The work objective is to obtain an integral equation by which, using the known fundamental solution to the other equation, it is possible to find a fundamental solution to the linear elliptic equation. The concept of a numerical fundamental solution (NFS) is introduced. The so obtained numerical fundamental solutions (NFS) can be used for solving boundary value problems for N-dimensional elliptic equations by the field point source method (PSM). The re-search result is the development of the effective numerical method for solving boundary value problems using the NFS. This allows expanding the range of solvable problems using PSM, making PSM a universal numerical method for solving boundary value problems for linear elliptic equations. It admits solutions to various types of boundary value problems. Especially effective is the use of the proposed method for solving three-dimensional Dirichlet problems for equations with spherically symmetric fundamental solutions. The Schrödinger equation for a one-dimensional quantum oscillator is solved by the proposed method as a test problem. It is shown that it is possible to find the eigenvalues and eigenfunctions of the quantum oscillator using numerically obtained fundamental solutions to the Schrödinger equation. The oscillator eigenfunctions obtained by the proposed method are in good agreement with the known analytical solutions to quantum problems. Then, as another test example, a two-dimensional boundary value problem for the Helmholtz equation is solved. In this case, it is necessary to obtain a numerical fundamental solution to the Helmholtz equation first. Dependences of the numerical solution error on the nodes number in the problem solution domain are calculated. Upon the results obtained, the following conclusion is made. The results of solving test problems confirm the efficiency of the proposed numerical method.
fundamental solution, method of fundamental solutions, point source method, integrated sources method, discrete sources.
Метод точечных источников поля (МТИ) является одним из эффективных методов моделирования физических полей (например, электрических и магнитных) в технических (в том числе электромеханических) устройствах [1–4]. Для указанного метода характерна высокая точность численного решения и чрезвычайная простота компьютерной реализации [5–9]. Наилучшие результаты получены при использовании МТИ для моделирования физических полей, описываемых однородными линейными уравнениями эллиптического типа с известными фундаментальными решениями, задаваемыми аналитически. Такими уравнениями являются уравнение Лапласа, уравнение Гельмгольца, бигармоническое уравнения, некоторые другие типы уравнений. Применение МТИ в этом случае позволило решить значительное число прикладных задач по моделированию, например: стационарных электрических, магнитных [8–10], тепловых, концентрационных полей [11–14], полей упругих напряжений [15–17]. МТИ успешно применяется также при численном решении краевых задач для неоднородных уравнений, таких как уравнение Пуассона [18–19], неоднородное уравнение Гельмгольца [19–21]. Однако во всех случаях использования МТИ предполагается известным фундаментальное решение соответствующего уравнения математической физики. Это резко ограничивает круг решаемых с помощью МТИ задач. Тем не менее, и в этом случае возможно решение краевых задач с помощью МТИ, если предварительно найти численные значения фундаментальных решений при определенных значениях параметров. Назовем фундаментальное решение, заданное численно, численным фундаментальным решением (ЧФР). Ниже описан метод нахождения ЧФР для линейных уравнений эллиптического типа и показывается возможность использования этих решений в МТИ.
1. Alexidze, М.А. Fundamental´nye funktsii v priblizhennykh resheniyakh granichnykh zadach. [Fundamental func-tions in approximate solutions of boundary value problems.] Moscow: Nauka, 1991, 352 p. (in Russian).
2. Fairweather, G., Karageorghis, A. The method of fundamental solutions for elliptic boundary value problems. Ad-vances in Computational Mathematics, 1998, vol. 9, pp. 69–95.
3. Alves, C.-J.-S., Chen, C.-S. A new method of fundamental solutions applied to nonhomogeneous elliptic problems. Advances in Computational Mathematics, 2005, vol. 2, pp. 125–142.
4. Knyazev, S.Yu. Ustoychivost´ i skhodimost´ metoda tochechnykh istochnikov polya pri chislennom reshenii kraevykh zadach dlya uravneniya Laplasa. [Stability and Convergence of Point-Source Field Method at Numerical Solution to Boundary Value Problems for Laplace Equation.] Russian Electromechanics, 2010, no. 3, pp. 3–12 (in Russian).
5. Bakhvalov, Y.A., et al. Pogreshnost´ metoda tochechnykh istochnikov pri modelirovanii potentsial´nykh poley v oblastyakh s razlichnoy konfiguratsiey. [Errors of Point Source Method under Simulation of Potential Fields in Areas with Dif-ferent Shape Configuration.] Russian Electromechanics, 2012, no. 5, pp. 17–21 (in Russian).
6. Knyazev, S.Yu., Shcherbakova, E.E., Zaichenko, A.N. SRAVNITEL´NYY ANALIZ DVUKH VARIANTOV METODA KOLLOKATSIY PRI CHISLENNOM MODELIROVANII POTENTSIAL´NYKH POLEY. [A Comparative Analysis of Two Variants of the Collocation in Numerical Modeling of Potential Fields.] Russian Electromechanics, 2014, no. , pp. 17–19 (in Russian).
7. Knyazev, S.Yu., Shcherbakova, E.E. Reshenie trekhmernykh kraevykh zadach dlya uravneniy Laplasa s pomoshch´yu metoda diskretnykh istochnikov polya. [The Decision of the Three-Dimensional Boundary Value Problems for the Laplace Equation Using the Method of Discrete Sources of the Field.] Russian Electromechanics, 2015, no. 5, pp. 25–30 (in Russian).
8. Knyazev, S.Yu. Metod tochechnykh istochnikov dlya komp´yuternogo modelirovaniya fizicheskikh poley v zadachakh s podvizhnymi granitsami : dis. … d-ra tekhn. nauk. [Point-source method for computer modeling of physical fields in problems with moving boundaries: Dr.Sci. (Eng.) diss.] Novocherkassk, 2011, 342 p. (in Russian).
9. Knyazev, S.Yu., Shcherbakova, E.E., Shcherbakov, A.A. Komp´yuternoe modelirovanie potentsial´nykh poley metodom tochechnykh istochnikov. [Computer modeling of potential fields by point-source method.] Rostov-on-Don: DSTU Publ. Centre, 2012, 156 p. (in Russian).
10. Bakhvalov, Yu.A., Knyazev, S.Yu., Shcherbakov, A.A. Matematicheskoe modelirovanie fizicheskikh poley metodom tochechnykh istochnikov. [Mathematical modeling of physical fields by a method of dot sources.] Bulletin of the Russian Academy of Sciences: Physics, 2008, vol. 72, no. 9, pp. 1259–1261 (in Russian).
11. Knyazev, S.Yu., Shcherbakova, E.E. Reshenie zadach teplo- i massoperenosa s pomoshch´yu metoda tochechnykh istochnikov polya. [Solution to heat and mass transfer problems by field point source method.] University News. North-Caucasian region. Technical Sciences Series, 2006, no. 4, pp. 43–47 (in Russian).
12. Knyazev, S.Yu., Shcherbakova, E.E. Chislennoe issledovanie stabil´nosti termomigratsii ploskikh zon. [Numerical study of stability of flat bands thermomigration.] Russian Electromechanics, 2007, no. 1, pp. 14–19 (in Russian).
13. Knyazev, S.Yu., Shcherbakova, E.E., Shcherbakov, A.A. Sravnitel´nyy analiz razlichnykh variantov ispol´zovaniya metoda tochechnykh istochnikov polya pri modelirovanii temperaturnykh poley. [Comparative analysis of different use cases of field point source method under simulation of temperature fields.] Fiziko-matematicheskoe modelirovanie sistem: mat-ly XII mezhdunar. seminara. [Physico- mathematical system modeling: Proc. XII Int. Workshop.] Voronezh: Voronezh State Technical University, 2014, pp. 52–56 (in Russian).
14. Lunin, L.S., et al. Issledovanie stabil´nosti termomigratsii ansamblya lineynykh zon s pomoshch´yu trekhmernoy komp´yuternoy modeli, postroennoy na osnove metoda tochechnykh istochnikov polya. [The study of stability of thermomigration of an ensemble of linear zones using a three-dimensional computer model constructed on the basis of the field point sources method.] Vestnik SSC RAS, 2015, vol. 11, no. 4, pp. 9–15 (in Russian).
15. Knyazev, S.Yu., Pustovoyt, V.N., Shcherbakova, E.E. Modelirovanie poley uprugikh deformatsiy s primeneniem metoda tochechnykh istochnikov. [Modeling the elastic strain fields by point-source method.] Vestnik of DSTU, 2015, vol. 15, no. 1 (80), pp. 29–38 (in Russian).
16. Knyazev, S.Yu. , et al. Modelirovanie trekhmernykh poley uprugikh deformatsiy s pomoshch´yu metoda tochech-nykh istochnikov.[Modeling of three-dimensional elastic strain fields by point-source method.] Vestnik of DSTU, 2015, vol. 15, no. 4 (83), pp. 13–23 (in Russian).
17. Knyazev, S.Yu., Shcherbakova, E.E., Shcherbakov, A.A. Matematicheskoe modelirovanie poley uprugikh defor-matsiy metodom tochechnykh istochnikov polya. [Mathematical modeling of elastic deformation fields by field point source method.] Mathematical Methods in Engineering and Technologies-MMTT, 2015, no. 5 (75), pp. 21–23 (in Russian).
18. Knyazev, S.Yu., Shcherbakova, E.E., Yengibaryan, A.A. Chislennoe reshenie kraevykh zadach dlya uravneniya Puassona metodom tochechnykh istochnikov polya. [Numerical solution to boundary problems for Poisson equation by point-source method.] Vestnik of DSTU, 2014, vol. 14, no. 2 (77), pp. 15–20 (in Russian).
19. Knyazev, S.Yu. Chislennoe reshenie uravneniy Puassona i Gel´mgol´tsa s pomoshch´yu metoda tochechnykh isto-chnikov. [Numerical solution to Poisson and Helmholtz equations using point source method.] Russian Electromechanics, 2007, no. 2, pp. 77–78 (in Russian).
20. Knyazev, S.Yu., Shcherbakova, E.E., Zaichenko, A.N. Chislennoe reshenie kraevykh zadach dlya neodnorodnykh uravneniy Gel´mgol´tsa metodom tochechnykh istochnikov polya. [Numerical Solution of the Boundary Problems with Non-Homogeneous Helmholtz Equation by Field Point-Source Method.] Russian Electromechanics, 2014, no. 4, pp. 14–19 (in Russian).
21. Knyazev, S.Yu., Shcherbakova, E.E. Primenenie metoda tochechnykh istochnikov polya pri chislennom reshenii zadach na sobstvennye znacheniya dlya uravneniya Gel´mgol´tsa. [The Numerical Eigenvalue Problems Solution for the Helmholtz Equation Using the Point Sources Method] Russian Electromechanics, 2016, no. 3 (545), pp. 11–17 (in Russian).
22. Landau, L.D., Lifshits, E.M. Kvantovaya mekhanika. Nerelyativistskaya teoriya. [Quantum Mechanics. Nonrela-tivistic theory.] Moscow: Nauka, 1963, 703 p. (in Russian).
23. Vladimirov, V.S., Zharinov, V.V. Uravneniya matematicheskoy fiziki. [Equations of Mathematical Physics] 2nd ed. Moscow: FIZMATLIT, 2004, 400 p. (in Russian).
