Discrete Nonlinear Equation Research Articles

Numerical solution of nonlinear Klein–Gordon equation using the element-free kp-Ritz method

This paper presents a numerical analysis of the one-dimensional Klein–Gordon equation with quadratic and cubic nonlinearity, using the element-free reproducing kernel particle Ritz method (kp-Ritz method). Approximation of the wave displacement is expressed according to a set of meshfree kernel particle functions. Based on the established functional corresponding to the nonlinear Klein–Gordon equation, a system of nonlinear discrete equations can be obtained by applying the Ritz minimization procedure. The Newmark integration scheme combined with an iterative technique is applied to the resulting nonlinear system equations. Numerical examples are tested to validate the proposition that the presented numerical solutions are in good agreement with analytical results available in extant literature.

Collective coordinates theory for discrete soliton ratchets in the sine-Gordon model.

A collective coordinate theory is developed for soliton ratchets in the damped discrete sine-Gordon model driven by a biharmonic force. An ansatz with two collective coordinates, namely the center and the width of the soliton, is assumed as an approximated solution of the discrete nonlinear equation. The dynamical equations of these two collective coordinates, obtained by means of the generalized travelling wave method, explain the mechanism underlying the soliton ratchet and capture qualitatively all the main features of this phenomenon. The numerical simulation of these equations accounts for the existence of a nonzero depinning threshold, the nonsinusoidal behavior of the average velocity as a function of the relative phase between the harmonics of the driver, the nonmonotonic dependence of the average velocity on the damping, and the existence of nontransporting regimes beyond the depinning threshold. In particular, it provides a good description of the intriguing and complex pattern of subspaces corresponding to different dynamical regimes in parameter space.

Numerical investigation of the dynamics for low tension marine cables

This paper presents a numerical investigation into the dynamics of marine cables which are extensively used in offshore industry. In this numerical study, the Euler-Bernoulli beam model is adopted to develop the governing equations of the cable. Bending stiffness is considered to cope with the low tension problem in local area of towing cable, and thus a more accurate solution with the consideration of the axial elongation can be given. The derived strongly-coupled and nonlinear governing equations are solved by a second-order accurate, implicit, and large time step stable central finite difference method. The quadratically convergent Newton-Raphson iteration method is applied to solving the discrete nonlinear algebraic equations. Then a towed array sonar system (TASS) problem is studied. The numerical solutions agree reasonably well with the experimental data and the simulated results of the references. The specified program of the present paper shows great robustness with high efficiency.

Approximate Solution of Nonlinear Discrete Equations of Convolution Type

By the method of potential monotone operators we prove global theorems on existence, uniqueness, and ways to find a solution for different classes of nonlinear discrete equations of convolution type with kernels of special form both in weighted and in weightless real spaces lp. Using the property of potentiality of the operators under consideration, in the case of space l2 and in the case of a weighted space lp(ϱ) with a generic weight ϱ, we prove that a discrete equation of convolution type with an odd power nonlinearity has a unique solution and it (the main result) can be found by gradient method.

Nonlinear discrete fractional mixed type sum-difference equation boundary value problems in Banach spaces

This paper is concerned with the existence of a unique solution to a nonlinear discrete fractional mixed type sum-difference equation boundary value problem in a Banach space. Under certain suitable nonlinear growth conditions imposed on the nonlinear term, the existence and uniqueness result is established by using the Banach contraction mapping principle. Additionally, two representative examples are presented to illustrate the effectiveness of the main result. MSC:26A33, 39A05, 39A10, 39A12.

Clausius inequality and [formula omitted]-theorems for some models of random wealth exchange

We discuss a possibility of deriving an H-theorem for nonlinear discrete time evolution equation that describes random wealth exchanges. In such kinetic models economical agents exchange wealth in pairwise collisions just as particles in a gas exchange their energy. It appears useful to reformulate the problem and represent the dynamics as a combination of two processes. The first is a linear transformation of a two-particle distribution function during the act of exchange while the second one corresponds to new random pairing of agents and plays a role of some kind of feedback control. This representation leads to a Clausius-type inequality which suggests a new interpretation of the exchange process as an irreversible relaxation due to a contact with a reservoir of a special type. Only in some special cases when equilibrium distribution is exactly a gamma distribution, this inequality results in the H-theorem with monotonically growing ‘entropy’ functional which differs from the Boltzmann entropy by an additional term. But for arbitrary exchange rule the evolution has some features of relaxation to a non-equilibrium steady state and it is still unclear if any general H-theorem could exist.

Multiscale expansions in discrete world

In this paper, we show the attainability of KdV equation from some types of nonlinear Schrodinger equation by using multiscale expansions discretely. The power of this manageable method is confirmed by applying it to two selected nonlinear Schrodinger evolution equations. This approach can also be applied to other nonlinear discrete evolution equations. All the computations have been made with Maple computer packet program.

Numerical analysis of generalized regularized long wave equation using the element-free kp-Ritz method

A numerical analysis of the generalized regularized long wave (GRLW) equation by application of the element-free kp-Ritz method is performed. To approximate the displacement field, an element-free kernel particle estimation is employed and a system of nonlinear discrete equations is achieved by using the Ritz minimization procedure to the energy expressions. Validation of the present numerical solutions with the exact results has been conducted by numerical examples and evidenced a good agreement.

Existence and multiplicity of wave trains in 2D lattices

We study the existence and branching patterns of wave trains in a two-dimensional lattice with linear and nonlinear coupling between nearest particles and a nonlinear substrate potential. The wave train equation of the corresponding discrete nonlinear equation is formulated as an advanced-delay differential equation which is reduced by a Lyapunov–Schmidt reduction to a finite-dimensional bifurcation equation with certain symmetries and an inherited Hamiltonian structure. By means of invariant theory and singularity theory, we obtain the small amplitude solutions in the Hamiltonian system near equilibria in non-resonance and p:q resonance, respectively. We show the impact of the direction θ of propagation and obtain the existence and branching patterns of wave trains in a one-dimensional lattice by investigating the existence of traveling waves of the original two-dimensional lattice in the direction θ of propagation satisfying tanθ is rational.

Oscillation Criteria for a Class of Discrete Nonlinear Fractional Equations

In this paper, we study the oscillatory behavior of the fractionaldifference equations of the following form (see in paper) denotes the Riemann-Liouville difference operator and η>0 is a quotient of odd positive integers. We establish some oscillation criteria forthe equation by using Riccati transformation technique and some inequalities. Anexample is shown to illustrate our main results.

Third-Order Continuous-Discrete Filtering for a Nonlinear Dynamical System

Approximate higher-order filters are more attractive and popular in control and signal processing literature in contrast to the exact filter, since the analytical and numerical solutions of the nonlinear exact filter are not possible. The filtering model of this paper involves stochastic differential equation (SDE) formalism in combination with a nonlinear discrete observation equation. The theory of this paper is developed by adopting a unified systematic approach involving celebrated results of stochastic calculus. The Kolmogorov–Fokker–Planck equation in combination with the Kolmogorov backward equation plays the pivotal role to construct the theory of this paper “between the observations.” The conditional characteristic function is exploited to develop “filtering” at the observation instant. Subsequently, the efficacy of the filtering method of this paper is examined on the basis of its comparison with extended Kalman filtering and true state trajectories. This paper will be of interest to applied mathematicians and research communities in systems and control looking for stochastic filtering methods in theoretical studies as well as their application to real physical systems.

Numerical Solution of Fractional Partial Differential Equations by Discrete Adomian Decomposition Method

AbstractIn this paper we find the solution of linear as well as nonlinear fractional partial differential equations using discrete Adomian decomposition method. Here we develop the discrete Adomian decomposition method to find the solution of fractional discrete diffusion equation, nonlinear fractional discrete Schrodinger equation, fractional discrete Ablowitz-Ladik equation and nonlinear fractional discrete Burger’s equation. The obtained solution is verified by comparison with exact solution whenα= 1.

On the dependence on parameters for second order discrete boundary value problems with the p(k)-Laplacian

In this paper we study the existence and the nonexistence of solutions for the boundary value problems of a class of nonlinear second-order discrete equations depending on a parameter. Variational (the mountain pass technique) and non-variational methods are applied.

Modeling asphaltene precipitation and flow behavior in the processes of CO2 flood for enhanced oil recovery

This paper focuses on the idea for enhancing oil recovery by CO2 injection into oil formation as well as reducing CO2 emission into the atmosphere. The mechanism of asphaltene flocculation during CO2 flooding in oil formations is analyzed and the regressive correlation between CO2 concentration and the amount of flocculated asphaltene for an oil sample is set up for coupling with the flow governing equations for the flocculated asphaltene transport in porous media. A three-dimensional multiphase mathematical model describing CO2 transport in oil reservoirs and asphaltene precipitation is presented for predicting CO2 flooding performances for enhanced oil recovery. The finite difference method and preconditioned conjugate gradient algorithm are used to solve the discrete nonlinear equation systems. A numerical simulation software is developed to study CO2 flooding performances and the effects of asphaltene precipitation on production behaviors. The numerical result indicates that water-cut decreases from initial 92.5% down to 40.3% after continuous CO2 injection for 10 years. It is also shown that 0.130 million tons of crude oil is displaced by CO2 injection in the 1 km2 of the reservoir within 10 years. Asphaltene precipitation leads to the decrease in permeability, and the decline in production rates.

Adaptive mesh refinement for simulation of thin film flows

We present a robust and accurate numer- ical method for simulating gravity-driven, thin-film flow problems. The convection term in the governing equation is treated by a semi-implicit, essentially non- oscillatory scheme. The resulting nonlinear discrete equation is solved using a nonlinear full approxima- tion storage multigrid algorithm with adaptive mesh refinement techniques. A set of representative numeri- cal experiments are presented. We show that the use of adaptive mesh refinement reduces computational time and memory compared to the equivalent uniform mesh results. Our simulation results are consistent with pre- vious experimental observations.

Trust-region based solver for nonlinear transport in heterogeneous porous media

We describe a new nonlinear solver for immiscible two-phase transport in porous media, where viscous, buoyancy, and capillary forces are significant. The flux (fractional flow) function, F, is a nonlinear function of saturation and typically has inflection points and can be non-monotonic. The non-convexity and non-monotonicity of F are major sources of difficulty for nonlinear solvers of coupled multiphase flow and transport in natural porous media. We describe a modified Newton algorithm that employs trust regions of the flux function to guide the Newton iterations. The flux function is divided into saturation trust regions delineated by the inflection, unit-flux, and end points. The updates are performed such that two successive iterations cannot cross any trust-region boundary. If a crossing is detected, the saturation value is chopped back to the appropriate trust-region boundary. The proposed trust-region Newton solver, which is demonstrated across the parameter space of viscous, buoyancy and capillary effects, is a significant extension of the inflection-point strategy of Jenny et al. (JCP, 2009) [5] for viscous dominated flows.We analyze the discrete nonlinear transport equation obtained using finite-volume discretization with phase-based upstream weighting. Then, we prove convergence of the trust-region Newton method irrespective of the timestep size for single-cell problems. Numerical results across the full range of the parameter space of viscous, gravity and capillary forces indicate that our trust-region scheme is unconditionally convergent for 1D transport. That is, for a given choice of timestep size, the unique discrete solution is found independently of the initial guess. For problems dominated by buoyancy and capillarity, the trust-region Newton solver overcomes the often severe limits on timestep size associated with existing methods. To validate the effectiveness of the new nonlinear solver for large reservoir models with strong heterogeneity, we compare it with state-of-the-art nonlinear solvers for two-phase flow and transport using the SPE 10 model. Compared with existing nonlinear solvers, our trust-region solver results in superior convergence performance and achieves reduction in the total Newton iterations by more than an order of magnitude together with a corresponding reduction in the overall computational cost.

Homoclinic solutions of a class of periodic difference equations with asymptotically linear nonlinearities

By using critical point theory and periodic approximations, new sufficient conditions are obtained on the existence and nonexistence of homoclinic solutions for a class of discrete nonlinear periodic equations with asymptotically linear nonlinearities. These results partially answer an open problem proposed by Pankov (2006) [2] under rather weaker conditions and greatly improve the related results before.

Dynamics of a nonlocal discrete Gross-Pitaevskii equation with defects

We study the dynamics of dipolar gas in deep lattices described by a nonlocal nonlinear discrete Gross-Pitaevskii equation. The stabilities and the propagation properties of traveling plane waves in the system with defects are discussed in detail. For a clean lattice, both energetic and dynamical stabilities of the traveling plane waves are studied. It is shown that the system with attractive local interaction can preserve the stabilities, i.e., the dipoles can stabilize the gas because of repulsive nonlocal dipole-dipole interactions. For a lattice with defects, within a two-mode approximation, the propagation properties of traveling plane waves in the system map onto a nonrigid pendulum Hamiltonian with quasimomentum-dependent nonlinearity (induced by the nonlocal interactions). Competition between defects, quasimomentum of the gas, and nonlocal interactions determines the propagation properties of the traveling plane waves. Critical conditions for crossing from a superfluid regime with propagation preserved to a normal regime with defect-induced damping are obtained analytically and confirmed numerically. In particular, the critical conditions for supporting the superfluidity strongly depend on the defect type and the quasimomentum of the plane waves. The nonlocal interaction can significantly enhance the superfluidity of the system.

Sylvester equations for Laguerre–Hahn orthogonal polynomials on the real line

Matrix Sylvester differential equations are introduced in the study of Laguerre–Hahn orthogonal polynomials. Matrix Sylvester differential systems are shown to yield representations for the Laguerre–Hahn orthogonal polynomials. Lax pairs are given, formed from the differential system and the recurrence relation, that yield discrete non-linear equations for the three term recurrence relation coefficients of the Laguerre–Hahn orthogonal polynomials.

STABILITY PROPERTIES IN NONLINEAR DISCRETE VOLTERRA EQUATIONS WITH UNBOUNDED DELAY

We study some stability properties in discrete Volterra equations by employing to change Yoshizawa's results in [13] for the nonlinear equations into results for the nonlinear discrete Volterra equations with unbounded delay.