An asymptotic solution to non-linear transmission lines

This paper explores an asymptotic approach to the solution of a non-linear transmission line model. The model is based on a set of non-linear partial differential equations without analytical solution. The perturbations method is used to reduce the system of non-linear equations to a single non-linear partial differential equation, the modified Korteweg–de Vries equation (KdV). By using the Laplace transform, the solution is represented in integral form in terms of Green's functions. The solution for the non-linear case is obtained by means of asymptotic methods. Thus, an approximate explicit analytical solution to the problem is obtained where the errors can be controlled. This allows us to analyze the non-linear behavior of the solution. This kind of information is difficult to obtain by means of numerical methods due to the fact that for large periods of time greater computational resources are required and also accumulated errors increase. For this reason, asymptotic methods have a great importance like a natural complement to numerical methods. Computer simulations support the developments presented.     

Age-dependent Population Growth

This paper considers a non-linear deterministic model, in whichthe death rate rises as the population grows. The model is inthe form of a semi-linear first-order partial differential equation,with respect to time and age. It is an age-dependent versionof a simple equation (the logistic equation) often used to describepopulations whose growth is controlled by limited resources.The problem reduces to the solution of a non-linear integralequation; it has a constant solution, which is proved to beglobally asymptotically stable. This implies that there areno steady oscillations, and that in the long run the populationsize and age-structure become fixed, independent of the initialconditions. Further details of the solution are discussed, includingnumerical results.     

The onset of stochastic pulsations at the early non-linear stage of the development of perturbations in the jet of an incompressible fluid propagating along a wall

The Korteweg-De Vries equation, which describes the non-linear propagation of perturbations in a jet of incompressible fluid emanating from a slit in a planar screen and propagating along a wall is considered. When account is taken of the natural vibrations of the wall, the equation becomes inhomogeneous. If an external action is specified in the form of a running wave, the particular solution of the inhomogeneous equation may be sought in an analogous form. As a result, the simplest problem in the theory of dynamical systems in the Hamiltonian formulation arises. As usual, the existence of a homoclinic structure in the neighbourhood of the separatrices is deduced from an analysis of a Poincaré transformation. Among the trajectories belonging to the homoclinic structure in the secant plane, there are some with properties which are formulated in terms of determinate chaos. A fundamentally important conclusion concerning the dual role of solitons at the non-linear stage of the wave motion of the fluid follows: on the one hand, they serve as the nuclei of large-scale coherent structures and, on the other hand, they are responsible for the onset of stochastic pulsations.     

MHD stagnation-point flow of an upper-convected Maxwell fluid over a stretching surface

The present analysis comprises the steady two-dimensional magnetohydrodynamic flow of an upper-convected Maxwell fluid near a stagnation-point over a stretching surface. The governing non-linear partial differential equation for the flow are reduced to an ordinary differential equation by using similarity transformations. The analytic solution of nonlinear system is constructed in the series form using Homotopy analysis method. Convergence of the obtained series is discussed explicitly. The effects of the sundry parameters on the velocity profile is shown through graphs. The values of skin-friction coefficient for different parameters is tabulated.     

Variational approach to a non-local identification problem related to non-linear ion transport

A variational approach to a non-linear non-local identification problem related to the non-linear transport equation is studied. Introducing a similarity transformation, the problem is formulated as an identification problem for a non-linear differential equation of second order with an additional non-local condition. For the solution of the forward problem stability in H¹-norm with respect to the identification parameter is obtained. Using this result the existence of a solution to the identification problem is proved. Some results of computational experiments are given. © 1998 B. G. Teubner Stuttgart—John Wiley & Sons, Ltd.     

A Note on the Numerical Solution of Quasi-linear Parabolic Partial Differential Equations with Error Estimates

A method is considered for the numerical solution of quasi-linearpartial differential equations. The partial differential equationis reduced to a set of ordinary differential equations usinga Chebyshev series expansion. The exact solution of this setof ordinary differential equations is shown to be the solutionof a perturbed form of the original equation. This enables errorestimates to be found for linear and mildly non-linear problems.     

The normal form of perturbations of non-linear oscillatory systems

The normal form of perturbations of a non-linear oscillatory system is defined. The system itself, called the generating system, is arbitrary in form. An algorithm is developed that enables one, without touching the generating system, to reduce a perturbation to normal form. The Campbell-Hausdorff series, the properties of the ring of asymptotics forms and the explicit solution of the homological equation are used to derive a one-dimensional recurrence formula of arbitrary approximation.     

Effect of viscosity on the conditions for the formation of centrifugal solitons in the translational-rotational flow of a liquid

The viscosity of the medium is taken into consideration in deriving an evolution equation describing the propagation of non-linear centrifugal waves along the free surface of a translational-rotational liquid flow. The result is the Burgers-Korteweg-de Vries (BKdV) equation, for which a steady solution is described in the form of a shock wave with soliton oscillators near the front. Estimates are presented for the effect of viscosity on the wave-front structure and the conditions of formation previously predicted by the author /1/ for centrifugal solitons, which play an important role in various atmosphere processes^**.     

Second-order hyperbolic s.p.d.e.'s driven by homogeneous Gaussian noise on a hyperplane

We study a class of hyperbolic stochastic partial differential equations in Euclidean space, that includes the wave equation and the telegraph equation, driven by Gaussian noise concentrated on a hyperplane. The noise is assumed to be white in time but spatially homogeneous within the hyperplane. Two natural notions of solutions are function-valued solutions and random field solutions. For the linear form of the equations, we identify the necessary and sufficient condition on the spectral measure of the spatial covariance for existence of each type of solution, and it turns out that the conditions differ. In spatial dimensions 2 and 3, under the condition for existence of a random field solution to the linear form of the equation, we prove existence and uniqueness of a random field solution to non-linear forms of the equation.

     

Free boundary problems with nonlinear source terms

Summary The method of lines is used to semi-discretize the non-linear Poisson equation over a domain with a free boundary. The resulting multipoint free boundary problem is solved with a line Gauss-Seidel method which is shown to converge monotonically. The method of lines solution is then shown to converge to the continuous solution of the variational inequality form of the obstacle problem. Some numerical results for the diffusion-reaction equation indicate that the method is applicable to more general free boundary problems for nonlinear elliptic equations.This research was supported by the U.S. Army Research Office under Contract DAAG-79-0145     

Renormalization-Group Transformation in a 2n-Component Fermionic Hierarchical Model

We study the 2N-component fermionic model on a hierarchical lattice and give explicit formulas for the renormalization-group transformation in the space of coefficients that determine a Grassmann-valued density of the free measure. We evaluate the inverse renormalization-group transformation. The de.nition of the renormalization-group fixed points reduces to a solution of a system of algebraic equations. We investigate solutions of this system for N = 1, 2, 3. For α = 1, we prove an analogue of the central limit theorem for fermionic 2N-component fields. We discover an interesting relation between renormalization-group transformations in bosonic and fermionic hierarchical models and show that one of these transformations is obtained from the other by replacing N with -N. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 146, No. 2, pp. 251–266, February, 2006.     

Gaussian estimates for the density of the non-linear stochastic heat equation in any space dimension

In this paper, we establish lower and upper Gaussian bounds for the probability density of the mild solution to the non-linear stochastic heat equation in any space dimension. The driving perturbation is a Gaussian noise which is white in time with some spatially homogeneous covariance. These estimates are obtained using tools of the Malliavin calculus. The most challenging part is the lower bound, which is obtained by adapting a general method developed by Kohatsu-Higa to the underlying spatially homogeneous Gaussian setting. Both lower and upper estimates have the same form: a Gaussian density with a variance which is equal to that of the mild solution of the corresponding linear equation with additive noise.     

A method for the approximate solution of quasi-static problems for hardening bodies

A method for the approximate solution of quasi-static problems for hardening elastoplastic bodies is proposed. The constitutive relation of the model is taken in the form of a variational inequality. An approximate solution of the initial problem is constructed in time steps and, by means of the finite element method, is reduced to the solution of a system of two variational inequalities in corresponding finite-dimensional space. It is shown that the solution of this system is equivalent to finding the saddle point of the corresponding quadratic functional. To find the saddle point, Udzawa's algorithm is used, by means of which the process of finding the velocity vector and stress tensor reduces to the successive calculation of these quantities: the velocity vector is determined from the variational inequality corresponding to the equilibrium equation, and the stress tensor is determined from the variational inequality corresponding to the constitutive relation. The latter inequality is reduced to a certain non-linear equation containing the operation of projection onto a closed convex set corresponding to the elastic strains of the medium. In turn, the solution of the non-linear equation is constructed using the method of successive approximations. To illustrate the use of the proposed method, the one-dimensional problem of the quasi-static deformation of a cylindrical tube under a load applied to its internal surface is considered.     

The relation between the mechanics of dissipative finite-dimensional systems with heredity and the mechanics of infinite-dimensional Hamiltonian systems

It is shown that the dynamics of a non-linear multi-dimensional oscillator interacting with a field of harmonic oscillators, continuously distributed with respect to frequency, is governed by a non-linear integro-differential equation. The investigation centres on the possibility of an inverse transition from a non-linear oscillator with heredity to be embeddable in a larger Hamiltonian system is the usual condition that the entropy production should not be negative. Existence and uniqueness theorems are proved and several a priori estimates are found for the solution. It is also proved that, subject to certain restrictions on the relaxation kernel, the solution converges to one of the critical points of the effective potential.     

Stochastic optimal control of DC pension funds

In this paper, we study the portfolio problem of a pension fund manager who wants to maximize the expected utility of the terminal wealth in a complete financial market with the stochastic interest rate. Using the method of stochastic optimal control, we derive a non-linear second-order partial differential equation for the value function. As it is difficult to find a closed form solution, we transform the primary problem into a dual one by applying a Legendre transform and dual theory, and try to find an explicit solution for the optimal investment strategy under the logarithm utility function. Finally, a numerical simulation is presented to characterize the dynamic behavior of the optimal portfolio strategy.     

Comparison of Two Methods for Superreplication

Abstract

We compare two methods for superreplication of options with convex pay-off functions. One method entails the overestimation of an unknown covariance matrix in the sense of quadratic forms. With this method the value of the superreplicating portfolio is given as the solution of a linear Black–Scholes BS-type equation. In the second method, the choice of quadratic form is made pointwise. This leads to a fully non-linear equation, the so-called Black–Scholes–Barenblatt (BSB) equation, for the value of the superreplicating portfolio. In general, this value is smaller for the second method than for the first method. We derive estimates for the difference between the initial values of the superreplicating strategies obtained using the two methods.     

Torsional buckling and post-buckling of columns made of aluminium alloy

The paper concerns torsional buckling and the initial post-buckling of axially compressed thin-walled aluminium alloy columns with bisymmetrical cross-section. It is assumed that the column material behaviour is described by the Ramberg–Osgood constitutive equation in non-linear elastic range. The stationary total energy principle is used to derive the governing non-linear differential equation. An approximate solution of the equation determined by means of the perturbation approach allows to determine the buckling loads and the initial post-buckling behaviour. Numerical examples dealing with simply supported I-column are presented and the effect of material elastic non-linearity on the critical loads and initial post-buckling behaviour are compared to the linear solution.     

Hopf bifurcation analysis of a general non-linear differential equation with delay

This work represents Hopf bifurcation analysis of a general non-linear differential equation involving time delay. A special form of this equation is the Hutchinson–Wright equation which is a mile stone in the mathematical modeling of population dynamics and mathematical biology. Taking the delay parameter as a bifurcation parameter, Hopf bifurcation analysis is studied by following the theory in the book by Hazzard et al. By analyzing the associated characteristic polynomial, we determine necessary conditions for the linear stability and Hopf bifurcation. In addition to this analysis, the direction of bifurcation, the stability and the period of a periodic solution to this equation are evaluated at a bifurcation value by using the Poincaré normal form and the center manifold theorem. Finally, the theoretical results are supported by numerical simulations.     

A Finite-difference Solution of the Ginzburg–Landau Equation

Two finite-difference methods, which differ only in the way that they approximate the derivative boundary conditions, are developed for solving a particular form of the complex Ginzburg–Landau equation of superconductivity. The non-linear term in this equation is linearized in a way familiar to readers of Professor Mickens' work, and the numerical solution is obtained at each time step by solving a linear algebraic system. Consistency and stability are discussed and some numerical results are reported.