On Solving the Regular Problem of Two-Phase Filtration Taking Into Account The Energy and one Model Problem of Filtration in Potentials by Monte Carlo Methods

d5d24e55-e242-4d34-8fe9-679dd1234bb020240308055802212wseas:wseasmdt@crossref.orgMDT DepositWSEAS TRANSACTIONS ON MATHEMATICS2224-28801109-276910.37394/23206http://wseas.org/wseas/cms.action?id=40511220241220242310.37394/23206.2024.23https://wseas.com/journals/mathematics/2024.phpOn Solving the Regular Problem of Two-Phase Filtration Taking Into Account The Energy and one Model Problem of Filtration in Potentials by Monte Carlo MethodsM. G.TastanovKostanay Regional University named after A.Baitursynuly, Kostanay, REPUBLIC OF KAZAKHSTANA. A.UtemissovaKostanay Regional University named after A.Baitursynuly, Kostanay, REPUBLIC OF KAZAKHSTANF. F.MaiyerKostanay Regional University named after A.Baitursynuly, Kostanay, REPUBLIC OF KAZAKHSTAND. S.KenzhebekovaKostanay Regional University named after A.Baitursynuly, Kostanay, REPUBLIC OF KAZAKHSTANN. M.TemirbekovKostanay Regional University named after A.Baitursynuly, Kostanay, REPUBLIC OF KAZAKHSTANIt should be noted that in some practical tasks, it is impossible not to take into account the temperature change. In this case, the energy equation should be added to the filtration equations. The algorithms of «random walk by spheres» and «random walk along boundaries» of Monte Carlo methods are used to solve regular degenerate filtration problems of two immiscible inhomogeneous incompressible liquids in a porous medium. The derivatives of the solution are evaluated using Monte Carlo methods. A model problem of filtration of a two-phase incompressible liquid with capillary forces is considered.38202438202415415918https://wseas.com/journals/mathematics/2024/a365106-011(2024).pdf10.37394/23206.2024.23.18https://wseas.com/journals/mathematics/2024/a365106-011(2024).pdfShakenov K.K. (2007) Solution of problem for one model of relaxational filtration by probability–difference and Monte Carlo methods. Polish Academy of Sciences. Committee of Mining. Archives of Mining Sciences. Vol. 52, Issue 2, Krakow, 2007. P. 247 – 255. Antontsev S.N., Kazhikhov A.V., Monakhov V.N. (1983). "Boundary value problems of mechanics of inhomogeneous fluids" (Kraevye zadachi mekhaniki neodnorodnykh zhidkostey). Novosibirsk: Nauka. Loitsyansky L.G. (2003) Mechanics of liquid and gas.7th edition, revised. – M.: Bustard, 2003, p.840. 10.1515/9783112319291-fmH.Amann, G.P. Galdi, K. Pllecas and V.A. Solonnikov (1997), Navier-Stokes Equations and Related Nonlinear Problems. Proceedings of the Sixth International Conference NSEC-6, Palanga, Lithuania, May 22–29, p.440. 10.1007/978-3-540-70942-8Todor Boyanov, Stefka Dimova, Krassimir Georgiev, Geno Nikolov. (2007) Numerical Methods and Applications, Springer Nature, p.732. New Directions in Matematical Fluid Mechanics. The Alexander V. Kazhikhov Mtmorial Volume. (2010) Springer Science and Business Media, p.414. 10.1007/978-94-009-2243-37.Ermakov S.M., Nekrutkin V.V., Sipin A.A. (1984), Random processes for solving classical equations of mathematical physics. Nauka, –M., 1989, p.208. 10.1016/j.jde.2022.10.015Shoujun Xu, Hao Wu. The Gevrey normalization for quasi-periodic systems under Siegel type small divisor conditions (2016), Journal of Differential Equations, Vol. 284, p.1223-1243. 10.1007/978-3-540-74496-2Alexander Keller, Stefan Heinrich, Harald Niederreiter Editors. Monte Carlo and Quasi-Monte Carlo Methods 2006 2008th Edition, Kindle Edition (2008). Springer, p.708. Mikhailov G.A. (1987). Optimization of Monte Carlo weight methods. Nauka, – M., 1987, p.239. Mikhailov G.A. (1974). Some questions of the theory of Monte Carlo methods. Nauka, Novosibirsk, 1974, p.145. Dynkin E.B. (1963) Markov processes. Fizmatgiz, –M., 1963, p.432. Dynkin E.B., Yushkevich A.A. (1967) Theorems and problems about Markov processes. Nauka, –M., 1967, p.232. Sobol I.M. (1973) Numerical Monte Carlo methods. Nauka, –M., 1973, p.312. Meyer P.A. (1976) Probability and potentials. Mir, –M., 1976, p.436. Ermakov S.M. (1975) The Monte Carlo method and related issues. Second edition, expanded. Nauka, –M., 1975, p.471. Yelepov B.S., Kronberg A.A., Mikhailov G.A., Sabelfeld K.K. (1980) Solving boundary value problems by the Monte Carlo method. Nauka, Novosibirsk, 1980, p.176. Mikhailov G.A. Voitishek A.V. (2006) Numerical statistical modeling. Monte Carlo methods. –M.: Publishing Center «Academy» 2006, p.368. 10.37394/23206.2022.21.46M. Tastanov, A. Utemissova, F. Maiyer, R. Ysmagul. (2022) On a Solution to the Stationary Problem of Two-Phase Filtration by theMonte-Carlo Method.WSEAS Transactions on Mathematics Volume 21, 2022, pp.386-394 DOI: 10.37394/23206.2022.21.88