Complex Lie symmetries for scalar second-order ordinary differential equations

A scalar complex ordinary differential equation can be considered as two coupled real partial differential equations, along with the constraint of the Cauchy–Riemann equations, which constitute a system of four equations for two unknown real functions of two real variables. It is shown that the resulting system possesses those real Lie symmetries that are obtained by splitting each complex Lie symmetry of the given complex ordinary differential equation. Further, if we restrict the complex function to be of a single real variable, then the complex ordinary differential equation yields a coupled system of two ordinary differential equations and their invariance can be obtained in a non-trivial way from the invariance of the restricted complex differential equation. Also, the use of a complex Lie symmetry reduces the order of the complex ordinary differential equation (restricted complex ordinary differential equation) by one, which in turn yields a reduction in the order by one of the system of partial differential equations (system of ordinary differential equations). In this paper, for simplicity, we investigate the case of scalar second-order ordinary differential equations. As a consequence, we obtain an extension of the Lie table for second-order equations with two symmetries.     

Semi‐invariant solutions of the Navier‐Stokes equations

We present new exact solutions and reduced differential systems of the Navier‐Stokes equations of incompressible viscous fluid flow. We apply the method of semi‐invariant manifolds, introduced earlier as a modification of the Lie invariance method. We show that many known solutions of the Navier‐Stokes equations are, in fact, semi‐invariant and that the reduced differential systems we derive using semi‐invariant manifolds generalize previously obtained results that used ad hoc methods. Many of our semi‐invariant solutions solve decoupled systems in triangular form that are effectively linear. We also obtain several new reductions of Navier‐Stokes to a single nonlinear partial differential equation. In some cases, we can solve reduced systems and generate new analytic solutions of the Navier‐Stokes equations or find their approximations, and physical interpretation.     

A relationship between λ‐symmetries and first integrals for ordinary differential equations

In this paper, we provide some geometric properties of λ‐symmetries of ordinary differential equations using vector fields and differential forms. According to the corresponding geometric representation of λ‐symmetries, we conclude that first integrals can also be derived if the equations do not possess enough symmetries. We also investigate the properties of λ‐symmetries in the sense of the deformed Lie derivative and differential operator. We show that λ‐symmetries have the exact analogous properties as standard symmetries if we take into consideration the deformed cases.     

Conditional Lie‐Bäcklund Symmetry of Evolution System and Application for Reaction‐Diffusion System

Conditional Lie‐Bäcklund symmetry (CLBS) method is developed to study system of evolution equations. It is shown that reducibility of a system of evolution equations to a system of ordinary differential equations can be fully characterized by the CLBS of the considered system. As an application of the approach, a class of two‐component nonlinear diffusion equations is studied. The governing system and the admitted CLBS can be identified. As a consequence, exact solutions defined on the polynomial, exponential, trigonometric, and mixed invariant subspaces are constructed due to the corresponding symmetry reductions.     

Symmetry group analysis and similarity solutions for the (2+1)‐dimensional coupled Burger's system

Symmetry group analysis and similarity reduction of nonlinear system of coupled Burger equations in the form of nonlinear partial differential equation are analyzed via symmetry method. The symmetry method has led to similarity reductions of this equation to solvable form to third‐order partial differential equation. The infinitesimal, similarity variables, dependent variables, and reduction have been tabulated. The search for solutions of these systems by using the improved tanh method has yielded certain exact solutions expressed by rational functions. Some figures are given to show the properties of the solutions. Copyright © 2013 John Wiley & Sons, Ltd.     

Bianchi Permutability for the Anti‐Self‐Dual Yang‐Mills Equations

The anti‐self‐dual Yang‐Mills equations are known to have reductions to many integrable differential equations. A general Bäcklund transformation (BT) for the anti‐self‐dual Yang‐Mills (ASDYM) equations generated by a Darboux matrix with an affine dependence on the spectral parameter is obtained, together with its Bianchi permutability equation. We give examples in which we obtain BTs of symmetry reductions of the ASDYM equations by reducing this ASDYM BT. Some discrete integrable systems are obtained directly from reductions of the ASDYM Bianchi system.     

Type-II hidden symmetries through weak symmetries for nonlinear partial differential equations

The Type-II hidden symmetries are extra symmetries in addition to the inherited symmetries of the differential equations when the number of independent and dependent variables is reduced by a Lie point symmetry. In [B. Abraham-Shrauner, K.S. Govinder, Provenance of Type II hidden symmetries from nonlinear partial differential equations, J. Nonlinear Math. Phys. 13 (2006) 612-622] Abraham-Shrauner and Govinder have analyzed the provenance of this kind of symmetries and they developed two methods for determining the source of these hidden symmetries. The Lie point symmetries of a model equation and the two-dimensional Burgers' equation and their descendants were used to identify the hidden symmetries. In this paper we analyze the connection between one of their methods and the weak symmetries of the partial differential equation in order to determine the source of these hidden symmetries. We have considered the same models presented in [B. Abraham-Shrauner, K.S. Govinder, Provenance of Type II hidden symmetries from nonlinear partial differential equations, J. Nonlinear Math. Phys. 13 (2006) 612-622], as well as the WDVV equations of associativity in two-dimensional topological field theory which reduces, in the case of three fields, to a single third order equation of Monge-Ampère type. We have also studied a second order linear partial differential equation in which the number of independent variables cannot be reduced by using Lie symmetries, however when is reduced by using nonclassical symmetries the reduced partial differential equation gains Lie symmetries.     

The four‐dimensional Martínez Alonso–Shabat equation: nonlinear self‐adjointness and conservation laws

We show that the four‐dimensional Martínez Alonso–Shabat equation is nonlinearly self‐adjoint with differential substitution and the required differential substitution is just the admitted adjoint symmetry and vice versa. By means of computer algebra system, we obtain a number of local and nonlocal symmetries admitted by the equations under study. Then such symmetries are used to construct conservation laws of the equation under study and its reductions. Copyright © 2016 John Wiley & Sons, Ltd.     

A study of integrability and symmetry for the (p + 1)th Boltzmann equation via Painlevé analysis and Lie‐group method

In this paper,we applied the Painlevé property test on Krook‐Wu model of the nonlinear Boltzmann equation (p = 1). As a result, by using Bäcklund transformation, we obtained three solutions two of them were known earlier, while the third one is new and more general, we have also two reductions one of them is Abel's equation. Also, Lie‐group method is applied to the (p + 1)th Boltzmann equation. The complete Lie algebra of infinitesimal symmetries is established. Three nonequivalent sub‐algebraic of the complete Lie algebra are used to investigate similarity solutions and similarity reductions in the form of nonlinear ordinary equations for (p + 1)th Boltzmann equation; we obtained two general solutions for (p + 1)th Boltzmann equation and new solutions for Krook‐Wu model of Boltzmann equation (p = 1). Copyright © 2014 John Wiley & Sons, Ltd.     

Oscillation criteria for fourth‐order differential equations with middle term

Oscillation criteria for self‐adjoint fourth‐order differential equations were established for various conditions on the coefficients r(x) > 0, q(x) and p(x). However, most of these results deal with the case when lim_{x → ∞}∫^x₁q(s) ds < +∞. In this note we give a new oscillation criterion in the case when this condition is not fulfilled, in particular when q(x)↗ + ∞ (even with exponential growth).     

Integrability of Equations Admitting the Nonsolvable Symmetry Algebra so(3,R)

If an ordinary differential equation admits the nonsolvable Lie algebra     , and we use any of its generators to reduce the order, the reduced equation does not inherit the remaining symmetries. We prove here how the lost symmetries can be recovered as   C ^∞  -symmetries of the reduced equation. If the order of the last reduced equation is higher than one, these   C ^∞  -symmetries can be used to obtain new order reductions. As a consequence, a classification of the third-order equations that admit     as symmetry algebra is given and a step-by-step method to solve the equations is presented.     

Exact solutions of the time fractional nonlinear Schrödinger equation with two different methods

In the present paper, exact solutions of fractional nonlinear Schrödinger equations have been derived by using two methods: Lie group analysis and invariant subspace method via Riemann‐Liouvill derivative. In the sense of Lie point symmetry analysis method, all of the symmetries of the Schrödinger equations are obtained, and these operators are applied to find corresponding solutions. In one case, we show that Schrödinger equation can be reduced to an equation that is related to the Erdelyi‐Kober functional derivative. The invariant subspace method for constructing exact solutions is presented for considered equations.     

Integrating Scalar Ordinary Differential Equations with Symmetry Revisited

The process of integrating an nth-order scalar ordinary differential equation with symmetry is revisited in terms of Pfaffian systems. This formulation immediately provides a completely algebraic method to determine the initial conditions and the corresponding solutions which are invariant under a one parameter subgroup of a symmetry group. To determine the noninvariant solutions the problem splits into three cases. If the dimension of the symmetry groups is less than the order of the equation, then there exists an open dense set of initial conditions whose corresponding solutions can be found by integrating a quotient Pfaffian system on a quotient space, and integrating an equation of fundamental Lie type associated with the symmetry group. If the dimension of the symmetry group is equal to the order of the equation, then there exists an open dense set of initial conditions whose corresponding solutions are obtained either by solving an equation of fundamental Lie type associated with the symmetry group, or the solutions are invariant under a one-parameter subgroup. If the dimension of the symmetry group is greater than the order of the equation, then there exists an open dense set of initial conditions where the solutions can either be determined by solving an equation of fundamental Lie type for a solvable Lie group, or are invariant. In each case the initial conditions, the quotient Pfaffian system, and the equation of Lie type are all determined algebraically. Examples of scalar ordinary differential equations and a Pfaffian system are given.     

Invariant approach to optimal investment–consumption problem: the constant elasticity of variance (CEV) model

The optimal investment–consumption problem under the constant elasticity of variance (CEV) model is solved using the invariant approach. Firstly, the invariance criteria for scalar linear second‐order parabolic partial differential equations in two independent variables are reviewed. The criteria is then employed to reduce the CEV model to one of the four Lie canonical forms. It is found that the invariance criteria help in transforming the original equation to the second Lie canonical form and with a proper parameter selection; the required transformation converts the original equation to the first Lie canonical form that is the heat equation. As a consequence, we find some new classes of closed‐form solutions of the CEV model for the case of reduction into heat equation and also into second Lie canonical form. The closed‐form analytical solution of the Cauchy initial value problems for the CEV model under investigation is also obtained. Copyright © 2016 John Wiley & Sons, Ltd.     

Existence of solutions for some non‐Fredholm integro‐differential equations with the bi‐Laplacian

We prove the existence of solutions for some semilinear elliptic equations in the appropriate H⁴ spaces using the fixed‐point technique where the elliptic equation contains fourth‐order differential operators with and without Fredholm property, generalizing the previous results.     

Conserved Forms derived from Symmetries

For first order scalar ordinary differential equations, a well–known result of Sophus Lie states that a Lie point symmetry can be used to construct an integrating factor and conversely. However, there exist higher order equations without Lie point symmetries that admit integrating factors or that are exact. We present a method based on λ-symmetries to calculate integrating factors. An example of a second order equation without Lie point symmetries illustrates how the method works in practice and how the computations that appear in other methods may be simplified. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

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.     

Space‐time spectral collocation method for the one‐dimensional sine‐Gordon equation

In this article, we introduce a new space‐time spectral collocation method for solving the one‐dimensional sine‐Gordon equation. We apply a spectral collocation method for discretizing spatial derivatives, and then use the spectral collocation method for the time integration of the resulting nonlinear second‐order system of ordinary differential equations (ODE). Our formulation has high‐order accurate in both space and time. Optimal a priori error bounds are derived in the L²‐norm for the semidiscrete formulation. Numerical experiments show that our formulation have exponential rates of convergence in both space and time. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 670–690, 2015     

Computation of local symmetries of second-order ordinary differential equations by the Cartan equivalence method

The Cartan equivalence method is used to find out if a given equation has a nontrivial Lie group of point symmetries. In particular, we compute invariants that permit one to recognize equations with a three-dimensional symmetry group. An effective method to transform the Lie system (the system of partial differential equations to be satisfied by the infinitesimal point symmetries) into a formally integrable form is given. For equations with a three-dimensional symmetry group, the formally integrable form of the Lie system is found explicitly. Translated fromMatematicheskie Zametki, Vol. 60, No. 1, pp. 75–91, July, 1996.     

General exact solution of the fin problem with the power law temperature‐dependent thermal conductivity

In this paper, the general exact implicit solution of the second‐order nonlinear ordinary differential equation governing heat transfer in rectangular fin is obtained using Lie point symmetry method. General relationship among the fin efficiency, the rate of heat transfer from the entire fin, the fin effectiveness, and the thermo‐geometric fin parameter is obtained for any value of the mode of heat transfer n and the constant β. Copyright © 2015 John Wiley & Sons, Ltd.