs K.O. The identity of weak and strong extensions of differential operators. Transactions on the American Mathematical Society, 1944, vol. 55, pp. 132–151.

13. Friedrichs K.O. On the differentiability of solutions of linear elliptic differential equations. Communications on Pure and Applied Mathematics, 1953, vol. 6, pp. 299–326.

14. John F. Derivatives of continuous weak solutions of linear elliptic equations. Communications on Pure and Applied Mathematics, 1953, vol. 6, pp. 327–335.

15. Alifanov O.M., Artyukhin E.A., Rumyantsev S.V. Extreme methods for solving ill – posed problems with applications to inverse heat problems. New York: Begell House, 1995, 292 p.