Enhanced group analysis and conservation laws of variable coefficient reaction-diffusion equations with power nonlinearities. (English) Zbl 1160.35365
With Lie symmetry point of view (equivalence transformations, conservation laws, exact solution) the authors study nonlinear partial differential equations of general form \(f(x)u_t=(g(x)u^nu_x)_x+h(x)u^m,\;f,g\) and \(h\) are arbitrary smooth functions, \(f(x)g(x)\neq 0, n\neq 0\) and \(m\) are arbitrary constants. The complete group classification with respect to convenient equivalence groups and to the set of all local transformations is given. Local conservation laws are classified, some exact solutions are constructed. The case \(n=0\) is excluded as well-investigated.
Reviewer: Boris V. Loginov (Ul’yanovsk)
MSC:
| 35A30 | Geometric theory, characteristics, transformations in context of PDEs |
| 37L20 | Symmetries of infinite-dimensional dissipative dynamical systems |
| 35K55 | Nonlinear parabolic equations |
Keywords:
symmetry analysis; Lie group and algebra methods; nonlinear diffusion equations; equivalence transformations; Lie symmetries; conservation lawsReferences:
| [1] | Anco, S. C.; Bluman, G., Direct construction method for conservation laws of partial differential equations. I. Examples of conservation law classifications, European J. Appl. Math., 13, 545-566 (2002) · Zbl 1034.35070 |
| [2] | Anco, S. C.; Bluman, G., Direct construction method for conservation laws of partial differential equations. II. General treatment, European J. Appl. Math., 13, 567-585 (2002) · Zbl 1034.35071 |
| [3] | Basarab-Horwath, P.; Lahno, V.; Zhdanov, R., The structure of Lie algebras and the classification problem for partial differential equation, Acta Appl. Math., 69, 43-94 (2001) · Zbl 1054.35002 |
| [4] | Bluman, G.; Kumei, S., Symmetries and Differential Equations (1989), Springer: Springer New York · Zbl 0698.35001 |
| [5] | Bluman, G.; Anco, S. C., Symmetry and Integration Methods for Differential Equations (2002), Springer: Springer New York · Zbl 1013.34004 |
| [6] | Borovskikh, A. V., Group classification of the eikonal equations for a three-dimensional nonhomogeneous medium, Mat. Sb.. Mat. Sb., Sb. Math., 195, 3-4, 479-520 (2004), (in Russian). Translation in: · Zbl 1073.35056 |
| [7] | Cherniha, R.; Serov, M., Symmetries ansätze and exact solutions of nonlinear second-order evolution equations with convection terms, European J. Appl. Math., 9, 527-542 (1998) · Zbl 0922.35033 |
| [8] | Crank, J., The Mathematics of Diffusion (1979), Oxford: Oxford London · Zbl 0427.35035 |
| [9] | V.A. Dorodnitsyn, Group properties and invariant solutions of a nonlinear heat equation with a source or a sink, Keldysh Institute of Applied Mathematics of Academy of Sciences USSR, Moscow, 1979, preprint N. 57; V.A. Dorodnitsyn, Group properties and invariant solutions of a nonlinear heat equation with a source or a sink, Keldysh Institute of Applied Mathematics of Academy of Sciences USSR, Moscow, 1979, preprint N. 57 |
| [10] | Dorodnitsyn, V. A., On invariant solutions of non-linear heat equation with a source, Zh. Vychisl. Mat. Mat. Fiz., 22, 1393-1400 (1982), (in Russian) · Zbl 0535.35040 |
| [11] | V.A. Dorodnitsyn, S.R. Svirshchevskii, On Lie-Bäcklund groups admitted by the heat equation with a source, Keldysh Institute of Applied Mathematics of Academy of Sciences USSR, Moscow, 1983, preprint N. 101; V.A. Dorodnitsyn, S.R. Svirshchevskii, On Lie-Bäcklund groups admitted by the heat equation with a source, Keldysh Institute of Applied Mathematics of Academy of Sciences USSR, Moscow, 1983, preprint N. 101 |
| [12] | Fushchich, W. I.; Shtelen, W. M.; Serov, N. I., Symmetry Analysis and Exact Solutions of Equations of Nonlinear Mathematical Physics (1993), Kluwer: Kluwer Dordrecht · Zbl 0838.58043 |
| [13] | Ibragimov, N. H., Transformation Groups Applied to Mathematical Physics, Math. Appl. (Soviet Ser.) (1985), Reidel: Reidel Dordrecht · Zbl 0558.53040 |
| [14] | (Ibragimov, N. H., Lie Group Analysis of Differential Equations—Symmetries, Exact Solutions and Conservation Laws, vol. 1 (1994), Chemical Rubber Company: Chemical Rubber Company Boca Raton, FL) · Zbl 0864.35001 |
| [15] | Ibragimov, N. H., Elementary Lie Group Analysis and Ordinary Differential Equations (1999), Wiley: Wiley New York · Zbl 1047.34001 |
| [16] | Ivanova, N. M.; Sophocleous, C., On the group classification of variable coefficient nonlinear diffusion-convection equations, J. Comput. Appl. Math., 197, 322-344 (2006) · Zbl 1103.35007 |
| [17] | N.M. Ivanova, R.O. Popovych, C. Sophocleous, Conservation laws of variable coefficient diffusion-convection equations, in: Proc. of 10th International Conference in Modern Group Analysis (MOGRAN X) (Larnaca, Cyprus, 2004), 2005, pp. 108-115, math-ph/0505015; N.M. Ivanova, R.O. Popovych, C. Sophocleous, Conservation laws of variable coefficient diffusion-convection equations, in: Proc. of 10th International Conference in Modern Group Analysis (MOGRAN X) (Larnaca, Cyprus, 2004), 2005, pp. 108-115, math-ph/0505015 |
| [18] | N.M. Ivanova, R.O. Popovych, C. Sophocleous, Enhanced group classification and conservation laws of variable coefficient diffusion-convection equations, in preparation; N.M. Ivanova, R.O. Popovych, C. Sophocleous, Enhanced group classification and conservation laws of variable coefficient diffusion-convection equations, in preparation · Zbl 1257.35018 |
| [19] | Kamin, S.; Rosenau, P., Nonlinear thermal evolution in an inhomogeneous medium, J. Math. Phys., 23, 1385-1390 (1982) · Zbl 0499.76111 |
| [20] | Khamitova, R. S., The structure of a group and the basis of conservation laws, Teoret. Mat. Fiz., 52, 244-251 (1982) · Zbl 0492.35049 |
| [21] | Kingston, J. G.; Sophocleous, C., On point transformations of a generalised Burgers equation, Phys. Lett. A, 155, 15-19 (1991) · Zbl 0737.35109 |
| [22] | Kingston, J. G.; Sophocleous, C., On form-preserving point transformations of partial differential equations, J. Phys. A, 31, 1597-1619 (1998) · Zbl 0905.35005 |
| [23] | Kingston, J. G.; Sophocleous, C., Symmetries and form-preserving transformations of one-dimensional wave equations with dissipation, Internat. J. Non-Linear Mech., 36, 987-997 (2001) · Zbl 1345.35063 |
| [24] | S.P. Kurdyumov, S.A. Posashkov, A.V. Sinilo, On the invariant solutions of the heat equation with the coefficient of heat conduction allowing the widest group of transformations, Keldysh Institute of Applied Mathematics of Academy of Sciences USSR, Moscow, 1989, preprint N. 110; S.P. Kurdyumov, S.A. Posashkov, A.V. Sinilo, On the invariant solutions of the heat equation with the coefficient of heat conduction allowing the widest group of transformations, Keldysh Institute of Applied Mathematics of Academy of Sciences USSR, Moscow, 1989, preprint N. 110 |
| [25] | Lahno, V. I.; Spichak, S. V.; Stognii, V. I., Symmetry Analysis of Evolution Type Equations (2002), Institute of Mathematics of NAS of Ukraine: Institute of Mathematics of NAS of Ukraine Kyiv |
| [26] | Meleshko, S. V., Homogeneous autonomous systems with three independent variables, Prikl. Mat. Mekh.. Prikl. Mat. Mekh., J. Appl. Math. Mech., 58, 857-863 (1994), (in Russian). Translation in: · Zbl 0923.76272 |
| [27] | Murray, J. D., Mathematical Biology I: An Introduction (2002), Springer: Springer New York · Zbl 1006.92001 |
| [28] | Murray, J. D., Mathematical Biology II: Spatial Models and Biomedical Applications (2003), Springer: Springer New York · Zbl 1006.92002 |
| [29] | Nikitin, A. G., Group classification of systems of non-linear reaction-diffusion equations, Ukr. Mat. Visn., 2, 149-200 (2005), ) · Zbl 1211.35147 |
| [30] | Olver, P., Applications of Lie Groups to Differential Equations (1986), Springer: Springer New York · Zbl 0588.22001 |
| [31] | Ovsiannikov, L. V., Group properties of nonlinear heat equation, Dokl. Akad. Nauk SSSR, 125, 492-495 (1959), (in Russian) · Zbl 0092.09903 |
| [32] | Ovsiannikov, L. V., Group Analysis of Differential Equations (1982), Academic Press: Academic Press New York · Zbl 0485.58002 |
| [33] | Patera, J.; Winternitz, P., Subalgebras of real three and four-dimensional Lie algebras, J. Math. Phys., 18, 1449-1455 (1977) · Zbl 0412.17007 |
| [34] | Peletier, L. A., Applications of Nonlinear Analysis in the Physical Sciences (1981), Pitman: Pitman London · Zbl 0474.35079 |
| [35] | Polyanin, A. D.; Zaitsev, V. F., Handbook of Nonlinear Equations of Mathematical Physics (2002), Fizmatlit: Fizmatlit Moscow · Zbl 1031.35001 |
| [36] | Popovych, R. O.; Cherniha, R. M., Complete classification of Lie symmetries of system of non-linear two-dimensional Laplace equations, Proc. Inst. Math., 36, 212-221 (2001) · Zbl 1003.35022 |
| [37] | R.O. Popovych, H. Eshraghi, Admissible point transformations of nonlinear Schrödinger equations, in: Proc. of 10th International Conference in Modern Group Analysis (MOGRAN X) (Larnaca, Cyprus, 2004), 2005, pp. 168-176; R.O. Popovych, H. Eshraghi, Admissible point transformations of nonlinear Schrödinger equations, in: Proc. of 10th International Conference in Modern Group Analysis (MOGRAN X) (Larnaca, Cyprus, 2004), 2005, pp. 168-176 |
| [38] | Popovych, R. O.; Ivanova, N. M., New results on group classification of nonlinear diffusion-convection equations, J. Phys. A, 37, 7547-7565 (2004) · Zbl 1067.35006 |
| [39] | Popovych, R. O.; Ivanova, N. M., Hierarchy of conservation laws of diffusion-convection equations, J. Math. Phys., 46, 043502 (2005) · Zbl 1067.37083 |
| [40] | Popovych, R. O.; Ivanova, N. M.; Eshraghi, H., Group classification of \((1 + 1)\)-dimensional Schrödinger equations with potentials and power nonlinearities, J. Math. Phys., 45, 3049-3057 (2004) · Zbl 1071.35117 |
| [41] | Prokhorova, M., The structure of the category of parabolic equations, 24 p |
| [42] | Touloukian, Y. S.; Powell, P. W.; Ho, C. Y.; Klemens, P. G., Thermodynamics Properties of Matter, vol. 1 (1970), Plenum: Plenum New York |
| [43] | Zharinov, V. V., Conservation laws of evolution systems, Teoret. Mat. Fiz., 68, 163-171 (1986) · Zbl 0625.35071 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
