Symbolic computation of conservation laws for nonlinear partial differential equations in multiple space dimensions

A method for symbolically computing conservation laws of nonlinear partial differential equations (PDEs) in multiple space dimensions is presented in the language of variational calculus and linear algebra. The steps of the method are illustrated using the Zakharov–Kuznetsov and Kadomtsev–Petviashvili equations as examples.The method is algorithmic and has been implemented in Mathematica. The software package, ConservationLawsMD.m, can be used to symbolically compute and test conservation laws for polynomial PDEs that can be written as nonlinear evolution equations.The code ConservationLawsMD.m has been applied to multi-dimensional versions of the Sawada–Kotera, Camassa–Holm, Gardner, and Khokhlov–Zabolotskaya equations.     

Oscillation of solutions for second-order nonlinear difference equations with nonlinear neutral term

Some Riccati type difference inequalities are given for the second-order nonlinear difference equations with nonlinear neutral term.

and using these inequalities, we obtain some oscillation criteria for the above equation.     

An automated tanh-function method for finding solitary wave solutions to non-linear evolution equations

The tanh-function method for finding explicit travelling solitary wave solutions to non-linear evolution equations is described. The method is usually extremely tedious to use by hand. We present a Mathematica package ATFM that deals with the tedious algebra and outputs directly the required solutions. The use of the package is illustrated by applying it to a variety of equations; not only are previously known solutions recovered but in some cases more general forms of solution are obtained.     

Oscillation criteria of certain nonlinear partial difference equations

This paper is concerned with the nonlinear partial difference equation with continuous variables

,where a, σ_i, τ_i are positive numbers, h_i(x, y, u) ε C(R⁺ × R⁺ × R, R), uh_i(x, y, u) > 0 for u ≠ 0, h_i is nondecreasing in u, i = 1, …, m. Some oscillation criteria of this equation are obtained.     

Instability in supercritical nonlinear wave equations: Theoretical results and symplectic integration

Nonlinear wave evolutions involve a dynamical balance between linear dispersive spreading of the waves and nonlinear self-interaction of the waves. In sub-critical settings, the dispersive spreading is stronger and therefore solutions are expected to exist globally in time. We show that in the supercritical case, the nonlinear self-interaction of the waves is much stronger. This leads to some sort of instability of the waves. The proofs are based on the construction of high frequency approximate solutions. Preliminary numerical simulations that support these theoretical results are also reported.     

Discrete-time nonlinear observer design using functional equations

The present research work proposes a new approach to the discrete-time nonlinear observer design problem. Based on the early ideas that influenced the development of the linear Luenberger observer, the proposed approach develops a nonlinear analogue. The formulation of the discrete-time nonlinear observer design problem is realized via a system of first-order linear nonhomogeneous functional equations, and a rather general set of necessary and sufficient conditions for solvability is derived using results from functional equations theory. The solution to the above system of functional equations can be proven to be locally analytic and this enables the development of a series solution method, that is easily programmable with the aid of a symbolic software package.     

Exponential stability of nonlinear differential delay equations

The aim of this paper is to investigate the exponential stability of a nonlinear differential delay equation ) Introduce the corresponding differential equation (without delay) ) and assume it is exponentially stable. It will be shown in this paper that the differential delay equation will remain exponentially stable provided the time lag τ is small enough.     

Numerical solution of nonlinear differential equations using computer algebra

This paper describes a numerical method for finding periodic solutions to nonlinear ordinary differential equations. The solution is approximated by a trigonometric series. The series is substituted into the differential equation using the FORMAC computer algebra system for the resulting lengthy algebraic manipulations. This lead to a set of nonlinear algebraic equations for the series coefficients. Modern search methods are used to solve for the coefficients. The method is illustrated by application to Duffing’ equation.     

Incomplete Jacobian Newton method for nonlinear equations

This paper presents a new modified Newton method for nonlinear equations. This method uses a part of elements of the Jacobian matrix to obtain the next iteration point and is refereed to as the incomplete Jacobian Newton (IJN) method. The IJN method may be fit for solving large scale nonlinear equations with dense Jacobian. The conditions of linear, superlinear and quadratic convergence of the IJN method are given and the local convergence results are analyzed and proved. Some special IJN algorithms are designed and numerical experiments are given. The results show that the IJN method is promising.     

Numerical methods for integrating chemical kinetic rate equations

In this paper we present a numerical method which is suitable for the integration of chemical rate equations. These equations are normally extremely stiff due to large differences in the kinetic rate coefficients. The method takes advantage of the fact that the Jacobian matrix is readily obtainable for this type of problem. Stability analysis will also be discussed in a general framework.     

A Shamanskii-like self-adaptive Levenberg–Marquardt method for nonlinear equations

In this paper, we are concerned with the Shamanskii-like self-adaptive Levenberg–Marquardt methods for nonlinear equations. We consider two choices of Levenberg–Marquardt parameter. One of them is the standard self-adaptive Levenberg–Marquardt parameter, the other is nonmonotone self-adaptive Levenberg–Marquardt parameter by using the nonmonotone technique of Grippo, Lampariello and Lucidi. Under the error bound condition which is weaker than nonsingularity, we show that the Shamanskii-like self-adaptive Levenberg–Marquardt methods converge with $Q$-order $(m+1)$. Numerical experiments show the new algorithms are efficient and promising.     

An extended integrating dual-slope analogue-to-digital converter suitable for continuous bipolar operation

The most accurate analogue-to-digital converter design presently known integrates the input voltage and employs the dual-slope principle for digitisation. However, without additional external components it is only suitable for unipolar operation, and continuous integrations are not possible, because some time is needed for the conversion process after every integration. An extended design, overcoming these two shortcomings, is given in the present paper. Furthermore, its application in multichannel measuring systems is discussed, using star-shaped and bus connections to the controlling unit, with only a single transmission line each.     

Solution of the incompressible Navier-Stokes equations by the nonlinear Galerkin method

Our aim in this article is to study a new method for the approximation of the Navier-Stokes equations, and to present and discuss numerical results supporting the method. This method, called the nonlinear Galerkin method, uses nonlinear manifolds which are close to the attractor, while in the usual Galerkin method, we look for solutions in a linear space, i.e., whose components are independent. The equation of the manifold corresponds to an interaction law between small and large eddies and it is derived by asymptotic expansion from the exact equation. We consider here the two- and three-dimensional space periodic cases in the context of a pseudo-spectral discretization of the equation. We notice however that the method applies as well to more general flows, in particular nonhomogeneous flows.     

Impulsive boundary value problem for nonlinear differential equations of fractional order

In this paper, we prove the existence and uniqueness of solutions for the boundary value problem of nonlinear impulsive differential equations of fractional order q∈(1,2]. Our results are based on Altman’s fixed point theorem and Leray-Schauder’s fixed point theorem.     

Solving nonlinear systems of functional equations with fuzzy adaptive simulated annealing

This paper proposes a method for finding solutions of arbitrarily nonlinear systems of functional equations through stochastic global optimization. The original problem (equation solving) is transformed into a global optimization one by synthesizing objective functions whose global minima, if they exist, are also solutions to the original system. The global minimization task is carried out by the stochastic method known as fuzzy adaptive simulated annealing, triggered from different starting points, aiming at finding as many solutions as possible. To demonstrate the efficiency of the proposed method, solutions for several examples of nonlinear systems are presented and compared with results obtained by other approaches. We consider systems composed of n   equations on Euclidean spaces ?n${?}^n$ (n variables: x₁, x₂, x₃, ? , x_n).     

Some existence results for impulsive nonlinear fractional differential equations with mixed boundary conditions

This paper investigates the existence and uniqueness of solutions for an impulsive mixed boundary value problem of nonlinear differential equations of fractional order α∈(1,2]. Our results are based on some standard fixed point theorems. Some examples are presented to illustrate the main results.     

An asynchronous parallel mixed algorithm for linear and nonlinear equations

In this paper we propose an asynchronous parallel mixed algorithm for solving linear and nonlinear equations. This algorithm can be used not only on serial and parallel computers, but also on MIMD multiprocessor systems. The convergence of the algorithm has been proved under certain conditions. This paper gives some special cases of the algorithm which are known to us as efficient iterative methods. Numerical experiments are given to illustrate the method.     

An efficient adaptive trust-region method for systems of nonlinear equations

We present an adaptive trust-region algorithm to solve systems of nonlinear equations. Using the nonmonotone technique of Grippo, Lampariello and Lucidi, we introduce a new adaptive radius to decrease the total number of iterations and function evaluations. In contrast with the pervious methods, the new adaptive radius ensures that the size of radius is not too large or too small. We show that the sequence generated by the proposed adaptive radius is decreasing, so it prevents the production of too large radius as possible. Furthermore, it is shown that this sequence is reduced slowly, so it prevents the production of the intensely small radius. The global and quadratic convergence of the proposed approach are proved. Preliminary numerical results of our algorithm are also reported which indicate the promising behaviour of the new procedure to solve systems of nonlinear equations.