Classification of scalar third order ordinary differential equations linearizable via generalized contact transformations

Whereas Lie had linearized scalar second order ordinary differential equations (ODEs) by point transformations, and later Chern had extended to the third order by using contact transformation, till recently no work had been done for higher order (or systems) of ODEs. Lie had found a unique class defined by the number of infinitesimal symmetry generators but the more general ODEs were not so classified. Recently, classifications of higher order and systems of ODEs were provided. In this paper we relate contact symmetries of scalar ODEs with point symmetries of reduced systems. We define a new type of transformation that builds upon this relation and obtain equivalence classes of scalar third order ODEs linearizable via these transformations. Four equivalence classes of such equations are seen to exist.     

Reduction of fourth order ordinary differential equations to second and third order Lie linearizable forms

Meleshko presented a new method for reducing third order autonomous ordinary differential equations (ODEs) to Lie linearizable second order ODEs. We extended his work by reducing fourth order autonomous ODEs to second and third order linearizable ODEs and then applying the Ibragimov and Meleshko linearization test for the obtained ODEs. The application of the algorithm to several ODEs is also presented.     

Travelling Wave Solutions and Conservation Laws of the (2+1)-dimensional Broer-Kaup-Kupershmidt Equatio

The travelling wave solutions and conservation laws of the (2+1)-dimensional Broer-Kaup-Kupershmidt (BKK) equation are considered in this paper. Under the travelling wave frame, the BKK equation is transformed to a system of ordinary differential equations (ODEs) with two dependent variables. Therefore, it happens that one dependent variable $u$ can be decoupled into a second order ODE that corresponds to a Hamiltonian planar dynamical system involving three arbitrary constants. By using the bifurcation analysis, we obtain the bounded travelling wave solutions $u$, which include the kink, anti-kink and periodic wave solutions. Finally, the conservation laws of the BBK equation are derived by employing the multiplier approach.     

A high‐order finite difference method for 1D nonhomogeneous heat equations

In this article a sixth‐order approximation method (in both temporal and spatial variables) for solving nonhomogeneous heat equations is proposed. We first develop a sixth‐order finite difference approximation scheme for a two‐point boundary value problem, and then heat equation is approximated by a system of ODEs defined on spatial grid points. The ODE system is discretized to a Sylvester matrix equation via boundary value method. The obtained algebraic system is solved by a modified Bartels‐Stewart method. The proposed approach is unconditionally stable. Numerical results are provided to illustrate the accuracy and efficiency of our approximation method along with comparisons with those generated by the standard second‐order Crank‐Nicolson scheme as well as Sun‐Zhang's recent fourth‐order method. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

Generalization of the simplest equation method for nonlinear non-autonomous differential equations

It is known that the simplest equation method is applied for finding exact solutions of autonomous nonlinear differential equations. In this paper we extend this method for finding exact solutions of non-autonomous nonlinear differential equations (DEs). We applied the generalized approach to look for exact special solutions of three Painlevé equations. As ODE of lower order than Painlevé equations the Riccati equation is taken. The obtained exact special solutions are expressed in terms of the special functions defined by linear ODEs of the second order.     

Classification of discrete equations linearizable by point transformation on a square lattice

We provide a complete set of linearizability conditions for nonlinear partial difference equations defined on four points and, using them, we classify all linearizable multilinear partial difference equations defined on four points up to a Möbious transformation.     

Quadratic Störmer‐type methods for the solution of the Boussinesq equation by the method of lines

We study the numerical treatment of Boussinesq PDE equation using the method of lines. For the space discretization, we choose either classical finite differences or Fourier pseudospectral methods. Both cases result in a system of second‐order ordinary differential equations (ODEs) that is quadratic. In order to take advantage of this special feature, we choose to solve the ODE system using a new type of hybrid Numerov method specially constructed for such problems. Other efficient ODE solvers taken from the literature are used to solve the system of ODEs as well. By taking all the combinations of space discretization methods and ODE solvers, we discuss the stability and accuracy features revealed from the numerical tests. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2008     

Classification of discrete systems on a square lattice

We consider the classification up to a M?bius transformation of real linearizable and integrable partial difference equations with dispersion defined on a square lattice by the multiscale reduction around their harmonic solution. We show that the A₁, A₂, and A₃ linearizability and integrability conditions constrain the number of parameters in the equation, but these conditions are insufficient for a complete characterization of the subclass of multilinear equations on a square lattice.     

Spectral Deferred Correction Methods for Ordinary Differential Equations

We introduce a new class of methods for the Cauchy problem for ordinary differential equations (ODEs). We begin by converting the original ODE into the corresponding Picard equation and apply a deferred correction procedure in the integral formulation, driven by either the explicit or the implicit Euler marching scheme. The approach results in algorithms of essentially arbitrary order accuracy for both non-stiff and stiff problems; their performance is illustrated with several numerical examples. For non-stiff problems, the stability behavior of the obtained explicit schemes is very satisfactory and algorithms with orders between 8 and 20 should be competitive with the best existing ones. In our preliminary experiments with stiff problems, a simple adaptive implementation of the method demonstrates performance comparable to that of a state-of-the-art extrapolation code (at least, at moderate to high precision).Deferred correction methods based on the Picard equation appear to be promising candidates for further investigation.     

ODE Methods in Non-Local Equations

Non-local equations cannot be treated using classical ODE theorems. Nevertheless, several new methods have been introduced in the non-local gluing scheme of our previous article; we survey and improve those, and present new applications as well. First, from the explicit symbol of the conformal fractional Laplacian, a variation of constants formula is obtained for fractional Hardy operators. We thus develop, in addition to a suitable extension in the spirit of Caffarelli–Silvestre, an equivalent formulation as an infinite system of second order constant coefficient ODEs. Classical ODE quantities like the Hamiltonian and Wrońskian may then be utilized. As applications, we obtain a Frobenius theorem and establish new Pohožaev identities. We also give a detailed proof for the non-degeneracy of the fast-decay singular solution of the fractional Lane–Emden equation.     

On RF-pairs, Bäcklund transformations and linearization of nonlinear equations

Some classes of nonlinear equations of mathematical physics are described that admit order reduction through the use of a hydrodynamic-type transformation, where the unknown function is taken as a new independent variable and an appropriate partial derivative is taken as the new dependent variable. RF-pairs and associated Bäcklund transformations are constructed for evolution equations of general form. The results obtained are used for order reduction of hydrodynamic equations (Navier-Stokes and boundary layer) and constructing exact solutions to these equations. A generalized Calogero equation and a number of other new linearizable nonlinear differential equations of the second, third and forth orders are considered. Some integro-differential equations are analyzed.     

Approximate methods based on order reduction for the periodic solutions of nonlinear third-order ordinary differential equations

Four approximate methods based on order reduction, the introduction of a book-keeping parameter and power series expansions for the solution and the frequency of oscillation are used to analyze three autonomous, nonlinear, third-order ordinary differential equations which have analytical periodic solutions. The first method introduces the velocity in both sides of the equation if this (linear) term is not present. The second one is based on the first one but employs a new independent variable, whereas, in the third and fourth techniques, a term equal to the velocity times the square of the unknown frequency of oscillation is introduced in both sides of the equation. The third method uses the original independent variable, whereas the fourth one employs a new independent variable which depends linearly on the unknown frequency of oscillation. It is shown that the second method provides accurate solutions only for initial velocities close to unity, whereas the third one is found to yield very accurate results for the first and second equations considered here and only for large initial velocities for the third one. The fourth technique provides as accurate results as or more accurate results than parameter-perturbation techniques which deal with the third-order equations directly and are based on the expansion of certain constants that appear in the differential equations in terms of a book-keeping parameter.     

Galerkin projections for state-dependent delay differential equations with applications to drilling

A Galerkin projection scheme to obtain low dimensional approximations of delay differential equations (DDEs) involving state-dependent delays is developed. The current scheme is an extension of a similar, recently proposed scheme for DDEs with constant delays in the publication by P. Wahi, A. Chatterjee 2005. The resulting ordinary differential equations (ODEs) from the Galerkin scheme are easier to integrate using commercial ODE solvers, and are amenable to stability and bifurcation analysis using standard techniques. First, the application of the formulation is demonstrated through a scalar delay differential equation, and the performance of the formulation is assessed. Next, the scheme is applied to a two degrees-of-freedom model describing the coupled axial and torsional vibrations of oil well drill-strings. In both cases, the Galerkin approximations show an excellent agreements with the direct numerical simulations of the original systems.     

<Emphasis Type="Italic">A</Emphasis>-Orbital Linearization of Affine Systems

The problem of transformation of an affine system into a linear controllable system is considered. For affine systems with a single control, the notion of A-orbital linearizability is introduced, which generalizes the notion (well known for affine systems) of orbital linearizability to the case where the control-dependent changes of independent variable are used. A necessary and sufficient condition for the A-orbital linearizability is proved, and an algorithm for determining linearizable transformations is proposed based on the construction of the derived series of the codistribution associated with the original system.     

Linearization of two-dimensional systems of ODEs without conditions on small denominators

We study the problem of linearizability for two-dimensional systems of ODEs in a neighborhood of the saddle type singular point with rationally incommensurable eigenvalues. It is shown that if the linearizing transformation is convergent in one of the variables, then it is absolutely convergent.     

Numerical solution of nonlinear Jaulent–Miodek and Whitham–Broer–Kaup equations

In this work we investigate the numerical solution of Jaulent–Miodek (JM) and Whitham–Broer–Kaup (WBK) equations. The proposed numerical schemes are based on the fourth-order time-stepping schemes in combination with discrete Fourier transform. We discretize the original partial differential equations (PDEs) with discrete Fourier transform in space and obtain a system of ordinary differential equations (ODEs) in Fourier space which will be solved with fourth order time-stepping methods. After transforming the equations to a system of ODEs, the linear operator in JM equation is diagonal but in WBK equation is not diagonal. However for WBK equation we can also implement the methods such as diagonal case which reduces the CPU time. Comparing numerical solutions with analytical solutions demonstrates that those methods are accurate and readily implemented.     

Momentum-space approach to asymptotic expansion for stochastic filtering

This paper develops an asymptotic expansion technique in momentum space for stochastic filtering. It is shown that Fourier transformation combined with a polynomial-function approximation of the nonlinear terms gives a closed recursive system of ordinary differential equations (ODEs) for the relevant conditional distribution. Thanks to the simplicity of the ODE system, higher-order calculation can be performed easily. Furthermore, solving ODEs sequentially with small sub-periods with updated initial conditions makes it possible to implement a substepping method for asymptotic expansion in a numerically efficient way. This is found to improve the performance significantly where otherwise the approximation fails badly. The method is expected to provide a useful tool for more realistic financial modeling with unobserved parameters and also for problems involving nonlinear measure-valued processes.     

Interfacial long traveling waves in a two‐layer fluid with variable depth

Long wave propagation in a two‐layer fluid with variable depth is studied for specific bottom configurations, which allow waves to propagate over large distances. Such configurations are found within the linear shallow‐water theory and determined by a family of solutions of the second‐order ordinary differential equation (ODE) with three arbitrary constants. These solutions can be used to approximate the true bottom bathymetry. All such solutions represent smooth bottom profiles between two different singular points. The first singular point corresponds to the point where the two‐layer flow transforms into a uniform one. In the vicinity of this point nonlinear shallow‐water theory is used and the wave breaking criterion, which corresponds to the gradient catastrophe is found. The second bifurcation point corresponds to an infinite increase in water depth, which contradicts the shallow‐water assumption. This point is eliminated by matching the “nonreflecting” bottom profile with a flat bottom. The wave transformation at the matching point is described by the second‐order Fredholm equation and its approximated solution is then obtained. The results extend the theory of internal waves in inhomogeneous stratified fluids actively developed by Prof. Roger Grimshaw, to the new solutions types.     

l-solutions and stability analysis of difference equations using the Kummer’s test

We address the p-summability and asymptotic stability properties in nonautonomous linear difference equations. We focus our discussion on two kind of difference equations. The first one is a first order system of linear nonautonomous difference equations, and our discussion involves the use of Kummer’s convergence test. The second one is a linear nonautonomous scalar higher order difference equation. In this case our discussion is based on a recently introduced transformation of a higher order system into a first-step recursion, where the companion matrices are well treatable from our point of view. We give insight on our ideas that are behind our methods, prove new results, and show applications.     

(3+1)-维耦合高阶非线性Schrödinger方程的光学多畸形波

本文研究带有高阶项、时间色散项和非线性系数项的复杂（3+1）-维高阶耦合非线性Schrödinger（3DHCNLSE）方程的精确解. 首先,利用相似变换将非自治的方程转化为自治的耦合Hirota 方程; 其次,采用Darboux 变换方法得到耦合Hirota 方程带有任意常数的有理解; 最后,给出变系数3DHCNLSE方程带有任意常数的1 阶和2 阶多畸形波解. 本文获得的（3+1）-维（3D）多畸形波解可以用来描述深海动力学波和非线性光学纤维中出现的一些物理现象.