Giải pháp chính xác của phương trình Richards tổng quát với phi tuyến tính theo quy luật lũy thừa

Differential Equations - Tập 56 - Trang 1119-1129 - 2020
A. A. Kosov1, E. I. Semenov1
1Matrosov Institute for System Dynamics and Control Theory, Siberian Branch, Russian Academy of Sciences, Irkutsk, Russia

Tóm tắt

Chúng tôi xem xét một phương trình Richards tổng quát với các phi tuyến tính theo quy luật lũy thừa mô hình hóa quá trình thẩm thấu trong môi trường xốp. Các điều kiện được rút ra, theo đó bài toán có thể được giảm về phương trình nhiệt phân tuyến hoặc các phương trình phi tuyến có nghiệm đã biết. Các gia đình nghiệm chính xác có thể biểu diễn qua các hàm lượng giác hoặc hàm $$W$$ của Lambert được tìm thấy. Một số ví dụ minh họa cho các kết quả được cung cấp.

Từ khóa

#Phương trình Richards tổng quát #phi tuyến tính #quá trình thẩm thấu #giải pháp chính xác #hàm Lambert

Tài liệu tham khảo

Barry, D.A., Parlange, J.Y., Sander, G.C., and Sivaplan, M., A class of exact solutions for Richard’s equation, J. Hydrol., 1993, vol. 142, pp. 29–46. Corless, R.M., Gonnet, G.H., Hare, D.E., Jeffrey, D.J., and Knuth, D.E., On the Lambert W function, Adv. Comput. Math., 1996, vol. 5, pp. 329–359. Zlotnik, V.A., Wang, T., Nieber, J.L., and šimunek, J., Verification of numerical solutions of the Richards equation using a traveling wave solution, Adv. Water Resour., 2007, vol. 30, pp. 1973–1980. Akhmetzyanov, A.V., Kushner, A.G., and Lychagin, V.V., Attractors in models of porous media flow, Dokl. Math., 2017, vol. 95, pp. 72–75. Witelski, T.P., Intermediate asymptotics for Richard’s equation in a finite layer, J. Eng. Math., 2003, vol. 45, pp. 379–399. Broadbridge, P., Daly, E., and Goard, J., Exact solutions of the Richards equation with nonlinear plant-root extraction, Water Resour. Res., 2017, vol. 53, pp. 9679–9691. Xi Chen and Ying, Dai., An approximate analytical solution of Richards equation with finite boundary, Boundary Value Probl., 2017, vol. 2017, no. 1, article ID 167. Barari, A., Omidvar, M., Ghotbi, A.R., and Ganji, D.D., Numerical analysis of Richard’s problem for water penetration in unsaturated soils, Hydrol. Earth Syst. Sci., 2009, vol. 6, pp. 6359–6385. Farthing, M.W. and Ogden, F.L., Numerical solution of Richard’s equation: A review of advances and challenges, Soil Sci. Soc. Am. J., 2017, vol. 81, pp. 1257–1269. Yuanyuan Zha, Jinzhong Yang, Jicai Zeng, Chak-Hau, M., Wenhi Zeng, and Liangsheng, Shi., Review of numerical solution of Richardson–Richards equation for variably saturated flow in soils, Wiley Interdiscip. Rev.: Water Resour., 2019, vol. 6, no. 5, pp. 1–23. Yung, C.M., Verburg, K., and Baveye, P., Group classification and symmetry reductions of the non-linear diffusion–convection equation \(u_{t}=(D(u)u_x)_x+K^{\prime }(u)u_x\), Int. J. Non-Linear Mech., 1994, vol. 29, no. 3, pp. 273–278. Gandarias, M.L., Romero, J.L., and Diaz, J.M., Nonclassical symmetry reductions of a porous medium equation with convection, J. Phys. A: Math. Gen., 1999, vol. 32, pp. 1461–1473. Popovych, R.O. and Ivanova, N.M., New results on group classification of nonlinear diffusion–convection equations, J. Phys. A: Math. Gen., 2004, vol. 37, no. 30, pp. 7547–7565. Ivanova, N.M., Exact solutions of diffusion–convection equations, Dyn. PDE, 2008, vol. 5, no. 2, pp. 139–171. Ovsyannikov, L.V., Group properties of nonlinear heat equations, Dokl. Akad. Nauk SSSR, 1959, vol. 125, no. 3, pp. 492–495. Polyanin, A.D. and Zaitsev, V.F., Spravochnik po nelineinym uravneniyam matematicheskoi fiziki. Tochnye resheniya (Handbook of Nonlinear Equations of Mathematical Physics. Exact Solutions), Moscow: Fizmatlit, 2002. Polyanin, A.D. and Zaitsev, V.F., Nelineinye uravneniya matematicheskoi fiziki. V 2-x ch. Ch. 1 (Nonlinear Equations of Mathematical Physics. In 2 Parts. Part 1), Moscow: Yurait, 2017. Polyanin, A.D. and Zaitsev, V.F., Nelineinye uravneniya matematicheskoi fiziki. V 2-x ch. Ch. 2 (Nonlinear Equations of Mathematical Physics. In 2 Parts. Part 2), Moscow: Yurait, 2017. Cherniha, R., Serov, M., and Pliukhin, O., Nonlinear Reaction–Diffusion–Convection Equations: Lie and Conditional Symmetry, Exact Solutions and Their Applications, Boca Raton: Chapman & Hall/CRC Press, 2018. Polyanin, A.D., Comparison of the effectiveness of different methods for constructing exact solutions to nonlinear PDEs. Generalizations and new solutions, Mathematics, 2019, vol. 7, no. 386, pp. 1–19. Galaktionov, V.A. and Posashkov, S.A., New exact solutions of parabolic equations with quadratic nonlinearities, Zh. Vychisl. Mat. Mat. Fiz., 1989, vol. 29, no. 4, pp. 497–506. Polyanin, A.D., Zaitsev, V.F., and Zhurov, A.I., Metody resheniya nelineinykh uravnenii matematicheskoi fiziki i mekhaniki (Methods for Solving Nonlinear Equations of Mathematical Physics and Mechanics), Moscow: Fizmatlit, 2005. Galaktionov, V.A and Svirshchevskii, S.R., Exact Solutions and Invariant Subspaces of Nonlinear Partial Differential Equations in Mechanics and Physics, Boca Raton–London–New York: Chapman & Hall/CRC Press, 2007. Rudykh, G.A. and Semenov, E.I., Construction of exact solutions of the one-dimensional nonlinear diffusion equation by the method of nonlinear invariant subspaces,Izv. Irkutsk. Gos. Univ. Ser. Mat., 2013, vol. 6, no. 4, pp. 69–84. Samarskii, A.A., Galaktionov, V.A., Kurdyumov, S.P., and Mikhailov, A.P., Rezhimy s obostreniem v zadachakh dlya kvazilineinykh parabolicheskikh uravnenii (Blow-Up Regimes in Problems for Quasilinear Parabolic Equations), Moscow: Nauka, 1987. Dubinov, A.E., Dubinova, I.D., and Saikov, S.K., \(W \)-funktsiya Lamberta i ee primenenie v matematicheskikh zadachakh fiziki (Lambert’s \(W \)-Function and Its Application to Mathematical Problems in Physics), Sarov: FGUP “TFYaTs-VNIIEF”, 2006. Kosov, A.A. and Semenov, E.I., The Lambert function and exact solutions of nonlinear parabolic equations, Russ. Math., 2019, vol. 63, no. 8, pp. 10–16. Barenblatt, G.I., On limit self-similar motions in the theory of nonstationary gas filtration in a porous medium and the boundary layer theory, Prikl. Mat. Mekh., 1954, vol. 18, no. 4, pp. 409–414. Barenblatt, G.I., Podobie, avtomodel’nost’, promezhutochnaya asimptotika (Similarity, Self-Similarity, and Intermediate Asymptotics), Leningrad: Gidrometeoizdat, 1978. Rudykh, G.A. and Semenov, E.I., New exact solutions to the one-dimensional equation of nonlinear diffusion, Sib. Mat. Zh., 1997, vol. 38, no. 5, pp. 1130–1139. Rudykh, G.A. and Semenov, E.I., On new exact solutions to the one-dimensional equation of nonlinear diffusion with a source (sink), Zh. Vychisl. Mat. Mat. Fiz., 1998, vol. 38, no. 6, pp. 971–977.