Showing papers on "Partial differential equation published in 1976"

Introduction to partial differential equations

Abstract: The Description for this book, Introduction to Partial Differential Equations. (MN-17): , will be forthcoming.

Nonlinear differential–difference equations and Fourier analysis

Abstract: The conceptual analogy between Fourier analysis and the exact solution to a class of nonlinear differential–difference equations is discussed in detail. We find that the dispersion relation of the associated linearized equation is prominent in developing a systematic procedure for isolating and solving the equation. As examples, a number of new equations are presented. The method of solution makes use of the techniques of inverse scattering. Soliton solutions and conserved quantities are worked out.

The Korteweg–deVries Equation: A Survey of Results

Abstract: The Korteweg–de Vries equation \[ u_t + uu_x + u_{xxx} = 0\] is a nonlinear partial differential equation arising in the study of a number of different physical systems, e.g., water waves, plasma physics, anharmonic lattices, and elastic rods. It describes the long time evolution of small-but-finite amplitude dispersive waves. From detailed studies of properties of the equation and its solutions, the concept of solitons was introduced and the method for exact solution of the initial-value problem using inverse scattering theory was developed. A survey of these and other results for the Korteweg–deVries equation are given, including conservation laws, an alternate method for exact solution, soliton solutions, asymptotic behavior of solutions, Backlund transformation, and a nonlinear WKB method. The recent literature contains many extensions of these ideas to a number of other nonlinear evolution equations of physical interest and to other classes of equations. Some of these equations and results are indica...

A derivation of equations for wave propagation in water of variable depth

Abstract: Within the scope of the three-dimensional theory of homogeneous incompressible inviscid fluids, this paper contains a derivation of a system of equations for propagation of waves in water of variable depth. The derivation is effected by means of the incompressibility condition, the energy equation, the invariance requirements under superposed rigid-body motions, together with a single approximation for the (three-dimensional) velocity field.

A derivation of equations for wave propagation in water of variable depth

Abstract: Within the scope of the three-dimensional theory of homogeneous incompressible inviscid fluids, this paper contains a derivation of a system of equations for propagation of waves in water of variable depth. The derivation is effected by means of the incompressibility condition, the energy equation, the invariance requirements under superposed rigid-body motions, together with a single approximation for the (three-dimensional) velocity field.

Molecular theory of surface tension

Abstract: A molecular theory of surface tension is developed for a liquid–gas interface of a one component system. The Helmholtz free energy, the quantity minimized in the van der Waals approach, is obtained here from a rigorous expansion in powers of derivatives of density ρ and is minimized by the calculus of variations. The coefficient A (ρ) of the term in the square of the density gradient is (kT/6) Fdr r2C (r,ρ), C being the direct correlation function. In the case in which ρ varies in one direction x only, the solution of the Euler–Lagrange differential equation is analyzed in detail. This describes the cases of a single phase and of two coexisting phases and leads to the equal area Maxwell construction. The effect of an external field on the solution is discussed. The Euler–Lagrange differential equation provides a differential statement of Bernoulli’s theorem. In a three dimensional treatment the stress tensor formula is obtained from the corresponding Euler–Lagrange partial differential equation. A (differ...

A Variety of Nonlinear Network Equations Generated from the Bäcklund Transformation for the Toda Lattice

Abstract: A Backlund transformation in the bilinear form is presented for the Toda equation. The Backlund transformation generates an important class of nonlinear evolution equations that exhibits N-soliton solutions. The equation reduces, in the special cases, to the Toda equation itself, the nonlinear self-dual network equation, the equation describing a Volterra system and a discrete Korteweg·de Vries equation. Physical meanings and properties of solitons of these equations are examined in detail. Special solutions are also given to the generated equation. Moreover, a relation between the Backlund transformation and the inverse scattering method, and a nonlinear transformation relating the Toda equation and the generated equation are presented.

Linear Quadratic Optimal Stochastic Control with Random Coefficients

Abstract: The purpose of this paper is to apply the methods developed in [1] and [2] to solve the problem of optimal stochastic control for a linear quadratic system.After proving some preliminary existence results on stochastic differential equations, we show the existence of an optimal control.The introduction of an ad joint variable enables us to derive extremality conditions: the control is thus obtained in random “feedback” form. By using a method close to the one used by Lions in [4] for the control of partial differential equations, a priori majorations are obtained.A formal Riccati equation is then written down, and the existence of its solution is proved under rather general assumptions.For a more detailed treatment of some examples, the reader is referred to [1].

Nonlinear evolution equations solvable by the inverse spectral transform.—I

Abstract: This paper is the first of a series based on a general method to discover and investigate nonlinear partial differential equations solvable via the inverse spectral transform technique. The results of this paper are those that obtain applying this method to the generalized Zakharov-Shabat linear problem. We give a class of nonlinear evolution equations solvable by the inverse spectral transform, that is more general than that introduced by Ablowitz, Kaup, Newell and Segur because it includes equations involving more than one space variable and containing coefficients that are not constant. We also report a very general class of Backlund transformations that includes all such transformations previously considered and clarifies their significance. And we produce, for a somewhat less general class of nonlinear evolution equations (involving only one space variable), a remarkable functional equation that relates the solution at timet to the same solution at timet′. This paper is focussed on a general presentation of the approach and the proof of the main results (some of which had been previously reported without proof). Although the analysis of special equations and special solutions is deferred to subsequent papers of this series, there are here also a few results of this kind, including the explicit display of the exact nonsoliton solution of the sine-Gordon equation corresponding to a double pole of the associated spectral parameter.

Introduction to partial differential equations with applications

Abstract: I would highly recommend strauss's book is the purpose. This is to the reference section has always. 2004 an excellent guide for self study by a prominent? They succeed admirably they will be investigated this. This is a textbook for motivated reader with lots of differential equations. A large number of the more, than in theory sobolev spaces has. Even though my reference of mathematicians and explaining their impact on a very readable. Thesis in pure and feels like a variety? This book is the best available, in pure mathematics this book. This book for the notation given in general. The modeling of functional analysis partial differential equations 2nd ed. G the modern books about, partial differential equations this text contains a variety. This is a prominent place in the pioneering work on physicist. Monatshefte fr angewandte mathematik und mechanik vol gary. This book meant for the skills, elsewhere desire to this book. 1072 the new examples and riemann partial differential equations. Lebesgue integration is extremely well written for mathematics geometry poincare conjecture this book. You will discuss the basis of science new problems modern theory. The book for postgraduate maths I still one of mathematical theories concerning partial differential equations. Gary consequently the reader, subject. There is apparently written for students. I would highly recommend strauss's book, is written for a prominent place. You will be used as complex function theory.

On the dynamics of a continuum spin system

Abstract: For a one-dimensional system of classical spins with nearest neighbour Heisenberg interaction we derive the equation of motion for each three-dimensional spin vector. In the continuum limit where the spins lie dense on a line this set of equations reduces to a nonlinear partial differential equation. In addition to spin-wave solutions we obtain some other special solutions of this equation. In particular we find solitary waves having total energy localised in a finite region, with velocity of propagation inversely proportional to the width of this region. Solutions of still another type are shown to have a diffusive character. The stability of such solutions and the possibility of interaction of two or more solitary waves have not yet been studied.

Non-linear saturation of the dissipative trapped-ion mode by mode coupling

Abstract: The non-linear saturation of the dissipative trapped-ion mode is analysed. The basic mechanism considered is the process whereby energy in long-wavelength unstable modes is non-linearly coupled via × convection to shortwavelength modes stabilized by Landau damping due to both circulating and trapped ions. In the usual limit of the mode frequency being small relative to the effective electron collision frequency, a one-dimensional non-linear partial differential equation for the potential can be derived, as was first shown by LaQuey, Mahajan, Tang, and Rutherford. The stability and accessibility of the possible equilibria for this equation are examined in detail, both analytically and numerically. The equilibrium emphasized by LaQuey et al. is shown to be unstable. However, a class of non-linear saturated states which are stable to linear perturbations is found. Included in the analysis are the effects of both ion collisions and dispersion due to finite-ion-banana-width effects. Cross-field transport is estimated and the scaling of the results is considered for tokamak parameters (specifically those for the Princeton Large Torus). It is concluded that the anomalous cross-field transport can be much lower than the estimate of Kadomtsev and Pogutse, for relevant parameters near marginal stability for the linear modes.

Mathematical methods for physics

Abstract: This classic book helps students learn the basics in physics by bridging the gap between mathematics and the basic fundamental laws of physics. With supplemental material such as graphs and equations, Mathematical Methods for Physics creates a strong, solid anchor of learning. The text has three parts: Part I focuses on the use of special functions in solving the homogeneous partial differential equations of physics, and emphasizes applications to topics such as electrostatics, wave guides, and resonant cavities, vibrations of membranes, heat flow, potential flow in fluids, plane and spherical waves. Part II deals with the solution of inhomogeneous differential equations with particular emphasis on problems in electromagnetism, Green's functions for Poisson's equation, the wave equation and the diffusion equation, and the solution of integral equations by iteration, eigenfunction expansion and the Fredholm series. Finally, Part II explores complex variable techniques, including evalution of itegrals, dispersion relations, special functions in the complex plane, one-sided Fourier transforms, and Laplace transforms.

Partial Differential Equations: Theory and Technique

Abstract: The Diffusion Equation. Laplace Transform Methods. The Wave Equation. The Potential Equation. Classification of Second Order Equations. First Order Equations. Extensions. Perturbations. Green's Functions. Variational Methods. Eigenvalue Problems. More on First Order Equations. More on Characteristics. Finite-Difference Equations and Numerical Methods. More on Transforms. Singular Perturbation Methods. Index.

A review of least-squares methods for solving partial differential equations

Abstract: Continuous (integral) and discrete (point-matching) least-squares methods are presented for linear and non-linear problems in boundary-value, eigenvalue, and initial-value form. The history is traced, and important theoretical and practical results are summarized. A comprehensive sample of the literature is presented, indexed to show type of application, version of least squares used, and results of comparison studies. The advantages of least-squares methods are discussed, including convenience in formulation and error evaluation, generality of mixed and local (finite element) versions, and performance that is competitive with other methods.

Vector Variational Formulation of Electromagnetic Fields in Anisotropic Media

Abstract: Maxwell's equations can be cast into a basic differential operator equation, the curlcurl equation, which lends itself easily to variational treatment. Various forms of this equation are associated with problems of practical importance. The formulation includes the treatment of loss-free anisotropic media. The boundary conditions associated with electromagnetic-field problems are treated in detail and the uniqueness of the solution is discussed. A functional is derived for the curlcurl equation in Cartesian and cylindrical coordinates.

The occurrence of parametric instabilities in finite-amplitude internal gravity waves

Abstract: The parametric instability of a plane internal gravity wave is considered. When the two-dimensional equations of vorticity and mass conservation are linearized in the disturbance quantities, partial differential equations with periodic coefficients result. Substitution of a perturbation of the form dictated by Floquet theory into these equations yields compatibility conditions which, when evaluated numerically, give the curves of neutral stability and constant disturbance growth rate. These results reveal that, for an internal wave of even infinitesimal amplitude, disturbance waves can begin to grow in amplitude. Moreover, these parametric instabilities are shown to reduce to the classical case of the nonlinear resonant interaction in the limit of vanishingly small basic-state amplitude. The fact that these unstable disturbances can exist for an internal wave of any amplitude suggests that this phenomenon may be an important mechanism for extracting energy from an internal gravity wave.

Symmetry and separation of variables for the Helmholtz and Laplace equations

Abstract: This paper is one of a series relating the symmetry groups of the principal linear partial differential equations of mathematical physics and the coordinate systems in which variables separate for these equations. In particular, we mention [1] and paper [2] which is a survey of and introduction to the series. Here we apply group-theoretic methods to study the separable coordinate systems for the Helmholtz equation.

A Comparison of Several Bandwidth and Profile Reduction Algorithms

Abstract: : This paper compares and analyzes six algorithms which have been suggested recently for use in reducing, by permutations, the bandwidth and profile of sparse matrices. This problem arises in many different areas of scientific computation such as in the finite element method for approximating solutions of partial differential equations and in analyzing large-scale power transmission systems.

An analysis of the numerical solution of the transport equation

Abstract: Various finite difference and finite element methods for solving the one-dimensional convective-dispersive equation are investigated. A Fourier series analysis is performed to determine the dissipative and dispersive characteristics of these numerical methods. The analysis indicates that the commonly observed phenomenon of overshoot of a concentration pulse is due to the inability of the numerical schemes to propagate the small wavelengths which are important to the description of the front. Furthermore, the numerical smearing of a sharp front is due to dissipation of these small wavelengths. The finite element method was found to be superior to finite difference methods for solution of the convective-dispersive equation.

One-dimensional mathematical model for large-strain consolidation

Abstract: A mathematical model is developed to represent the one-dimensional large-strain consolidation of a fully saturated clay. The fluid limit is postulated to be that water content associated with a ‘stress-free’ condition of the soil, and it is taken as the reference state from which strains are measured. Experimental results from a series of permeability tests suggest that the relationship between the logarithm of the coefficient of permeability and the void ratio is not a straight line for the entire range of void ratio considered. In addition, the variation of the constrained modulus as consolidation progresses is taken into account. The resulting boundary value problem involves a non-linear partial differential equation with void ratio as the dependent variable, and the numerical solution is accomplished by a step-by-step procedure combined with a weighted residual technique which leads to a finite element discretization in the spatial variable and a finite difference discretization in the time variable. ...

Stochastic theory of the kinetics of phase transitions

Abstract: We present a stochastic theory of the kinetics of phase transitions in univariant, nonuniform systems. We assume a master equation and a relation of the transition probability to the free energy [J. S. Langer, Ann. Phys. (N.Y) 65, 53 (1971)]. The free energy is taken to be of the Cahn–Hilliard form. By means of path integral methods we obtain a formal solution from which we derive a deterministic differential equation for the most probable variation of the density distribution in time. This equation is of the Landau–Ginzburg type. We show the existence of a Liapunoff function for this kinetic equation, which is then used to derive simply the classical result of nucleation theory for the critical radius of a droplet. Next we show from the kinetic equation that if the structure of the free energy is such that a phase transition occurs and metastable states are possible, then the kinetics of the decay to the stable and metastable states is nonlinear, of the cubic type. Finally we derive from the kinetic equa...

Calculation of eigenvalues of the Helmholtz equation by an integral equation

Abstract: A method is presented to calculate the eigenvalues of the Helmholtz equation Δ2o + k2o = 0 in a two-dimensional area when o vanishes on the boundary. The method is based on an integral equation, which can be easily solved numerically. Results obtained for circular and rectangular geometries are also given and compared to the exact values.

Development of the Theory of a Nonlinear Lattice

Abstract: An extensive account of the exact theory of wave propagation in the one-dimensional nonlinear lattice with exponential interaction between nearest neighbor particles is given. A brief review of the development, useful particular solutions, the general method of solving the equations of motion and the relation between the discrete lattice and the con­ tinuous Korteweg-de Vries system are given, with some future aspect of the problems of nonlinear lattices. In this article the author wishes to present the problems related to wave propagation in nonlinear lattices with special emphasis on the one-dimensional lattice of particles with the nearest neighbor interaction of the exponential type (the exp-lattice or the Toda lattice). The development of the theory of the nonlinear lattice will be briefly reviewed in §§l and 2, and the charac­ teristic features of waves will be presented in §3 by showing particular solutions to the equations of motion for the lattice. In §§ 4 and 5, general theory of the exp-lattice will be followed by the general method for solving the equations of motion. If one takes the continuum limit under certain restrictions, one sees that the time evolution of the wave can be approximated by a partial differential equation which was found by Korteweg and de Vries to describe shallow­ water waves (KdV equation). In §6 the relation between the exp-lattice and the KdV equation will be discussed. In the final section, some remarks on further problems will be presented. 1-l. Nonlz'near lattice The equations of motion for the one-dimensional lattice of particles with nearest neighbor interaction can be written, when no external force is present, as