Solving Mathematical Physics Problems in the Wolfram Mathematica System
Abstract
Mathematical models of various processes are built on the base of partial differential equations, which are called equations of mathematical physics. The investigation of such an equation is a very difficult problem. At the same time increase of volumes of problems of modeling demands the readiness of many specialists, working in applied fields, to investigate such class of models. Modern scientific software, in particular, Mathematica system, helps to solve this problem. The article is devoted to the methods of usage of this system for the investigation of the mathematical models, leading to the need for the solution of such equations. These methods help to increase the effectiveness of learning of disciplines, connected with mathematical modeling and solution of problems of scientific and research character. The examples of the solution for linear and nonlinear partial differential equations are given.
References
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[19] Kogan I.L. Construction of Mikusinski operational calculus based on the convolution algebra of distributions. Methods for solving mathematical physics problems. Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences. 2018; 22(2):236-253. (In Russ., abstract in Eng.) DOI: 10.14498/vsgtu1569
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[26] Yang X.J., Gao F., Tenreiro M.J.A., Baleanu D. Exact Travelling Wave Solutions for Local Fractional Partial Differential Equations in Mathematical Physics. In: Taş K., Baleanu D., Machado J. (eds) Mathematical Methods in Engineering. Nonlinear Systems and Complexity, vol 24. Springer, Cham, 2019, pp. 175-191. (In Eng.) DOI: 10.1007/978-3-319-90972-1_12
[27] Abu Arqub O. Numerical solutions for the Robin time-fractional partial differential equations of heat and fluid flows based on the reproducing kernel algorithm. International Journal of Numerical Methods for Heat & Fluid Flow. 2018; 28(4):828-856. (In Eng.) DOI: 10.1108/HFF-07-2016-0278
[2] Samarsky A.A., Mikhailov A.P. Matematicheskoe modelirovanie. Idei. Metody`. Primery` [Mathematical Modeling: Ideas, Methods, Examples]. 2nd edn. Fizmatlit, Moscow, 2001. (In Russ.)
[3] Tikhonov A.N., Samarskii A.A. Uravneniya matematicheskoj fiziki [Equations of Mathematical Physics]. 6 ed. MSU, Moscow, 1999. (In Russ.)
[4] Polyanin A.D., Zhurov A.I. Functional Separable Solutions of Two Classes of Nonlinear Mathematical Physics Equations. Doklady Mathematics. 2019; 99(3):321-324. (In Eng.) DOI: 10.1134/S1064562419030128
[5] Karimov M.F., Mukimov V.R. Mezhdisciplinarnoe izuchenie differencial'nyh uravnenij i kvantovoj fiziki studentami vysshej shkoly [Interdisciplinary study of differential equations and quantum physics by students of higher education]. In: Prospects for the development of science and education. Proceedings. AR-Konsalt, Lyubertsy, 2018, pp. 29-31. Available at: https://www.elibrary.ru/item.asp?id=35188588 (accessed 14.07.2019). (In Russ.)
[6] Belyaev V.A. The collocation and least squares method for solving problems of mathematical physics in noncanonical areas. In: Kozlov V.V. (ed) Problems of mechanics: theory, experiment and new technologies. Abstracts of the XII All-Russian Conference of Young Scientists. Proceedings. Parallel Publ., Novosibirsk, 2018, pp. 16-17. Available at: https://www.elibrary.ru/item.asp?id=32846750 (accessed 14.07.2019). (In Russ.)
[7] Anikonov D.S., Konovalova D.S. Direct and inverse problems for a wave equation with discontinuous coefficients. St. Petersburg Polytechnical University Journal: Physics and Mathematics. 2018; 11(2):61-72. (In Russ., abstract in Eng.) DOI: 10.18721/JPM.11206
[8] Golubeva L.A., Gorshunov V.S., Ilyin V.P., Erdyniev E.B. Software package for solving 3-dimensional problems of mathematical physics based on the BSM concep. In: Proceedings of the International Conference "Computational Mathematics and Mathematical Geophysics". ICMMG SB RAS, Novosibirsk, 2018, pp. 126-132. Available at: https://www.elibrary.ru/item.asp?id=37014757 (accessed 14.07.2019). (In Russ.)
[9] Rakhmelevich I.V. On the solutions of two-dimensional partial differential equation of the second order of quasy-polynomial type. In: Modern scientific research: topical issues, achievements and innovations. Proceedings. Nauka i Prosveshchenie Publ., Penza, 2018, pp. 12-14. Available at: https://www.elibrary.ru/item.asp?id=32591741 (accessed 14.07.2019). (In Russ.)
[10] Pastuhov D., Pastuhov Y., Volosova N. To question about of the lumpy marginal problem dirihle for wave equation on length. HeraldofPolotskStateUniversity.SeriesC,FUNDAMENTALSCIENCES. 2018; (12):60-74. Available at: https://www.elibrary.ru/item.asp?id=36527816 (accessed 14.07.2019). (In Russ., abstract in Eng.)
[11] Kurdyumov V.P., Khromov A.P., Khalova V.A. A Mixed Problem for a Wave Equation with a Nonzero Initial Velocity. Izvestiya of Saratov University. New Series. Series: Mathematics. Mechanics. Informatics. 2018; 18(2):157-171. (In Russ., abstract in Eng.) DOI: 10.18500/1816-9791-2018-18-2-157-171
[12] Guzhov I.M. Postroenie yavnoj ustojchivoj skhemy dlya resheniya odnomernogo kvazilinejnogo uravneniya teploprovodnosti [The construction of an explicit stable scheme for solving the one-dimensional quasilinear heat equation]. SUSU, Chelyabinsk, 2018. Available at: https://dspace.susu.ru/xmlui/handle/0001.74/23424?show=full (accessed 14.07.2019). (In Russ.)
[13] Voskovskaya N.I., Zvereva M.B., Kamenskii M.I. On the Wave Equation with the Hysteresis Type Condition. Tambov University Reports. Series Natural and Technical Sciences. 2018; 23(122):235-242. (In Russ., abstract in Eng.) DOI: 10.20310/1810-0198-2018-23-122-235-242
[14] Khromov A.P. Mixed Problem for a Homogeneous Wave Equation with a Nonzero Initial Velocity. Computational Mathematics and Mathematical Physics. 2018; 58(9):1531-1543. (In Eng.) DOI: 10.31857/S004446690002535-9
[15] Pavlysh V.N., Tarabayeva I.V. The Mathematical Modeling of Gas-Air Mix Moving Process in Continuous Environment (With Coal Stratum As Example). Problems of Artificial Intelligence. 2018; (3):104-111. Available at: https://www.elibrary.ru/item.asp?id=36533215 (accessed 14.07.2019). (In Russ., abstract in Eng.)
[16] Tolstyh A.V., Doroshenko Yu.N., Il'inskij M.E. Testirovanie vychislitel'nogo paketa FlexPDE na tipovyh zadachah teploobmena dlya stroitel'noj teplofiziki [Testing the FlexPDE computing package on typical heat transfer tasks for building thermal physics]. Evrazijskoe Nauchnoe Ob'edinenie. 2018; 1(1):11-14. Available at: https://www.elibrary.ru/item.asp?id=32379894 (accessed 14.07.2019). (In Russ.)
[17] Kerefov M.A., Kerefov B.M. Boundary-Value Problems for a Wave Equation of Fractional Order”.Proceedings of the International Conference “Actual Problems of Applied Mathematics and Physics,” Kabardino-Balkaria, Nalchik, May 17–21, 2017. Itogi Nauki i Tekhniki. Seriya "Sovremennaya Matematika i ee Prilozheniya. Tematicheskie Obzory". 2018; 149:44-55. Available at: https://www.elibrary.ru/item.asp?id=35242549 (accessed 14.07.2019). (In Russ.)
[18] Mezaal N.A. Matematicheskoe modelirovanie vetroenergeticheskoj ustanovki (VEU) moshchnost'yu 1.5 MVt metodami vychislitel'noj gidrodinamiki (CFD) v ANSYS [Mathematical modeling of a wind power plant (wind turbine) with a capacity of 1.5 MW using computational fluid dynamics (CFD) in ANSYS]. SUSU, Chelyabinsk, 2018. Available at: https://dspace.susu.ru/xmlui/handle/0001.74/25335?show=full (accessed 14.07.2019). (In Russ.)
[19] Kogan I.L. Construction of Mikusinski operational calculus based on the convolution algebra of distributions. Methods for solving mathematical physics problems. Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences. 2018; 22(2):236-253. (In Russ., abstract in Eng.) DOI: 10.14498/vsgtu1569
[20] Algazin O.D. Similarity Method in Constructing Fundamental Solution of the Dirichlet Problem for Equation of Keldysh Type in Half-Space. Herald of the Bauman Moscow State Technical University. Series Natural Sciences. 2018; (1):4-15. (In Russ., abstract in Eng.) DOI: 10.18698/1812-3368-2018-1-4-15
[21] Yakovleva Yu.O. The cauchy problem for the hyperbolic differential equation of the third order. Vestnik of Samara University. Natural Science Series. 2018. Т. 24, № 3. С. 30-34. (In Russ., abstract in Eng.) DOI: 10.18287/2541-7525-2018-24-3-30-34
[22] Raissi M., Karniadakis G.E. Hidden physics models: Machine learning of nonlinear partial differential equations. Journal of Computational Physics. 2018; 357:125-141. (In Eng.) DOI: 10.1016/j.jcp.2017.11.039
[23] Han J., Jentzen A., Weinan E. Solving high-dimensional partial differential equations using deep learning. Proceedings of the National Academy of Sciences. 2018; 115(34):8505-8510. (In Eng.) DOI: 10.1073/pnas.1718942115
[24] Chaudhari P. et al. Deep relaxation: partial differential equations for optimizing deep neural networks. Research in the Mathematical Sciences. 2018; 5(3):30. (In Eng.) DOI:10.1007/s40687-018-0148-y
[25] Ruthotto L., Haber E. Deep neural networks motivated by partial differential equations. arXiv preprint arXiv:1804.04272. 2018. (In Eng.)
[26] Yang X.J., Gao F., Tenreiro M.J.A., Baleanu D. Exact Travelling Wave Solutions for Local Fractional Partial Differential Equations in Mathematical Physics. In: Taş K., Baleanu D., Machado J. (eds) Mathematical Methods in Engineering. Nonlinear Systems and Complexity, vol 24. Springer, Cham, 2019, pp. 175-191. (In Eng.) DOI: 10.1007/978-3-319-90972-1_12
[27] Abu Arqub O. Numerical solutions for the Robin time-fractional partial differential equations of heat and fluid flows based on the reproducing kernel algorithm. International Journal of Numerical Methods for Heat & Fluid Flow. 2018; 28(4):828-856. (In Eng.) DOI: 10.1108/HFF-07-2016-0278
Published
2019-12-23
How to Cite
KRISTALINSKII, Vladimir Romanovich; KRISTALINSKII, Roman Efimovich.
Solving Mathematical Physics Problems in the Wolfram Mathematica System.
Modern Information Technologies and IT-Education, [S.l.], v. 15, n. 4, p. 981-991, dec. 2019.
ISSN 2411-1473. Available at: <http://sitito.cs.msu.ru/index.php/SITITO/article/view/584>. Date accessed: 03 july 2024.
doi: https://doi.org/10.25559/SITITO.15.201904.981-991.
Section
Scientific software in education and science
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