Showing papers on "Partial differential equation published in 1981"

Numerical solution of the Euler equations by finite volume methods using Runge Kutta time stepping schemes

Abstract: A new combination of a finite volume discretization in conjunction with carefully designed dissipative terms of third order, and a Runge Kutta time stepping scheme, is shown to yield an effective method for solving the Euler equations in arbitrary geometric domains. The method has been used to determine the steady transonic flow past an airfoil using an O mesh. Convergence to a steady state is accelerated by the use of a variable time step determined by the local Courant member, and the introduction of a forcing term proportional to the difference between the local total enthalpy and its free stream value.

Solitons and the Inverse Scattering Transform

Abstract: : Under appropriate conditions, ocean waves may be modeled by certain nonlinear evolution equations that admit soliton solutions and can be solved exactly by the inverse scattering transform (IST). The theory of these special equations is developed in five lectures. As physical models, these equations typically govern the evolution of narrow-band packets of small amplitude waves on a long (post-linear) time scale. This is demonstrated in Lecture I, using the Korteweg-deVries equation as an example. Lectures II and III develop the theory of IST on the infinite interval. The close connection of aspects of this theory to Fourier analysis, to canonical transformations of Hamiltonian systems, and to the theory of analytic functions is established. Typical solutions, including solitons and radiation, are discussed as well. With periodic boundary conditions, the Korteweg-deVries equation exhibits recurrence, as discussed in Lecture IV. The fifth lecture emphasizes the deep connection between evolution equations solvable by IST and Painleve transcendents, with an application to the Lorenz model.

Introduction to perturbation techniques

Abstract: Algebraic Equations. Integrals. The Duffing Equation. The Linear Damped Oscillator. Self-Excited Oscillators. Systems with Quadratic and Cubic Nonlinearities. General Weakly Nonlinear Systems. Forced Oscillations of the Duffing Equation. Multifrequency Excitations. The Mathieu Equation. Boundary-Layer Problems. Linear Equations with Variable Coefficients. Differential Equations with a Large Parameter. Solvability Conditions. Appendices. Bibliography. Index.

Quantum Physics: A Functional Integral Point of View

Abstract: This book is addressed to one problem and to three audiences. The problem is the mathematical structure of modem physics: statistical physics, quantum mechanics, and quantum fields. The unity of mathemati cal structure for problems of diverse origin in physics should be no surprise. For classical physics it is provided, for example, by a common mathematical formalism based on the wave equation and Laplace's equation. The unity transcends mathematical structure and encompasses basic phenomena as well. Thus particle physicists, nuclear physicists, and con densed matter physicists have considered similar scientific problems from complementary points of view. The mathematical structure presented here can be described in various terms: partial differential equations in an infinite number of independent variables, linear operators on infinite dimensional spaces, or probability theory and analysis over function spaces. This mathematical structure of quantization is a generalization of the theory of partial differential equa tions, very much as the latter generalizes the theory of ordinary differential equations. Our central theme is the quantization of a nonlinear partial differential equation and the physics of systems with an infinite number of degrees of freedom. Mathematicians, theoretical physicists, and specialists in mathematical physics are the three audiences to which the book is addressed. Each of the three parts is written with a different scientific perspective."

Plane Waves and Spherical Means: Applied to Partial Differential Equations

Abstract: The author would like to acknowledge his obligation to all his (;Olleagues and friends at the Institute of Mathematical Sciences of New York University for their stimulation and criticism which have contributed to the writing of this tract. The author also wishes to thank Aughtum S. Howard for permission to include results from her unpublished dissertation, Larkin Joyner for drawing the figures, Interscience Publishers for their cooperation and support, and particularly Lipman Bers, who suggested the publication in its present form. New Rochelle FRITZ JOHN September, 1955 [v] CONTENTS Introduction...1 CHAPTER I Decomposition of an Arbitrary Function into Plane Waves Explanation of notation ...7 The spherical mean of a function of a single coordinate. 7 9 Representation of a function by its plane integrals . CHAPTER II Tbe Initial Value Problem for Hyperbolic Homogeneous Equations with Constant Coefficients Hyperbolic equations...15 Geometry of the normal surface for a strictly hyperbolic equation. 16 Solution of the Cauchy problem for a strictly hyperbolic equation . 20 Expression of the kernel by an integral over the normal surface. 23 The domain of dependence ...2 9 The wave equation ...32 The initial value problem for hyperbolic equations with a normal surface having multiple points ...36 CHAPTER III The Fundamental Solution of a Linear Elliptic Differential Equation witL Analytic Coefficients Definition of a fundamental solution ...43 The Cauchy problem ...45 Solution of the inhomogeneous equation with a plane wave function as right hand side ...49 The fundamental solution...

Applications of number theory to numerical analysis

Abstract: 1. Algebraic Number Fields and Rational Approximation.- 1.1. The units of algebraic number fields.- 1.2. The simultaneous Biophantine approximation of an integral basis.- 1.3. The real eyelotomie field.- 1.4. The units of a eyelotomie field.- 1.5. Continuation.- 1.6. The Drriehlet field.- 1.7. The cubic field.- Notes.- 2. Recurrence Relations and Rational Approximation.- 2.1. The recurrence formula for the elementary symmetric fonction.- 2.2. The generalization of Sn.- 2.3. PV numbers.- 2.4. The roots of the equation F(x) = 0.- 2.5. The roots of the equation G(x) = 0.- 2.6. The roots of the equation E(x) = 0.- 2.7. The irreducibility of a polynomial.- 2.8. The rational approximations of ?, ?, ?.- Notes.- 3. Uniform Distribution.- 3.1. Uniform distribution.- 3.2. Vinogradov's lemma.- 3.3. The exponential sum and the discrepancy.- 3.4. The number of solutions to the congruence.- 3.5. The solutions of the congruence and the discrepancy.- 3.6. The partial summation formula.- 3.7. The comparison of discrepancies.- 3.8. Eational approximation and the solutions of the congruence.- 3.9. The rational approximation and the discrepancy.- 3.10. The lower estimate of discrepancy.- Notes.- 4. Estimation of Discrepancy.- 4.1. The set of equi-distribution.- 4.2. The Halton theorem.- 4.3. The p set.- 4.4. The gp set.- 4.5. The eonstruetion of good points.- 4.6. The ?s set.- 4.7. The ? set.- 4.8. The ease s = 2.- 4.9. The glp set.- Notes.- 5. Uniform Distribution and Numerical Integration.- 5.1. The function of bounded variation.- 5.2. Uniform distribution and numerical integration.- 5.3. The lower estimation for the error term of quadrature formula.- 5.4. The quadrature formulas.- Notes.- 6. Periodic Functions.- 6.1. The classes of functions.- 6.2. Several lemmas.- 6.3. The relations between Hs?(C), Qs?(C) and Es?(C).- 6.4. Periodic functions.- 6.5. Continuation.- Notes.- 7. Numerical Integration of Periodic Functions.- 7.1. The set of equi-distribution and numerical integration.- 7.2. The p set and numerical integration.- 7.3. The gp set and numerical integration.- 7.4. The lower estimation of the error term for the quadrature formula.- 7.5. The solutions of congruences and numerical integration.- 7.6. The glp set and numerical integration.- 7.7. The Sarygin theorem.- 7.8. The mean error of the quadrature formula.- 7.9. Continuation.- Notes.- 8. Numerical Error for Quadrature Formula.- 8.1. The numerical error.- 8.2. The comparison of good points.- 8.3. The computation of the ? set.- 8.4. The computation of the ?s set.- 8.5. Examples of other F s sets.- 8.6. The computation of a glp set.- 8.7. Several remarks.- 8.8. Tables.- 8.9. Some examples.- Notes.- 9. Interpolation.- 9.1. Introduction.- 9.2. The set of equi-distribution and interpolation.- 9.3. Several lemmas.- 9.4. The approximate formula of the function of E?s(C).- 9.5. The approximate formula of the function of Q?s(C).- 9.6. The Bernoulli polynomial and the approximate polynomial.- 9.7. The ? results.- Notes.- 10. Approximate Solution of Integral Equations and Differential Equations.- 10.1. Several lemmas.- 10.2. The approximate solution of the Fredholm integral equation of second type.- 10.3. The approximate solution of the Volterra integral equation of second type.- 10.4. The eigenvalue and eigenfunction of the Fredholm equation.- 10.5. The Cauehy problem of the partial differential equation of the parabolic type.- 10.6. The Diriehlet problem of the partial differential equation of the elliptic type.- 10.7. Several remarks.- Notes.- Appendix Tables.

Discrete Analogue of a Generalized Toda Equation

Abstract: A discrete analogue of a generalized Toda equation and its Backlund transformations are obtained. The equation is expressed with the bilinear form as follows \begin{aligned} [Z_{1} \exp (D_{1})+Z_{2} \exp (D_{2})+Z_{3} \exp (D_{3})]f \cdot f=0 \end{aligned} where Z i and D i for i =1, 2, 3, are an arbitrary parameter and a linear combination of the binary operators D t , D x , D y , D n , etc., respectively. The equation is very generic, namely appropriate combinations of parameters give various types of soliton equations including the Korteweg-de Vries equation, Kadomtsev-Petviashvili equation, modified KdV equation, sine-Gordon equation, nonlinear Klein-Gordon equation, Benjamin-Ono equation and various types of discrete analogues of soliton equations.

Group velocity in finite difference schemes

Abstract: The relevance of group velocity to the behavior of finite difference models of time-dependent partial differential equations is surveyed and illustrated. Applications involve the propagation of wave packets in one and two dimensions, numerical dispersion, the behavior of parasitic waves, and the stability analysis of initial boundary-value problems.

Advances in mathematical models for image processing

Abstract: Several state-of-the-art mathematical models useful in image processing are considered. These models include the traditional fast unitary transforms, autoregessive and state variable models as well as two-dimensional linear prediction models. These models introduced earlier [51], [52] as low-order finite difference approximations of partial differential equations are generalized and extended to higher order in the framework of linear prediction theory. Applications in several image Processing problems, including image restoration, smoothing, enhancement, data compression, spectral estimation, and filter design, are discussed and examples given.

One-sided difference approximations for nonlinear conservation laws

Abstract: We analyze one-sided or upwind finite difference approximations to hyperbolic partial differential equations and, in particular, nonlinear conservation laws. Second order schemes are designed for which we prove both nonlinear stability and that the entropy condition is satisfied for limit solutions. We show that no such stable approximation of order higher than two is possible. These one-sided schemes have desirable properties for shock calculations. We show that the proper switch used to change the direction in the upwind differencing across a shock is of great importance. New and simple schemes are developed for which we prove qualitative properties such as sharp monotone shock profiles, existence, uniqueness, and stability of discrete shocks. Numerical examples are given.

Theory of electronic transitions in slow atomic collisions

Abstract: This review deals with quantitative descriptions of electronic transitions in atom-atom and ion-atom collisions. In one type of description, the nuclear motion is treated classically or semiclassically, and a wave function for the electrons satisfies a time-dependent Schr\"odinger equation. Expansion of this wave function in a suitable basis leads to time-dependent coupled equations. The role played by electron-translation factors in this expansion is noted, and their effects upon transition amplitudes are discussed. In a fully quantum-mechanical framework there is a wave function describing the motion of electrons and nuclei. Expansion of this wave function in a basis which spans the space of electron variables leads to quantum-mechanical close-coupled equations. In the conventional formulation, known as perturbed-stationary-states theory, certain difficulties arise because scattering boundary conditions cannot be exactly satisfied within a finite basis. These difficulties are examined, and a theory is developed which surmounts them. This theory is based upon an intersecting-curved-wave picture. The use of rotating or space-fixed electronic basis sets is discussed. Various bases are classified by Hund's cases (a)-(e). For rotating basis sets, the angular motion of the nuclei is best described using symmetric-top eigenfunctions, and an example of partial-wave analysis in such functions is developed. Definitions of adiabatic and diabatic representations are given, and rules for choosing a good representation are presented. Finally, representations and excitation mechanisms for specific systems are reviewed. Processes discussed include spin-flip transitions, rotational coupling transitions, inner-shell excitations, covalent-ionic transitions, resonant and near-resonant charge exchange, fine-structure transitions, and collisional autoionization and electron detachment.

Implicit schemes and LU decompositions

Abstract: Implicit methods for hyperbolic equations are analyzed by constructing LU factorizations. It is shown that the solution of the resulting tridiagonal systems in one dimension is well conditioned if and only if the LU factors are diagonally dominant. Stable implicit methods that have diagonally dominant factors are constructed for hyperbolic equations in n space dimesnions. Only two factors are required even in three space dimensions. Acceleration to a steady state is analyzed. When the multidimensional backward Euler method is used with large time steps, it is shown that the scheme approximates a Newton-Raphson iteration procedure.

A partial differential equation with infinitely many periodic orbits: Chaotic oscillations of a forced beam

Abstract: This paper delineates a class of time-periodically perturved evolution equations in a Banach space whose associated Poincar´e map contains a Smale horseshoe. This implies that such systems possess periodic orbits with arbitrarily high period. The method uses techniques originally due to Melnikov and applies to systems of the form x˙ = f0(x) + "f1(x, t), where x˙ = f0(x) is Hamiltonian and has a homoclinic orbit. We give an example from structural mechanics: sinusoidally forced vibrations of a buckled beam.

Summation in Finite Terms

Abstract: Results which allow either the computatmn of symbolic solutions to first-order-linear difference equauons or the determination that solutions of a certain form do not exist are presented. Starting with a field of constants, larger fields may be constructed by the formal adjunctlon of symbols whtch behave hke solutions to first-order-linear equations (with a few restrictions) It IS in these extension fields that the difference equations may be posed and m which the solutions are requested. The principal apphcatmn of these results is In finding formulas for a broad class of finite sums or In showing the nonexistence of such formulas.

A distribution-moment approximation for kinetic theories of muscular contraction

Abstract: A rational mathematical procedure is proposed for approximating solutions to the partial differential equations of cross-bridge kinetics in theories of muscular contraction of the type first proposed by A. F. Huxley. The essence of the procedure is to approximate the exact bond-distribution functions by distributions of prescribed form, and this leads to a set of first-order ordinary differential equations on the low-order moments of the approximate distributions. Thus the procedure effectively results in a lumped-parameter model of muscle approaching the structural simplicity of the classic two-element model, but one which exhibits more realistic behavior. The approximation is worked out in detail for Huxley's original (1957) two-state model (modified slightly to produce a more realistic stretch response), compared with exact solutions of the model, and used to predict muscle behavior under various conditions. It is anticipated that this approximation, with its attendant conceptual and computational simplifications, will make recent theoretical advances in molecular contraction mechanics more accessible for applications in macroscopic muscle dynamics. Generalizations of the procedure to the case of length-dependent behavior are discussed.

A kinetic theory for polymer melts. I. The equation for the single‐link orientational distribution function

Abstract: In this series of papers we show how a kinetic theory of undiluted polymers can be developed using the Curtiss–Bird–Hassager phase‐space formulation. The polymer molecule is modeled as a Kramers freely jointed bead–rod chain. The objective is to obtain a molecular‐theory expression for the stress tensor from which the rheological properties of polymer melts can be obtained. This development is put forth as an alternative to the Doi–Edwards theory; using a very different approach, we have rederived some of their results and generalized or extended others. In this first paper we develop the partial differential equation for the chain configurational distribution function, and then proceed to get the equation for the orientational distribution function for a single link in the chain. A modification of Stokes’ law is introduced that includes a tensor drag coefficient, characterized by two scalar parameters ζ (the friction coefficient) and e (the link tension coefficient). In addition, to describe the increase of the drag force on a bead with chain length, at constant bead density, a ’’chain constraint exponent’’ β is used, which can vary from zero (the Doi–Edwards limit) to about 0.5. Solutions to the partial differential equation for the single‐link distribution function are given in several forms, including an explicit series solution to terms of third order in the velocity gradients.

Aquifer parameter identification with optimum dimension in parameterization

Abstract: This paper presents a systematic procedure whereby the functional coefficients imbedded in a two-dimensional partial differential equation which governs unsteady groundwater flow are optimally identified. The coefficients to be identified are transmissivities which vary spatially. Finite elements are used to represent the unknown transmissivity function parametrically in terms of nodal values over a suitable discretization of a flow region. A modified Gauss-Newton algorithm is used for parameter optimization. Covariance analysis is used to estimate the reliability of the estimated parameters. As the dimension of the unknown parameter increases, the modeling error represented by a least squares criterion will generally decrease, but errors in data would be propagated to a greater degree into the estimated parameters, thus reducing the reliability of estimation. The reliability of the estimated parameters is characterized by a norm of the covariance matrix. This information is used for the determination of the optimum dimension in parameterization.

The boundary element method applied to the analysis of two‐dimensional nonlinear sloshing problems

Abstract: This paper deals with an application of the boundary element method to the analysis of nonlinear sloshing problems, namely nonlinear oscillations of a liquid in a container subjected to forced oscillations. First, the problem is formulated mathematically as a nonlinear initial-boundary value problem by the use of a governing differential equation and boundary conditions, assuming the fluid to be inviscid and incompressible and the flow to be irrotational. Next, the governing equation (Laplace equation) and boundary conditions, except the dynamic boundary condition on the free surface, are transformed into an integral equation by employing the Galerkin method. Two dynamic boundary condition is reduced to a weighted residual equation by employing the Galerkin method. Two equations thus obtained are discretized by the use of the finite element method spacewise and the finite difference method timewise. Collocation method is employed for the discretization of the integral equation. Due to the nonlinearity of the problem, the incremental method is used for the numerical analysis. Numerical results obtained by the present boundary element method are compared with those obtained by the conventional finite element method and also with existing analytical solutions of the nonlinear theory. Good agreements are obtained, and this indicates the availability of the boundary element method as a numerical technique for nonlinear free surface fluid problems.

Implicit Degenerate Evolution Equations and Applications

Abstract: The initial-value problem is studied for evolution equations in Hilbert space of the general form d se(u)+N(u) l:, dt where and are maximal monotone operators. Existence of a solution is proved when1 is a subgradient and either is strongly monotone or 9 is coercive; existence is established also in the case where 1 is strongly monotone and is subgradient. Uniqueness is provedwhen one of or is continuous self-adjoint and thesum is strictlymonotone; examples of nonuniqueness are given. Applications are indicated for various classes of degenerate nonlinear partial differential equations or systems of mixed elliptic-parabolic-pseudo- parabolic types and problems with nonlocal nonlinearity.

Numerical Analysis of Laminar Natural Convection Between Concentric and Eccentric Cylinders

Abstract: A numerical analysis is carried out to investigate the local and overall heat transfer between concentric and eccentric horizontal cylinders. The numerical procedure, based on Stone's strongly Implicit method, is extended to the 3 × 3 coupled system of the governing partial differential equations describing the conservation of mass, momentum, and energy. This method allows finite-difference solutions of the governing equations without artificial viscosity, and conserves its great stability even for arbitrarily large time steps. The algorithm is written for a numerically generated, body-fitted coordinate system. This procedure allows the solution of the governing equations in arbitrarily shaped physical domains Numerical solutions were obtained for a Raylelgh number In the range 102-103, a Prandtl number of 0.7, and three different eccentric positions of the inner cylinder. The results are discussed in detail and are compared with previous experimental and theoretical results.

Optimal control and nonlinear filtering for nondegenerate diffusion processes

Abstract: A linear parabolic partial differential equation describing the pathwise filter for a nondegenerate diffusion is changed, by an exponential substitution. into the dynamic programming equation of an optimal stochastic control problem. This substitution is applied to obtain results about the rate of decay as of solutions p(x,t)to the pathwise filter equation, and for solutions of the corresponding Zakai equation