A Class of Quadratic Integral-Differential Equations

A nonlinear integro-ordinary differential equation built up by a linear ordinary differential operator of n th order with constant coefficients and a quadratic integral term is dealt with. The integral term represents the so-called autocorrelation of the unknown function. Applying the Fourier cosine transformation, the integral-differential equation is reduced to a quadratic boundary value problem for the complex Fourier transform of the solution in the upper half-plane. This problem in turn is reduced to a linear boundary value problem which can be solved in closed form. There are infinitely many solutions of the integral-differential equation depending on the prescribed zeros of a function related to the complex Fourier transform.     

A semi-analytical direct optimal control solution for strongly excited and dissipative Hamiltonian systems

A semi-analytical direct optimal control solution for strongly excited and dissipative Hamiltonian systems is proposed based on the extended Hamiltonian principle, the Hamilton-Jacobi-Bellman (HJB) equation and its variational integral equation, and the finite time element approximation. The differential extended Hamiltonian equations for structural vibration systems are replaced by the variational integral equation, which can preserve intrinsic system structure. The optimal control law dependent on the value function is determined by the HJB equation so as to satisfy the overall optimality principle. The partial differential equation for the value function is converted into the integral equation with variational weighting. Then the successive solution of optimal control with system state is designed. The two variational integral equations are applied to sequential time elements and transformed into the algebraic equations by using the finite time element approximation. The direct optimal control on each time element is obtained respectively by solving the algebraic equations, which is unconstrained by the system state observed. The proposed control algorithm is applicable to linear and nonlinear systems with the quadratic performance index, and takes into account the effects of external excitations measured on control. Numerical examples are given to illustrate the optimal control effectiveness.     

依赖凝聚函数求解非线性互补问题的一种微分方程方法

利用凝聚函数一致逼近非光滑极大值函数的性质,将非线性互补问题转化为参数化光滑方程组.然后,对此方程组给出了一种微分方程解法,并且证明了非线性互补问题的解是微分方程系统的渐进稳定平衡点.在适当的假设条件下,证明了所给出的算法具有二次收敛速度.数值结果表明了此算法的有效性.     

Direct search for exact solutions to the nonlinear Schrödinger equation

A five-dimensional symmetry algebra consisting of Lie point symmetries is firstly computed for the nonlinear Schrödinger equation, which, together with a reflection invariance, generates two five-parameter solution groups. Three ansätze of transformations are secondly analyzed and used to construct exact solutions to the nonlinear Schrödinger equation. Various examples of exact solutions with constant, trigonometric function type, exponential function type and rational function amplitude are given upon careful analysis. A bifurcation phenomenon in the nonlinear Schrödinger equation is clearly exhibited during the solution process.     

New soliton solutions for a Kadomtsev-Petviashvili (KP) like equation coupled to a Schrödinger equation

The repeated homogeneous balance is used to construct a new exact traveling wave solution of the Kadomtsev-Petviashvili (KP) like equation coupled to a Schrödinger equation, in which the homogeneous balance method is applied to solve the Riccati equation and the reduced nonlinear ordinary differential equation, respectively. Many new exact traveling wave solutions are successfully obtained, which contain rational and periodic-like solutions. This method is straightforward and concise, and it can be applied to other nonlinear evolution equations.     

Homotopy–perturbation method for pure nonlinear differential equation

In this paper, the homotopy–perturbation method proposed by J.-H. He is adopted for solving pure strong nonlinear second-order differential equation. For the oscillatory differential equation the initial approximate solution is assumed in the form of Jacobi elliptic function and the forementioned method is used for obtaining of the approximate analytic solution. Two types of differential equations are considered: with strong cubic and strong quadratic nonlinearity. The obtained solution is compared with exact numerical one. The difference between these solutions is negligible for a long time period. The method is found to work extremely well in the examples, but the theoretical reasons are not yet clear.     

Solitary wave solutions to the Tzitzeica type equations obtained by a new efficient approach

The properties of Tzitzeica equations in nonlinear optics have received a great attention of many recent studies. In this work, the so-called generalized exponential rational function method (GERFM) has been applied for finding the analytical solution of two nonlinear partial differential equations type of equations, namely Tzitzeica-Dodd-Bullough and Tzitzeica equation. The proposed method provides a wide range of closed-form travelling solutions leading to a very effective and simply-applied method by means of a symbolic computation system. The method not only provides a general form of solutions with some free parameters but also shows potential application to other types of nonlinear partial differential equations.     

Generalized method and its application in the higher-order nonlinear Schrodinger equation in nonlinear optical fibres

A generalized method, which is called the generally projective Riccati equation method, is presented to find more exact solutions of nonlinear differential equations based upon a coupled Riccati equation. As an application of the method, we choose the higher-order nonlinear Schrodinger equation to illustrate the method. As a result more new exact travelling wave solutions are found which include bright soliton solutions, dark soliton solution, new solitary waves, periodic solutions and rational solutions. The new method can be extended to other nonlinear differential equations in mathematical physics.     

New soliton and periodic solutions of (1 + 2)-dimensional nonlinear Schrödinger equation with dual-power law nonlinearity

With the aid of symbolic computation, the new generalized algebraic method is extended to the (1 + 2)-dimensional nonlinear Schrödinger equation (NLSE) with dual-power law nonlinearity for constructing a series of new exact solutions. Because of the dual-power law nonlinearity, the equation cannot be directly dealt with by the method and require some kinds of techniques. By means of two proper transformations, we reduce the NLSE to an ordinary differential equation that is easy to solve and find a rich variety of new exact solutions for the equation, which include soliton solutions, combined soliton solutions, triangular periodic solutions and rational function solutions. Numerical simulations are given for a solitary wave solution to illustrate the time evolution of the solitary creation. Finally, conditional stability of the solution in Lyapunov’s sense is discussed.     

Entire solutions of delay differential equations of Malmquist type

The celebrated Malmquist theorem states that a differential equation, which admits a transcendental meromorphic solution, reduces into a Riccati differential equation. Motivated by the integrability of difference equations, this paper investigates the delay differential equations of form $w(z+1)-w(z-1)+a(z)\frac{w''(z)}{w(z)}=R(z, w(z))(*),$ where $R(z, w(z))$ is an irreducible rational function in $w(z)$ with rational coefficients and $a(z)$ is a rational function. We characterize all reduced forms when the equation $(*)$ admits a transcendental entire solution with hyper-order less than one. When we compare with the results obtained by Halburd and Korhonen[Proc. Amer. Math. Soc. 145, no.6 (2017)], we obtain the reduced forms without the assumptions that the denominator of rational function $R(z,w(z))$ has roots that are nonzero rational functions in $z$. The value distribution and forms of transcendental entire solutions for the reduced delay differential equations are studied. The existence of finite iterated order entire solutions of the Kac-van Moerbeke delay differential equation is also detected.     

An efficient method for quadratic Riccati differential equation

In this paper, a new form of homotopy perturbation method (NHPM) has been adopted for solving the quadratic Riccati differential equation. In this technique, the solution is considered as a Taylor series expansion converges rapidly to the exact solution of the nonlinear equation. Having found the exact solution of the Riccati equation, the capability and the simplicity of the proposed technique is clarified.     

DEVELOPABLE SURFACE PATCHES BOUNDED BY NURBS CURVES

In this paper we construct developable surface patches which are bounded by two rational or NURBS curves, though the resulting patch is not a rational or NURBS surface in general. This is accomplished by reparameterizing one of the boundary curves. The reparameterization function is the solution of an algebraic equation. For the relevant case of cubic or cubic spline curves, this equation is quartic at most, quadratic if the curves are Bézier or splines and lie on parallel planes, and hence it may be solved either by standard analytical or numerical methods.     

Rational Jacobi elliptic solutions for nonlinear differential–difference lattice equations

In this work, we present a direct new method for constructing the rational Jacobi elliptic solutions for nonlinear differential–difference equations, which may be called the rational Jacobi elliptic function method. We use the rational Jacobi elliptic function method to construct many new exact solutions for some nonlinear differential–difference equations in mathematical physics via the lattice equation. The proposed method is more effective and powerful for obtaining the exact solutions for nonlinear differential–difference equations.     

Rational scaled generalized Laguerre function collocation method for solving the Blasius equation

In this paper we propose, a collocation method for solving the Blasius equation. The Blasius equation is a third-order nonlinear ordinary differential equation. This approach is based on a rational scaled generalized Laguerre function collocation method. We also present the comparison of this work with some well-known results and show that the present solution is accurate.     

Solution of a laminar boundary layer flow via a numerical method

In this paper, the numerical solution of the Blasius problem is obtained using the collocation method based on rational Chebyshev functions. The Blasius equation is a nonlinear ordinary differential equation which arises in the boundary layer flow. The method reduces solving the equation to solving a system of nonlinear algebraic equations. The results presented here demonstrate reliability and efficiency of the method.     

Application of the exp‐function method to nonlinear lattice differential equations for multi‐wave and rational solutions

In this paper, we extend the basic Exp‐function method to nonlinear lattice differential equations for constructing multi‐wave and rational solutions for the first time. We consider a differential‐difference analogue of the Korteweg–de Vries equation to elucidate the solution procedure. Our approach is direct and unifying in the sense that the bilinear formalism of the equation studied becomes redundant. Copyright © 2011 John Wiley & Sons, Ltd.     

Coupled Van der Pol oscillator with non-integer order connection

In this paper the dynamics of a system of two oscillators with strong nonlinear connection is considered. The two mass system connected with a spring with pure nonlinear force of any positive rational order (integer or noninteger), on which some additional small nonlinear forces act, is analyzed. The mathematical model of the system contains two coupled second order differential equations of oscillatory type with strong pure nonlinearity and small additional terms. In the paper an analytical solving procedure which introduces the periodical Ateb function is developed. The averaging solution method is adopted to this special function and gives the new type of averaged differential equations.The special attention is given to the steady-state motion of a two-degree-of-freedom Van der Pol oscillator system of positive rational order of nonlinearity. The influence of the order of nonlinearity on the motion of the system is analyzed. using the suggested approximate method three numerical examples are solved. The obtained results are much more accurate than those obtained by the already published methods based on the trigonometric functions.     

Interaction behavior associated with a generalized (2+1)-dimensional Hirota bilinear equation for nonlinear waves

In this paper, we focus on the interaction behavior associated with a generalized (2+1)-dimensional Hirota bilinear equation. With symbolic computation, two types of interaction solutions including lump-kink and lump-soliton ones are derived through mixing two positive quadratic functions with an exponential function, or two positive quadratic functions with a hyperbolic cosine function in the bilinear equation. The completely non-elastic interaction between a lump and a stripe is presented, which shows the lump is drowned or shallowed by the stripe. The interaction between lump and soliton is also given, where the lump moves from one branch to the other branch of the soliton. These phenomena exhibit the dynamics of nonlinear waves and the solutions are useful for the study on interaction behavior of nonlinear waves in shallow water, plasma, nonlinear optics and Bose–Einstein condensates.     

Sequential Linear-Quadratic Method for Differential Games with Air Combat Applications

We present a numerical method for computing a local Nash (saddle-point) solution to a zero-sum differential game for a nonlinear system. Given a solution estimate to the game, we define a subproblem, which is obtained from the original problem by linearizing its system dynamics around the solution estimate and expanding its payoff function to quadratic terms around the same solution estimate. We then apply the standard Riccati equation method to the linear-quadratic subproblem and compute its saddle solution. We then update the current solution estimate by adding the computed saddle solution of the subproblem multiplied by a small positive constant (a step size) to the current solution estimate for the original game. We repeat this process and successively generate better solution estimates. Our applications of this sequential method to air combat simulations demonstrate experimentally that the solution estimates converge to a local Nash (saddle) solution of the original game.     

Polynomial dynamical systems and ordinary differential equations associated with the heat equation

We consider homogeneous polynomial dynamical systems in n-space. To any such system our construction matches a nonlinear ordinary differential equation and an algorithm for constructing a solution of the heat equation. The classical solution given by the Gaussian function corresponds to the case n = 0, while solutions defined by the elliptic theta-function lead to the Chazy-3 equation and correspond to the case n = 2. We explicitly describe the family of ordinary differential equations arising in our approach and its relationship with the wide-known Darboux-Halphen quadratic dynamical systems and their generalizations.