Group classification of the perturbed nonlinear filtration equation is performed assuming that the perturbation is an arbitrary function of the dependent variable. The nonlinear self-adjointness of the equation under consideration is investigated. Using these results, the approximate conservation laws are constructed.
In the present paper, the recent method of conservation laws for constructing exact solutions for systems of nonlinear partial differential equations is applied to the gasdynamic equations describing one-dimensional and three-dimensional polytropic flows. In the one-dimensional case singular solutions are constructed in closed forms. In the threedimensional case several conservation laws are used simultaneously. It is shown that the method of conservation laws leads to particular solutions different from group invariant solutions.
The paper is devoted to investigation of group properties of a one-dimensional model of two-phase filtration in porous medium. Along with the general model, some of its particular cases widely used in oil-field development are discussed. The Buckley-Leverett model is considered in detail as a particular case of the one-dimensional filtration model. This model is constructed under the assumption that filtration is one-dimensional and horizontally directed, the porous medium is homogeneous and incompressible, the filtering fluids are also incompressible. The model of "chromatic fluid" filtration is also investigated. New conservation laws and particular solutions are constructed using symmetries and nonlinear self-adjointness of the system of equations.
We find the Lie point symmetries of a coupled variable-coefficient modified Korteweg–de Vries system in a two-layer fluid model. Then we establish its quasi self-adjointness and corresponding conservation laws.
It is shown that the Noether theorem can be extended for some equations associated (accompanying) with Euler-Lagrange equation. Each symmetry of Lagrangian yields a class of accompanying equations possessing conservation law (first integral). The generalization is done for canonical Hamiltonian equations as well.
In his extensive work of 1884 on the group classification of ordinary differential equations Lie performed, inter alia, the group classification of the particular type of the second-order equations y″ = F (x, y). In the present paper we extend Lie's classification to the third-order equations y‴ = F (x, y, y′).
It is known that the classification of third-order evolutionary equations with the constant separant possessing a nontrivial Lie-Bäcklund algebra (in other words, integrable equations) results in the linear equation, the KdV equation and the Krichever-Novikov equation. The first two of these equations are nonlinearly self-adjoint. This property allows to associate conservation laws of the equations in question with their symmetries. The problem on nonlinear self-adjointness of the Krichever-Novikov equation has not been solved yet. In the present paper we solve this problem and find the explicit form of the differential substitution providing the nonlinear self-adjointness.
A fourth-order non-linear evolutionary partial differential equation containing several arbitrary functions of the dependent variable is considered. This equation arises as a generalization of various non-linear models describing a non-linear heat diffusion, the dynamics of thin liquid films, etc. Equivalence transformations give more flexibility to the unified model. We determine the generators of the equivalence group and use them for specifying certain types of arbitrary functions when the model equation has additional symmetries, and hence admits non-trivial group invariant solutions. (c) 2006 Elsevier B.V. All rights reserved.
Four time-fractional generalizations of the Kompaneets equation are considered. Group analysis is performed for physically relevant approximations. It is shown that all approximations have nontrivial symmetries and conservation laws. The symmetries are used for constructing group invariant solutions, whereas the conservation laws allow to find non-invariant exact solutions. (C) 2014 Elsevier B.V. All rights reserved.
Process of heating of thin layer located between two vibrating surfaces is studied. Energy loss goes on due to viscous or dry friction. Optimal quantities of shear viscosity and friction corresponding to maximum energy loss are determined. Resonant behavior of loss must be taken into account in the description of "slow dynamics" of rocks and materials exposed to high-intensity seismic or acoustic irradiation as well as in various technologies. Bonding of materials by linear friction welding, widely used in propulsion engineering, can exemplify such a technology.
Euler generalized d'Alembert's solution to a wide class of linear hyperbolic equations with two independent variables. He introduced in 1769 the quantities that were rediscovered by Laplace in 1773 and became known as the Laplace invariants. The present paper is devoted to an extension of Euler's method to linear parabolic equations with two independent variables. The new method allows one to derive an explicit formula for the general solution of a wide class of parabolic equations. In particular, the general solution of the Black-Scholes equation is obtained. (c) 2008 Elsevier B.V. All rights reserved.
We consider three-dimensional free-boundary problem on the propagation of incompressible, homogeneous and inviscid fluid with zero surface tension confined in a channel of variable depth. Since for large-scale flows the fluid motion is affected by the rotation of the earth, the model is considered in rotating reference frame. Additionally, small atmospheric pressure variations across the channel are taken into account. It is shown that the non-trivial solution to the problem represents three-dimensional solitary wave which is given by the rotation modified Korteweg-de Vries equation (fKdV): b(1)xi(xxx) + b(2)xi xi(x) + b(3)(f)xi(x) = 0, where x is the down-channel coordinate and the coefficients b(i) (i = 1,2,3) of the resulting fKdV equation depend on the transverse topography of the channel and, additionally, b(3) depends on the Coriolis parameter f. It is also shown that if the vertical profile of the channel is symmetric about the vertical axis, the small atmospheric variations will not appear in the resulting fKdV equation. The effects of channel's cross-sectional geometry on the shape of the resulting three-dimensional wave profile in a longitudinal direction are studied numerically. Additionally, to better understand the effects of the Earth rotation, the above analysis is performed at different latitudes. (C) 2008 Elsevier B.V. All rights reserved.
Conservation laws and exact solutions of nonlinear differential equations describing diffusion phenomena in anisotropic media with external sources are constructed. The construction is based on the method of nonlinear self-adjointness. Numerous exact solutions are obtained by using the recent method of conservation laws. These solutions are different from group invariant solutions and can be useful for investigating diffusion phenomena in complex media, e.g. in oil industry.
Different approximations of the Kompaneets equation are studied using approximate symmetries, which allows consideration of the contributions of all terms of this equation previously neglected in the analysis of the limiting cases.
It is shown that Lie group analysis of differential equations provides the exact solutions of two-dimensional stratified rotating Boussinesq equations which are a basic model in geophysical fluid dynamics. The exact solutions are obtained as group invariant solutions corresponding to the translation and dilation generators of the group of transformations admitted by the equations. The comparison with the previous analytic studies and experimental observations confirms that the anisotropic nature of the wave motion allows to associate these invariant solutions with uni-directional internal wave beams propagating through the medium. It is also shown that the direction of internal wave beam propagation is in the transverse direction to one of the invariants which corresponds to a linear combination of the translation symmetries. Furthermore, the amplitudes of a linear superposition of wave-like invariant solutions forming the internal gravity wave beams are arbitrary functions of that invariant. Analytic examples of the latitude-dependent invariant solutions associated with internal gravity wave beams that have different general profiles along the obtained invariant and propagating in the transverse direction are considered. The behavior of the invariant solutions near the critical latitude is illustrated. © 2009 Elsevier B.V. All rights reserved.
Lie group analysis of nonlinear differential equations reveals existence of singularities provided by invariant solutions and invisible from the form of the equation in question. We call them internal singularities in contrast with external singularities manifested by the form of the equation. It is illustrated by way of examples that internal singularities are useful for analyzing a behaviour of solutions of nonlinear differential equations near external singularities.
The article is devoted to the Solution Of the invariants problem for the one-dimensional parabolic equations written in the two-coefficient canonical form used recently by N.H. Ibragimov: u(t) - u(xx) + a (t, x)u(x) + c(t, x)u = 0. A simple invariant condition is obtained for determining all equations that are reducible to the heat equation by the general group of equivalence transformations. The solution to the problem of invariants is given also in the one-coefficient canonical u(t) - u(xx) + c(t, x)u = 0. One of the main differences between these two canonical forms is that the equivalence group for the two-coefficient form contains the arbitrary linear transformation of the dependent variable whereas this group for the one-coefficient form contains only a special type of the linear transformations of the dependent variable. (C) 2008 Elsevier B.V. All rights reserved.
Lie's group classification of ODEs shows that the second-order equations can possess one, two, three or eight infinitesimal symmetries. The equations with eight symmetries and only these equations can be linearized by a change of variables. Lie showed that the latter equations are at most cubic in the first derivative and gave a convenient invariant description of all linearizable equations. Our aim is to provide a similar description of the equations with three symmetries. There are four different types of such equations. We present here the candidates for all four types. We give an invariant test for existence of three symmetries for one of these candidates.
The second-order ordinary differential equations can have one, two, three or eight independent symmetries. Sophus Lie showed that the equations with eight symmetries and only these equations can be linearized by a change of variables. Moreover he demonstrated that these equations are at most cubic in the first derivative and gave a convenient invariant description of all linearizable equations. We provide a similar description of the equations with three symmetries. There are four different types of such equations. Classes of equations belonging to one of these types were studied in N.H. Ibragimov and S.V. Meleshko, Invariants and invariant description of second-order ODEs with three infinitesimal symmetries. I, Communications in Nonlinear Science and Numerical Simulation, Vol. 12, No. 8, 2007, pp. 1370--1378. Namely, we presented there the candidates for all four types and studied one of these candidates.The present paper is devoted to other three candidates.
The paper is dedicated to construction of invariants for the parabolic equation u(t) + a(t, x)u(xx) + b(t, x)u(x) + c(t, x)u = 0. We consider the equivalence group given by point transformations and find all invariants up to seventh-order, i.e. the invariants involving the derivatives up to seventh-order of the coefficients a, b and c with respect to the independent variables t, x. (c) 2006 Elsevier B.V. All rights reserved.
We consider evolution equations of the form ut = f(x, u, ux)uxx + g(x, u, ux) and ut = uxx + g(x, u, ux). In the spirit of the recent work of Ibragimov [Ibragimov NH. Laplace type invariants for parabolic equations. Nonlinear Dynam 2002;28:125-33] who adopted the infinitesimal method for calculating invariants of families of differential equations using the equivalence groups, we apply the method to these equations. We show that the first class admits one differential invariant of order two, while the second class admits three functional independent differential invariants of order three. We use these invariants to determine equations that can be transformed into the linear diffusion equation.
Most of mathematical models describing spread of malignant tumours are formulated as systems of nonlinear partial differential equations containing, in general, several unknown functions of dependent variables. Determination of these unknown functions (called in group analysis arbitrary elements) is a complicated problem that challenges researchers. Our aim is to calculate the generators of the equivalence group for one of the known models and, using the equivalence generators, specify arbitrary elements, find additional symmetries and calculate group invariant solutions.
We apply the infinitesimal technique for calculating invariants for the family of nonlinear equations formulated in the title. We show that the infinite-dimensional equivalence Lie algebra has three functionally independent differential invariants of the second order. Knowledge of invariants of families of equations is essential for identifying distinctly different equations and therefore for the problem of group classification.
In the present paper a quantum drift–diffusion model describing semi-conductor devices is considered. New conservation laws for the model are computed and used to construct exact solutions.
In this paper the general magma equation modelling a melt flow in the Earth's mantle is discussed. Applying the new theorem on nonlocal conservation laws [Ibragimov NH. A new conservation theorem. J Math Anal Appl 2007;333(1):311-28] and using the symmetries of the model equation nonlocal conservation laws are computed. In accordance with Ibragimov [Ibragimov NH. Quasi-self-adjoint differential equations. Preprint in Archives of ALGA, vol. 4, BTH, Karlskrona, Sweden: Alga Publications; 2007. p. 55-60, ISSN: 1652-4934] it is shown that the general magma equation is quasi-self-adjoint for arbitrary m and n and self-adjoint for n = -m. These important properties are used for deriving local conservation laws. © 2008 Elsevier B.V. All rights reserved.
Solution of linearization problem of fourth-order ordinary differential equations Via contact transformations is presented in the paper. We show that all fourth-order ordinary differential equations that are linearizable by contact transformations are contained in the class of equations which is at most quadratic in the third-order derivative. We provide the linearization test and describe the procedure for obtaining the linearizing transformations as well as the linearized equation. Moreover, we obtain the general form of ordinary differential equations of order greater than four linearizable via contact transformations. (C) 2008 Elsevier B.V. All rights reserved.