Exact solutions of Calgero–Bogoyavlenskii–Schiff equation using the singular manifold method after Lie reductions

Calgero–Bogoyavlenskii–Schiff (CBS) equation is analytically solved through two successive reductions into an ordinary differential equation (ODE) through a set of optimal Lie vectors. During the second reduction step, CBS equation is reduced using hidden vectors. The resulting ODE is then analytically solved through the singular manifold method in three steps; First, a Bäcklund truncated series is obtained. Second, this series is inserted into the ODE, and finally, a seminal analysis leads to a Schwarzian differential equation in the eigenfunction φ(η). Solving this differential equation leads to new analytical solutions. Then, through two backward substitution steps, the original dependent variable is recovered. The obtained results are plotted for several Lie hidden vectors and compared with previous work on CBS equation using Lie transformations. Copyright © 2017 John Wiley & Sons, Ltd.     

Investigation of new solutions for an extended (2+ 1)-dimensional Calogero-Bogoyavlenskii-Schif equation

We investigate and concentrate on new infinitesimal generators of Lie symmetries for an extended (2+ 1)-dimensional Calogero-Bogoyavlenskii-Schif (eCBS) equation using the commutator table which results in a system of nonlinear ordinary differential equations (ODEs) which can be manually solved. Through two stages of Lie symmetry reductions, the eCBS equation is reduced to non-solvable nonlinear ODEs using different combinations of optimal Lie vectors. Using the integration method and the Riccati and Bernoulli equation methods, we investigate new analytical solutions to those ODEs. Back substituting to the original variables generates new solutions to the eCBS equation. These results are simulated through three- and two-dimensional plots.     

Lie symmetry analysis to Fisher''s equation with time fractional order

The aim of this letter is to apply the Lie group analysis method to the Fisher''s equation with time fractional order. We considered the symmetry analysis, explicit solutions to the time fractional Fisher''s(TFF) equations with Riemann-Liouville (R-L) derivative. The time fractional Fisher''s is reduced to respective nonlinear ordinary differential equation(ODE) of fractional order. We solve the reduced fractional ODE using an explicit power series method.     

An exact solution of a quasilinear Fisher equation in cylindrical coordinates

Lie symmetry method is applied to analyse Fisher equation in cylindrical coordinates. Symmetry algebra is found and symmetry invariance is used to reduce the equation to a first-order ODE. The first-order ODE is further analysed to obtain exact solution of Fisher equation in explicit form.     

Classification of ordinary differential equations by conditional linearizability and symmetry

Lie’s invariant criteria for determining whether a second order scalar equation is linearizable by point transformation have been extended to third and fourth order scalar ordinary differential equations (ODEs). By differentiating the linearizable by point transformation scalar second order ODE (respectively third order ODE) and then requiring that the original equation holds, what is called conditional linearizability by point transformation of third and fourth order scalar ODEs, is discussed. The result is that the new higher order nonlinear ODE has only two arbitrary constants available in its solution. One can use the same procedure for the third and fourth order extensions mentioned above to get conditional linearizability by point or other types of transformation of higher order scalar equations. Again, the number of arbitrary constants available will be the order of the original ODE. A classification of ODEs according to conditional linearizability by transformation and classifiability by symmetry are proposed in this paper.     

An exact solution of a quasilinear Fisher equation in cylindrical coordinates

Lie symmetry method is applied to analyse Fisher equation in cylindrical coordinates. Symmetry algebra is found and symmetry invariance is used to reduce the equation to a first-order ODE. The first-order ODE is further analysed to obtain exact solution of Fisher equation in explicit form.     

几种微分方程李对称决定组生成软件包的比较研究

根据微分方程李对称决定方程组生成算法,在计算机代数系统Maple上给出了具体实现GDS,并指出了现行Maple系统上关于李对称计算的软件包liesymm的缺陷,最后通过一些计算实例对三个Maple系统下的软件包GDS,liesymm以及Desolv进行了比较研究.     

Modified ‐expansion method for finding exact wave solutions of the coupled Klein–Gordon–Schrödinger equation

The coupled Klein–Gordon–Schrödinger equation is reduced to a nonlinear ordinary differential equation (ODE) by using Lie classical symmetries, and various solutions of the nonlinear ODE are obtained by the modified ‐expansion method proposed recently. With the aid of solutions of the nonlinear ODE, more explicit traveling wave solutions of the coupled Klein–Gordon–Schrödinger equation are found out. The traveling wave solutions are expressed by the hyperbolic functions, trigonometric functions, and rational functions. Copyright © 2012 John Wiley & Sons, Ltd.     

非线性偏微分方程的约化和精确解

§ 1　IntroductionSeeking the exact solutions of the nonlinear partial differential equation is one of thevery importantsubjectin PDE research.Up to now,many methods offinding the exact so-lutions for NLPDE are constructed,such as inverse scattering transformation(IST) [1 ] ,Liepoint symmetry and similar reductions[2 ,3] ,B cklund[4— 6] and Cole-Hofe transformations,Hirota s bilinear method[7] ,the homogeneous balance method[8,9] ,tanh function method[1 0 ]and so on.In this paper,we giv…     

Whittaker vectors and conical vectors

A holomorphic family of differential operators of infinite order is constructed that transforms conical vectors for principal series representations of quasi-split, linear, semi-simple Lie groups into Whittaker vectors. Using this transform, it is shown that algebraic Whittaker vectors (as studied by Kostant) extend to ultradistributions of Gevrey type on principal series representations. For each element of the small Weyl group, a meromorphic family of Whittaker vectors is constructed from this transform and the Kunze-Stein intertwining integrals. An explict formula is derived for the smooth Whittaker vector (discovered by Jacquet), in terms of these families of ultradistribution Whittaker vectors. In particular, this gives new proofs of Jacquet's analytic continuation of the smooth Whittaker vector and its functional equation (Jacquet and Schiffman). Applications of the transform are also given to the theory of Verma modules.     

Nonlinear asymptotic analysis of a system of two free coupled oscillators with cubic nonlinearities

In this paper a two degrees of freedom undamped nonlinear system of two unforced coupled oscillators with cubic nonlinearities is analyzed. Through a decoupling procedure and using admissible functional transformations it is proved that this system can be reduced to an intermediate second order nonlinear ordinary differential equation (ODE) connecting both displacements to each other. By nonlinear asymptotic approximations the above equation can be further reduced to new nonlinear ODE that can be analytically solved. The solutions in the physical plane are extracted in parametric form. As generalization, the model of a damped system of two masses connected with stiffness with linear and nonlinear coefficient of rigidities respectively is analyzed and exact analytical solutions are extracted. Finally an application has been given in the case of a two mass system with cubic strong non-linearity.     

Invariant analysis of time fractional generalized Burgers and Korteweg–de Vries equations

A systematic investigation to derive Lie point symmetries to time fractional generalized Burgers as well as Korteweg–de Vries equations is presented. Using the obtained Lie point symmetries we have shown that each of them has been transformed into a nonlinear ordinary differential equation of fractional order with a new independent variable. The derivative corresponding to time fractional in the reduced equation is usually known as the Erdélyi–Kober fractional derivative.     

On limiting values of stochastic differential equations with small noise intensity tending to zero

When the right-hand side of an ordinary differential equation (ODE in short) is not Lipschitz, neither existence nor uniqueness of solutions remain valid. Nevertheless, adding to the differential equation a noise with nondegenerate intensity, we obtain a stochastic differential equation which has pathwise existence and uniqueness property. The goal of this short paper is to compare the limit of solutions to stochastic differential equation obtained by adding a noise of intensity ε to the generalized Filippov notion of solutions to the ODE. It is worth pointing out that our result does not depend on the dimension of the space while several related works in the literature are concerned with the one dimensional case.     

Lie symmetries and differential Galois groups of linear equations

For a linear ordinary differential equation the Lie algebra of its infinitesimal Lie symmetries is compared with its differential Galois group. For this purpose an algebraic formulation of Lie symmetries is developed. It turns out that there is no direct relation between the two above objects. In connection with this a new algorithm for computing the Lie symmetries of a linear ordinary differential equation is presented.     

Continuous dependence theorems on solutions of uncertain differential equations

In ordinary differential equation (ODE) and stochastic differential equation (SDE), the solution continuously depends on initial value and parameter under some conditions. This paper investigates the analogous continuous dependence theorems in uncertain differential equation (UDE). It proves two continuous dependence theorems, a basic one and a general one.     

Optimal solutions of a (3 + 1)-dimensional B-Kadomtsev-Petviashvii equation

We explored and specialized new Lie infinitesimals for the (3 + 1)-dimensional B-Kadomtsev-Petviashvii (BKP) using the commutation product, which results a system of nonlinear ODEs manually solved. Through two stages of Lie symmetry reduction, (3 + 1)-dimensional BKP equation is reduced to nonsolvable nonlinear ODEs using various combinations of optimal Lie vectors. Using the integration and Riccati equation methods, we investigate new analytical solutions for these ODEs. Back substituting to the original variables generates new solutions for BKP. Some selected solutions illustrated through three-dimensional plots.     

Exact solutions of a novel integrable partial differential equation

We consider the derivation of exact solutions of a novel integrable partial differential equation (PDE). This equation was introduced with the aim that it mirror properties of the second Painlevé equation (P_II), and it has the remarkable property that, in addition to the usual kind of auto-Bäcklund transformation that one would expect of an integrable PDE, it also admits an auto-Bäcklund transformation of ordinary differential equation (ODE) type, i.e., a mapping between solutions involving shifts in coefficient functions, and which is an exact analogue of that of P_II with its shift in parameter.We apply three methods of obtaining exact solutions. First of all we consider the Lie symmetries of our PDE, this leading to a variety of solutions including in terms of the second Painlevé transcendent, elliptic functions and hyperbolic functions. Our second approach involves the use of our ODE-type auto-Bäcklund transformation applied to solutions arising as solutions of an equation analogous to the special integral of P_II. It turns out that our PDE has a second remarkable property, namely, that special functions defined by any linear second order ODE can be used to obtain a solution of our PDE. In particular, in the case of solutions defined by Bessel’s equation, iteration using our ODE-type auto-Bäcklund transformation is possible and yields a chain of solutions defined in terms of Bessel functions. We also consider the use of this transformation in order to derive solutions rational in x. Finally, we consider the standard auto-Bäcklund transformation, obtaining a nonlinear superposition formula along with one- and two-soliton solutions. Velocities are found to depend on coefficients appearing in the equation and this leads to a wide range of interesting behaviours.     

Completions of nonlinear DAE flows based on index reduction techniques and their stabilization

Differential algebraic equations (DAEs) define a differential equation on a manifold. A number of ways have been developed to numerically solve some classes of DAEs. Motivated by problems in control theory, numerical simulation, and the use of general purpose modeling environments, recent research has considered the embedding of the DAE solutions of a general DAE into the solutions of an ODE where the added dynamics have special properties. This paper both provides new results on the linear time-varying case and considers the important nonlinear case.     

Similarity analytical solutions for the Schrӧdinger equation with the Riesz fractional derivative in quantum mechanics

The present article deals with the similarity method to tackle the fractional Schrӧdinger equation where the derivative is defined in the Riesz sense. Moreover, the procedure of reducing a fractional partial differential equation (FPDE) into an ordinary differential equation (ODE) has been efficiently displayed by means of suitable scaled transform to the proposed fractional equation. Furthermore, the ODEs are treated effectively via the Fourier transform. The graphical solutions are also depicted for different fractional derivatives α .     

Elrod–Adams cavitation model for a new nonlinear Reynolds equation in piezoviscous hydrodynamic lubrication

In previous studies, different cavitation models have been incorporated into the classical Reynolds equation in piezoviscous regimes. The advantages of the Elrod–Adams cavitation model compared with the Reynolds model have been demonstrated in this classical framework. Recently, a new nonlinear Reynolds equation was rigorously justified [15] for lubricated line contact problems by introducing the piezoviscous Barus law into the departure Navier–Stokes equations before passing to the thin film limit. In addition, the corresponding nonlinear first order ordinary differential equation (ODE) has been proposed.In the present study, we incorporate the Elrod–Adams model for cavitation and we pose the free boundary problem associated with the nonlinear first order ODE, which involves a multivalued Heaviside operator for the relationship between the lubricant pressure and saturation. After analyzing the qualitative properties of the solution, we propose suitable numerical techniques for solving the problem as well as obtaining the lubricant pressure, saturation, and viscosity. Finally, we give some numerical results to illustrate the performance of the proposed numerical methods as well as comparisons with alternative models.