Approximate symmetry group classification for a nonlinear fractional filtration equation of diffusion-wave type

A one-dimensional nonlinear fractional filtration equation with the Riemann–Liouville time-fractional derivative is proposed for modeling fluid flow through a porous medium. This equation is derived under an assumption that the fluid has a fractional equation of state in which the fluid density depends on the time-fractional derivative of pressure. The obtained equation belongs to the diffusion-wave type of equations. A case when the order of fractional differentiation is close to an integer number is considered, and a small parameter is introduced into the fractional filtration equation under consideration. An expansion of the Riemann–Liouville time-fractional derivative into the series with respect to this small parameter is obtained. Using this expansion, a first-order approximation of the derived fractional filtration equation is performed. Next, the problem of approximate Lie point symmetry group classification for this approximate nonlinear filtration equation with a small parameter is studied. It is shown that approximate symmetry groups admitted by different realizations of the approximate filtration equation have much more dimensions than the corresponding exact Lie point symmetry groups admitted by unperturbed fractional diffusion-wave equations. Obtained classification results permit to construct approximate invariant solutions for the considered nonlinear time-fractional filtration equations.     

Invariant and partially invariant solutions of integro-differential equations for linear thermoviscoelastic aging materials with memory

A linear thermoviscoelastic model for homogeneous, aging materials with memory is established. A system of integro-differential equations is obtained by using two motions (a one-dimensional motion and a shearing motion) for this model. Applying the group analysis method to the system of integro-differential equations, the admitted Lie group is determined. Using this admitted Lie group, invariant and partially invariant solutions are found. The present paper gives a first example of application of partially invariant solutions to integro-differential equations.     

On group properties and conservation laws for second-order quasi-linear differential equations

A sufficient condition for the absence of tangent transformations admitted by second-order quasi-linear differential equations and a sufficient condition for linear autonomy of operators of the Lie group of transformations admitted by second-order weakly nonlinear differential equations are found. A theorem on the structure of the first-order conservation laws for second-order weakly nonlinear differential equations is proved. A classification of second-order linear differential equations with two independent variables in terms of first-order conservation laws is proposed. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 50, No. 3, pp. 64–70, May–June, 2009.     

Group analysis and similarity solutions of the compressible boundary layer equations

Summary In this paper the application of Lie's methods to the equations of the laminar boundary layer is discussed. The momentum and energy equations in Prandtl's form are considered for a steady, viscous, compressible laminar flow with non zero pressure gradient, variable viscosity and thermal conductivity. Group analysis yields similarity solutions for given pressure distributions and particular values of the invariance group parameters (group classification). Crocco's transformation is obtained for the infinite-dimensional group of the Lie's algebra admitted by the equations.

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Sommario In questa nota si applicano i metodi di Lie alle equazioni dello strato limite laminare nella forma di Prandtl per un fluido viscoso, compressibile, con gradiente di pressione non nullo e con viscosità e conducibilità termica variabili. L'analisi gruppale fornisce soluzioni di similarità per date distribuzioni di pressione e valori particolari dei parametri del gruppo di invarianza. La trasformazione di Crocco si ottiene in corrispondenza della parte infinito-dimensionale dell'algebra di Lie ammessa dalle equazioni.

     

Symmetries of the hyperbolic shallow water equations and the Green–Naghdi model in Lagrangian coordinates

The observation that the hyperbolic shallow water equations and the Green–Naghdi equations in Lagrangian coordinates have the form of an Euler–Lagrange equation with a natural Lagrangian allows us to apply Noether's theorem for constructing conservation laws for these equations. In this study the complete group analysis of these equations is given: admitted Lie groups of point and contact transformations, classification of the point symmetries and all invariant solutions are studied. For the hyperbolic shallow water equations new conservation laws which have no analog in Eulerian coordinates are obtained. Using Noether's theorem a new conservation law of the Green–Naghdi equations is found. The dependence of solutions on the parameter is illustrated by self-similar solutions which are invariant solutions of both models.     

Nonlocal symmetries of evolution equations

We suggest the method for group classification of evolution equations admitting nonlocal symmetries which are associated with a given evolution equation possessing nontrivial Lie symmetry. We apply this method to second-order evolution equations in one spatial variable invariant under Lie algebras of the dimension up to three. As a result, we construct the broad families of new nonlinear evolution equations possessing nonlocal symmetries which in principle cannot be obtained within the classical Lie approach.     

On group classification of the spatially homogeneous and isotropic Boltzmann equation with sources

In Nonenmacher (1984) [1] an admitted Lie group of transformations for the spatially homogeneous and isotropic Boltzmann equation with sources was studied. In fact, the author is Nonenmacher (1984) [1] considered the equation for a generating function of the power moments of the Boltzmann equation solution. However, this equation is still a non-local partial differential equation, and this property was not taken into account there. In the present paper the admitted Lie group of this equation is studied, using our original method developed for group analysis of equations with non-local operators (Grigoriev and Meleshko, 1986; Meleshko, 2005; Grigoriev et al., 2010 [2], [3], [4]). The Lie groups obtained are compared with Nonenmacher (1984) [1]. The deficiency of Nonenmacher (1984) [1] is corrected.     

Group analysis of integro-differential equations describing stress relaxation behavior of one-dimensional viscoelastic materials

In this paper a recently developed approach of the group analysis method is applied to a system of integro-differential equations describing the stress relaxation behavior of one-dimensional viscoelastic materials. An admitted Lie group is defined by solving determining equations of the system. Using an optimal system of one-dimensional subalgebras, all invariant solutions are obtained.     

Symmetry groups of integro-differential equations for linear thermoviscoelastic materials with memory

The group analysis method is applied to a system of integro-differential equations corresponding to a linear thermoviscoelastic model. A recently developed approach for calculating the symmetry groups of such equations is used. The general solution of the determining equations for the system is obtained. Using subalgebras of the admitted Lie algebra, two classes of partially invariant solutions of the considered system of integro-differential equations are studied.     

New Steady and Self-Similar Solutions of the Euler Equations

Exact steady and self-similar solutions of the Euler equations are considered, which possess the property of partial invariance with respect to a certain six-parameter Lie group. New examples of vortex motion of a swirled liquid in curved channels are presented. A classification is given for self-similar solutions of the reduced system with two independent variables, which admits a three-parameter group of extensions, whereas the initial system of the Euler equations possesses a two-parameter group.     

Preliminary group classification of quasilinear third-order evolution equations

Group classification of quasilinear third-order evolution equations is given by using the classical infinitesimal Lie method, the technique of equivalence transformations, and the theory of classification of abstract low-dimensional Lie algebras. We show that there are three equations admitting simple Lie algebras of dimension three. All non-equivalent equations admitting simple Lie algebras are nothing but these three. Furthermore, we also show that there exist two, five, twenty-nine and twenty-six non- equivalent third-order nonlinear evolution equations admitting one-, two-, three-, and four-dimensional solvable Lie algebras, respectively.     

General dynamical systems described by first order quasilinear PDEs reducible to homogeneous and autonomous form

Within the framework of Lie group analysis of differential equations, a theorem is determined stating necessary and sufficient conditions allowing one to recover an invertible point transformation mapping a general dynamical system described by nonhomogeneous and nonautonomous first order quasilinear partial differential equations to homogeneous and autonomous form. The proof of the theorem is constructive and the new independent and dependent variables are obtained by determining the canonical variables associated to a suitable subalgebra of the Lie algebra of point symmetries admitted by the source system. The theorem is applied by considering some examples of physical interest arising from different contexts.     

Group classification and exact solutions of a generalized Emden–Fowler equation

The aim of this work is to perform a complete symmetry classification of a generalized Emden-Fowler equation. The various forms of this equation are extensively studied in the literature and they have applications in astrophysical and physiological phenomena. The classical approach of group classification and the procedure based upon the Lie algebras of low dimension are employed for classification. Exact solutions of the invariant equations are derived.     

Symmetry classification and optimal systems of a non-linear wave equation

In this paper the complete Lie group classification of a non-linear wave equation is obtained. Optimal systems and reduced equations are achieved in the case of a hyperelastic homogeneous bar with variable cross section.     

On group classification of the spatially homogeneous and isotropic Boltzmann equation with sources II

A previous paper by the authors (Grigoriev and Meleshko, 2012 [4]) was devoted to group analysis of the equation for the power moment generating function of a solution of the Boltzmann kinetic equation with sources. An approach developed earlier by Grigoriev and Meleshko (1986 [2]) was employed for finding the admitted Lie group. This approach allowed to correct Nonenmacher׳s results (1984, [1]) and to perform a partial group classification of the considered equation with respect to a source function. The present paper completes this group classification by an efficient algebraic method.     

Application of group analysis to stochastic equations of fluid dynamics

Group analysis is used to study stochastic equations of fluid dynamics. Determining equations for admitted Lie groups of transformation involving independent and dependent variables and Wiener processes are obtained. It is shown that, as in the case of deterministic differential equations, admitted groups make it possible to reduce invariant solutions of stochastic differential equations to solutions with a smaller number of independent variables.     

Lie Symmetry Analysis of Differential Equations in Finance

Lie group theory is applied to differential equations occurring as mathematical models in financial problems. We begin with the complete symmetry analysis of the one-dimensional Black–Scholes model and show that this equation is included in Sophus Lie's classification of linear second-order partial differential equations with two independent variables. Consequently, the Black–Scholes transformation of this model into the heat transfer equation follows directly from Lie's equivalence transformation formulas. Then we carry out the classification of the two-dimensional Jacobs–Jones model equations according to their symmetry groups. The classification provides a theoretical background for constructing exact (invariant) solutions, examples of which are presented.     

Analysis of the von Karman equations by group methods

One of the systems of equations approximating the large deflection of plates consists of two coupled non-linear fourth order partial differential equations, known as the von Karman equations. The full symmetry group for the steady equations is a finitely generated Lie group with ten parameters. For the time-dependent system the full symmetry group is an infinite parameter Lie group. Several subgroups of the full group are used to generate exact solutions of the time-independent and the time-dependent systems. These include the dilatation group (similar solutions), rotation group, screw group and others. Physical implications and applications are discussed.     

Group classification and exact solutions of a radially symmetric porous-medium equation

In this paper we perform a group classification for the generalized radial porous-medium equation. We also classify symmetry reductions of the equation to first- or second-order ordinary differential equations (ODEs) and hence construct invariant solutions in a systematic manner. We show that the reduced second-order equations are invariant under either a two-parameter or one-parameter Lie groups. In the first case, they are completely integrated by a pair of quadratures. In the latter, they are often reduced to first-order ODEs of Abel type.     

Application of Lie transformation group methods to classical linear theories of rods and plates

In the present paper, a class of partial differential equations governing various rod and plate theories of Bernoulli–Euler and Poisson–Kirchhoff type is studied by Lie transformation group methods. A system of equations determining the generators of the admitted point Lie groups (symmetries) is derived and the general statement of the associated group-classification problem is given. A simple relation is deduced allowing to recognize easily the variational symmetries among the “ordinary” symmetries of a self-adjoint equation of the class examined. Explicit formulae for the conserved currents of the corresponding (via Bessel-Hagen’s extension of Noether’s theorem) conservation laws are suggested. Solutions of group-classification problems are given for subclasses of equations of the foregoing type governing stability and vibration of rods, fluid conveying pipes and plates resting on variable elastic foundations. The obtained group-classification results are used to derive conservation laws and group-invariant solutions readily applicable in rod dynamics and plate statics and dynamics. New generalized symmetries and conservation laws for the theories of Timoshenko beams, Reissner–Mindlin plates and three-dimensional elastostatics are presented.