On symmetries of the heat equation

Two aims are pursued in this article. The first one is methodological and consists of demonstrating the symmetry calculation techniques based on the commutation relation on the example of the quasi-linear heat equation. The second one consists of investigating the following open question: is the algebra of higher symmetries of the linear heat equation exhausted by those which may be obtained by means of the classical symmetries and well-known recursion operators? A positive solution of this question is given.     

Nonclassical symmetry of nonlinear diffusion equation and factorization method

We present the nonclassical symmetry of a nonlinear diffusion equation whose a nonlinear term is an arbitrary function. Generally, there is no guarantee that we can always determine the nonclassical symmetries admitted by the given equation because of the nonlinearity included in the determining equations. Accordingly, constructing invariant solutions is also generally difficult. In this paper, we apply the factorization method to nonclassical symmetry analysis for the nonlinear diffusion equation. Applying this method simplifies the determining equations and leads to their invariant solution automatically.     

Symmetry analysis of the two-dimensional diffusion equation with a source term

A symmetry analysis is performed on a (2+1)-dimensional linear diffusion equation with a nonlinear source term involving the dependent variable and its spatial derivatives. In the first part of the paper, we use the classical method to classify source terms where the original equation admits a nontrivial symmetry. In the second part of the paper, we use the nonclassical method and show that we simply recover the classical symmetries.     

Nonclassical symmetries as special solutions of heir-equations

In Phys. D 78 (1994) 124, we have found that iterations of the nonclassical symmetries method give rise to new nonlinear equations, which inherit the Lie point symmetry algebra of the given equation. In the present paper, we show that special solutions of the right-order heir-equation correspond to classical and nonclassical symmetries of the original equations. An infinite number of nonlinear equations which possess nonclassical symmetries are derived.     

Symmetries for a family of Boussinesq equations with nonlinear dispersion

In this paper, we make a full analysis of a family of Boussinesq equations which include nonlinear dispersion by using the classical Lie method of infinitesimals. We consider travelling wave reductions and we present some explicit solutions: solitons and compactons.For this family, we derive nonclassical and potential symmetries. We prove that the nonclassical method applied to these equations leads to new symmetries, which cannot be obtained by Lie classical method. We write the equations in a conserved form and we obtain a new class of nonlocal symmetries. We also obtain some Type-II hidden symmetries of a Boussinesq equation.     

Type-II hidden symmetry and nonlinear self-adjointness of Boiti–Leon–Pempinelli equation

This paper focuses on two aspects. Firstly, we convert Boiti–Leon–Pempinelli (BLP) equation to (1+1)-dimensional partial differential equation via similarity transformation, and then analyze hidden symmetry of BLP equations via studying classical and nonclassical symmetries of the (1+1)-dimensional equations. As a byproduct, some new invariant solutions of BLP equations are constructed. Secondly, we show that BLP equation is nonlinearly self-adjoint and give the general formula of conservation laws.     

Nonclassical Symmetries for Nonlinear Diffusion and Absorption

Nonclassical symmetry methods are used to study the nonlinear diffusion equation with a nonlinear source. In particular, exponential and power law diffusivities are examined and we obtain mathematical forms of the source term which permit nonclassical symmetry reductions. In addition to the known source terms obtained by classical symmetry methods, we find new source terms which admit symmetry reductions. We also deduce a class of nonclassical symmetries which are valid for arbitrary diffusivity and deduce corresponding new solution types. This is an important result since previously only traveling wave solutions were known to exist for arbitrary diffusivity. A number of examples are considered and new exact solutions are constructed.     

Group analysis,exact solutions and conservation laws of a generalized fifth order KdV equation

We study the generalized fifth order KdV equation using group methods and conservation laws. All of the geometric vector fields of the special fifth order KdV equation are presented. By using the nonclassical Lie group method, it is show that this equation does not admit nonclassical type symmetries. Then, on the basis of the optimal system, the symmetry reductions and exact solutions to this equation are constructed. For some special cases, we obtain additional nontrivial conservation laws and scaling symmetries.     

Symmetry analysis and exact solutions of semilinear heat flow in multi-dimensions

A symmetry group method is used to obtain exact solutions for a semilinear radial heat equation in n>1 dimensions with a general power nonlinearity. The method involves an ansatz technique to solve an equivalent first-order PDE system of similarity variables given by group foliations of this heat equation, using its admitted group of scaling symmetries. This technique yields explicit similarity solutions as well as other explicit solutions of a more general (non-similarity) form having interesting analytical behavior connected with blow up and dispersion. In contrast, standard similarity reduction of this heat equation gives a semilinear ODE that cannot be explicitly solved by familiar integration techniques such as point symmetry reduction or integrating factors.     

Nonclassical symmetry reductions for an inhomogeneous nonlinear diffusion equation

In this paper we consider a class of generalised diffusion equations which are of great interest in mathematical physics. For some of these equations model, fast diffusion nonclassical symmetries are derived. We find the connection between classes of nonclassical symmetries of the equation and of an associated system. These symmetries allow us to increase the number of solutions. Some of these solutions are unobtainable by classical symmetries and exhibit an interesting behaviour.     

Similarity variables and reduction of the heat equation on torus

The problem of symmetry classification for the heat equation on torus is studied by means of classical Lie group theory. The Lie point symmetries are constructed and Lie algebra is formed for equation under consideration. Then these algebras are used to classify its subalgebras up to conjugacy classes. In general the heat equation on torus admits one-, two-, three- and four-dimensional algebras. For one-dimensional algebra £₁ and £₂ the heat equation on torus is reduced in independent variables whereas in two-dimensional algebras £₃ and £₄ the considered heat equation is investigated by quadrature. While three- and four-dimensional algebras lead to a trivial solution.     

Lie group symmetries and Riemann function of Klein–Gordon–Fock equation with central symmetry

In the present paper Lie symmetry group method is applied to find new exact invariant solutions for Klein–Gordon–Fock equation with central symmetry. The found invariant solutions are important for testing finite-difference computational schemes of various boundary value problems of Klein–Gordon–Fock equation with central symmetry. The classical admitted symmetries of the equation are found. The infinitesimal symmetries of the equation are used to find the Riemann function constructively.     

Applying a new algorithm to derive nonclassical symmetries

In this paper we extend the procedure described for Bîlă and Niesen in [Bîlă N, Niesen J. On a new procedure for finding nonclassical symmetries. J Symbol Comp 2004;38:1523–33], to obtain the determining equations of the nonclassical symmetries associated with a partial differential equation system, to a different case. We offer some examples of how our method works. By using this procedure we obtain a new nonclassical symmetry for the 2 + 1-dimensional shallow water wave equation.     

Nonclassical symmetry solutions for reaction–diffusion equations with explicit spatial dependence

Nonclassical symmetry methods are used to study the linear diffusion equation with a nonlinear source term which includes explicit spatial dependence. Mathematical forms for the spatial dependence are found which enable strictly nonclassical symmetries to be admitted when the nonlinearity is cubic. A number of new exact solutions are constructed, and an application of one of these solutions to diploid population genetics is discussed.     

Symmetry reductions for a dissipation-modified KdV equation

In this paper we give a group classification for a dissipation-modified Korteweg-de Vries equation by means of the Lie method of the infinitesimals. We prove that, by using the nonclassical method, we get several new solutions which are unobtainable by Lie classical symmetries. We obtain nonclassical symmetries that reduce the dissipation-modified Korteweg-de Vries equation to ordinary equations with the Painlevé property. These solutions have not been derived elsewhere by the singular manifold method.     

Invariance properties of a general bond-pricing equation

We perform the group classification of a bond-pricing partial differential equation of mathematical finance to discover the combinations of arbitrary parameters that allow the partial differential equation to admit a nontrivial symmetry Lie algebra. As a result of the group classification we propose “natural” values for the arbitrary parameters in the partial differential equation, some of which validate the choices of parameters in such classical models as that of Vasicek and Cox-Ingersoll-Ross. For each set of these natural parameter values we compute the admitted Lie point symmetries, identify the corresponding symmetry Lie algebra and solve the partial differential equation.     

Steady one-dimensional heat flow in a longitudinal triangular and parabolic fin

We consider a heat transfer problem of a longitudinal fin with triangular and parabolic profiles. Both thermal conductivity and heat transfer coefficient are assumed to be temperature-dependent, and given by power laws. We construct exact solution when the problem is linearizable. In the other case, classical Lie symmetry techniques are employed to analyze the problem. The obtained exact solutions satisfy the realistic boundary conditions. The effects of the physical applicable parameters such as thermo-geometric fin parameter and the fin efficiency are analyzed.     

Solutions through nonclassical potential symmetries for a generalized inhomogeneous nonlinear diffusion equation

In this paper, we consider a class of generalized diffusion equations which are of great interest in mathematical physics. For some of these equations that model fast diffusion, nonclassical and nonclassical potential symmetries are derived. These symmetries allow us to increase the number of solutions. These solutions are unobtainable neither from classical nor from classical potential symmetries. Copyright © 2007 John Wiley & Sons, Ltd.     

Nonclassical symmetries of a class of Burgers' systems

The nonclassical symmetries of a class of Burgers' systems are considered. This study was initialized by Cherniha and Serov with a restriction on the form of the nonclassical symmetry operator. In this paper we remove this restriction and solve the determining equations to show that (1) a new form of a Burgers' system exists that admits a nonclassical symmetry and (2) a Burgers' system exists that is linearizable.     

Symmetry analysis and exact solutions of some Ostrovsky equations

We apply the classical Lie method and the nonclassical method to a generalized Ostrovsky equation (GOE) and to the integrable Vakhnenko equation (VE), which Vakhnenko and Parkes proved to be equivalent to the reduced Ostrovsky equation. Using a simple nonlinear ordinary differential equation, we find that for some polynomials of velocity, the GOE has abundant exact solutions expressible in terms of Jacobi elliptic functions and consequently has many solutions in the form of periodic waves, solitary waves, compactons, etc. The nonclassical method applied to the associated potential system for the VE yields solutions that arise from neither nonclassical symmetries of the VE nor potential symmetries. Some of these equations have interesting behavior such as “nonlinear superposition.”