On a method for solving a two-dimensional nonlinear integral equation of the second kind

In this article, the existence of at least one solution of a nonlinear integral equation of the second kind is proved. The degenerate method is used to obtain a nonlinear algebraic system, where the existence of at least one solution of this system is discussed. Finally, computational results with error estimates are obtained using Maple software.     

Multiple Solutions for a Class of Semilinear Elliptic Equations

We show that for a class of semilinear elliptic equations there are at least three nontrivial solutions. Existence of infinitely many solutions is also shown when the nonlinear term is odd. In our results, the nonlinear term can grow super-critically at infinity.

     

Long Nonlinear Water Waves in a Channel Near a Cut-off Frequency

A theoretical study is made of the free-surface flow induced by a wavemaker, performing torsional oscillations about a vertical axis, in a shallow rectangular channel near a cut-off frequency. Exactly at cut-off, linearized water-wave theory predicts a temporally unbounded response due to a resonance phenomenon. It is shown, through a perturbation analysis using characteristic variables, that the nonlinear response is governed by a forced Kadomtsev—Petviashvili (KP) equation with periodic boundary conditions across the channel. This nonlinear initial-boundary-value problem is investigated analytically and numerically. When surface-tension effects are negligible, the nonlinear response reaches a steady state and exhibits jump phenomena. On the other hand, in the high-surface-tension regime, no steady state is obtained. These results are discussed in connection with similar forced wave phenomena studied previously in a deepwater channel and related laboratory experiments.     

Solving Convex MINLP Optimization Problems Using a Sequential Cutting Plane Algorithm

In this article we look at a new algorithm for solving convex mixed integer nonlinear programming problems. The algorithm uses an integrated approach, where a branch and bound strategy is mixed with solving nonlinear programming problems at each node of the tree. The nonlinear programming problems, at each node, are not solved to optimality, rather one iteration step is taken at each node and then branching is applied. A Sequential Cutting Plane (SCP) algorithm is used for solving the nonlinear programming problems by solving a sequence of linear programming problems. The proposed algorithm generates explicit lower bounds for the nodes in the branch and bound tree, which is a significant improvement over previous algorithms based on QP techniques. Initial numerical results indicate that the described algorithm is a competitive alternative to other existing algorithms for these types of problems.     

Numerical model of a pine in a wind

Steady-state nonlinear differential equations govering the stem curve of a wind-loaded pine are derived and solved numerically. Comparison is made between the results computed and the data from photographs of a pine stem during strong wind. The pine breaking is solved at the end.     

Perturbation and stability analyses of flow in a mushy layer with permeable interface

We consider the problem of weakly nonlinear convective flow stability in a horizontal mushy layer with permeable interface. No assumption is made on the thickness of the mushy layer, and a number of simplifying assumptions made in previous related nonlinear analyses are lifted here in order to determine the results based on a more realistic model. Using perturbation and stability analyses and some numerical calculations, we determine the steady stable solutions to the weakly nonlinear problem under certain range of the parameter values where such analyses are valid. We find, in particular, that depending on the range of values of the parameters, the convective flow in the form of down-hexagons, with down-flow at the cells’ centers and up-flow at the cells’ boundaries, up-hexagons, with up-flow at the cells’ centers and down-flow at the cells’ boundaries, rectangles, squares and rolls can be possible.     

Blow-up of solutions of wave equation with a nonlinear boundary condition and interior focusing source of variable order of growth

In this paper, being investigated an initial-boundary value problem for a one-dimensional wave equation with a nonlinear source of variable order and nonlinear dissipation at the boundary. The existence of a local solution of the problem under consideration is proved. Then the question of the absence of global solutions is investigated. Depending on the relationship between the order of growth of the nonlinear source and the nonlinear boundary dissipation, different results are obtained on the blow-up of weak solutions in a finite time interval.     

Existence of solutions to a discrete fourth order periodic boundary value problem

This paper is concerned with periodic boundary value problems for a fourth order nonlinear difference equation. Via variational methods and critical point theory, sufficient conditions are given for the existence of at least one solution, two solutions, and nonexistence of solutions. Our conditions do not involve the primitive function of the nonlinear term. Examples are provided to illustrate the applicability of the results.     

A method of solving a nonlinear boundary value problem with a parameter for a loaded differential equation

A nonlinear loaded differential equation with a parameter on a finite interval is studied. The interval is partitioned by the load points, at which the values of the solution to the equation are set as additional parameters. A nonlinear boundary value problem for the considered equation is reduced to a nonlinear multipoint boundary value problem for the system of nonlinear ordinary differential equations with parameters. For fixed parameters, we obtain the Cauchy problems for ordinary differential equations on the subintervals. Substituting the values of the solutions to these problems into the boundary condition and continuity conditions at the partition points, we compose a system of nonlinear algebraic equations in parameters. A method of solving the boundary value problem with a parameter is proposed. The method is based on finding the solution to the system of nonlinear algebraic equations composed.     

Analytical construction of peaked solutions for the nonlinear evolution of an electromagnetic pulse propagating through a plasma

Abstract

We obtain analytical solutions, by way of the homotopy analysis method, to a nonlinear wave equation describing the nonlinear evolution of a vector potential of an electromagnetic pulse propagating in an arbitrary pair plasma with temperature asymmetry. As the method is analytical, we are able to construct peaked structures which propagate through the pair plasma, analogous to peakon solutions. These solutions are obtained through a novel matching of inner and outer homotopy solutions. In order to ensure that our analytical results are valid over the whole real line, we also discuss the convergence of the analytical results to the true solution, through minimization of the residual errors resulting from an approximate analytical solution. These results demonstrate the existence of peaked pulses propagating through a pair plasma. The algebraic decay rate of the pulses are determined analytically, as well. The method discussed here can be applied to approximate solutions to similar nonlinear partial differential equations of nonlinear Schr¨odinger type.     

Solvability for a third-order three-point BVP on time scales

We are concerned with the existence of a nontrivial solution to a nonlinear third-order three-point boundary-value problem on general time scales. Using the corresponding Green function, without a non-negative or monotone-type assumption on the nonlinearity, we obtain sufficient conditions for the existence of at least one nontrivial solution, using the Leray–Schauder nonlinear alternative theorem. This paper extends recent results in differential equations to difference equations, quantum equations, and general dynamic equations on time scales.     

Weakly Nonlinear Analysis of a Localized Disturbance in Poiseuille Flow

In this article, we investigate, via a perturbation analysis, some important nonlinear features related to the process of transition to turbulence in a wall-bounded flow subject to a spatially localized disturbance that is harmonic in time. We show that the perturbation expansion, truncated at second order, is able to capture the generation of streamwise vorticity as a weakly nonlinear effect. The results of the perturbation approach are discussed in comparison with direct numerical simulation data for a sample case by extracting the contribution of the different orders. The main aim is to provide a tool to select the most effective nonlinear interactions to enlighten the essential features of the transitional process.     

时域自适应精细算法求解二维非线性湿热耦合问题

采用时域自适应精细算法求解二维非线性湿热耦合问题．通过在离散时间段上将变量展开,更精确地描述变量的变化过程,将非线性时空耦合的初边值问题转化为一系列递推的线性边值问题,并利用有限元法进行求解．计算中不需任何附加假设和迭代过程,并提出了自适应计算的收敛判则,以保持时间步长选择不同时的计算精度．数值比较中,分别考虑了材料属性随温度变化、随湿度变化、以及随温度和湿度变化的情形,结果令人满意,体现了这种算法处理复杂非线性问题的能力．     

Second-order kinetics for EC' reactions at a spherical microelectrode

A second-order (nonlinear) model is derived for steady-statekinetics of an EC' (catalytic electrochemical) reaction at aspherical microelectrode in the case where the electron transferprocess is followed by a homogeneous chemical reaction regeneratingthe electroactive species. An asymptotic analysis of the modelis performed, and the asymptotic results are compared with thosefrom a numerical solution of the full nonlinear model. It isshown that in the fast reaction limit, where the current atthe electrode takes its maximum possible value, the concentrationsof the reactants are controlled by diffusion both close to andfar from the electrode, with significant chemical activity occurringonly in a narrow zone standing off the electrode. Also, it isshown that an equation obtained from a different asymptoticlimit may be used to predict the limiting current at the microelectrodein all circumstances. The reasons for the surprising measureof agreement at the surface of the electrode are discussed,the predictions from the model of the limiting current are compared(favourably) with experimental results, and the model is comparedwith the standard pseudo-first-order model, which, althoughalso based on a linearization of the governing equations, hasa restricted range of validity.     

On the numerical solution of the flow between a rotating and a stationary disk

The flow between two co-axial, infinite disks, one rotating with constant angular velocity and one stationary is treated in this paper. The problem is reduced to that of finding the solution of a two-point boundary value for a sixth order nonlinear ordinary differential equation and three boundary conditions at each of a finite interval. The numerical solutions are obtained by using a fourth order Runge-Kutta integration scheme in modification due to Gill and in conjunction with a modified shooting method to correct the initial guesses at one boundary. The numerical calculations for different Reynolds numbers are carried out. The results obtained by this method are compared with available results. The comparison shows excellent agreement.     

On the existence of infinitely many nonperturbative solutions in a transmission eigenvalue problem for nonlinear Helmholtz equation with polynomial nonlinearity

The paper focuses on a transmission eigenvalue problem for nonlinear Helmholtz equation with polynomial nonlinearity which describes the propagation of transverse electric waves along a dielectric layer filled with nonlinear medium. It is proved that even if the nonlinearity coefficients are small, the nonlinear problem has infinitely many nonperturbative solutions, whereas the corresponding linear problem always has a finite number of solutions. This results in the theoretical existence of a novel type of nonlinear guided waves that exist only in nonlinear guided systems. Asymptotic distribution of the eigenvalues is found and a comparison theorem is proved; periodicity of the eigenfunctions is proved, the exact formula for the period is found, and the zeros of the eigenfunctions are determined. The results found essentially extend the theory evolved earlier (particular cases for Kerr, cubic-quintic, septic nonlinearities, etc. are easily extracted from the general results found here). Numerical results are also presented.     

An SQP approach with line search for a system of nonlinear equations

In this paper, a line search sequential quadratic programming (SQP) approach to a system of nonlinear equations (SNE) is taken. In this method, the system of nonlinear equations is transformed into a constrained nonlinear programming problem at each step, which is then solved using SQP algorithms with a line search strategy. Furthermore, at each step, some equations, which are satisfied at the current point, are treated as constraints and the others act as objective functions. In essence, constrained optimization strategies are utilized to cope with the system of nonlinear equations.     

Nonlinear Waves in a Shear Flow with a Vorticity Discontinuity

We consider nonlinear finite-amplitude progressive shear-flow waves on a basic velocity profile consisting of two coflowing layers of inviscid equal-density fluid, each of uniform but different vorticity. The problem is formulated as a nonlinear integral equation describing the shape of the vorticity discontinuity in a frame of reference in which the flow is steady. Numerical solutions to this equation are presented for a range of values of the vorticity ratio Ω. For 1 > © ≥ ? 1 the theoretical maximum wave amplitude occurs when the wave crest forms a 90° corner which just touches the appropriate critical-layer stagnation point. The linearized stability of the progressive wave states to arbitrary subharmonic isovortical disturbances is studied numerically. The results indicate stability at moderate values of the wave amplitude.     

Nonlinear resonances of a semi-infinite cable on a nonlinear elastic foundation

The Multiple Time Scale (MTS) method is applied to the study of nonlinear resonances of a semi-infinite cable resting on a nonlinear elastic foundation, subject to a constant uniformly distributed load and to a linear viscous damping force. The zero order solution provides the static displacement, which is governed by a nonlinear equation which has been solved in closed form. The first order solution provides the linear resonances, which are seen to be functions of the nonlinearity parameter and of the static displacement at the finite boundary only. Although the first-order governing equation is linear, it has non constant coefficients and cannot be solved in closed form, so that a numerical solution is considered; the eigenfrequencies obtained in this way are also compared with the approximate eigenvalues obtained by the WKB method. At the second order of the MTS expansion, we see that the solution is independent of the intermediate time scale; some additional terms are present, including a time-independent shift of the average position of the oscillations. Finally, the nonlinear frequency–amplitude response curves, which are investigated in detail and which represent the main result of this work, are obtained from the solvability condition at the third order.     

双函数法及一类非线性发展方程的精确行波解

给出一种求解非线性发展方程精确行波解的新方法：双函数法。使用此方法，获得了一类非线性发展方程的许多精确行波解，其中包括孤波解和周期解，推广了文献用其它方法取得的结果，同时还获得了许多新的弧波解和周期解，借助于Mathemat-ica，此方法能部分地在计算机上实现。