Showing papers on "Partial differential equation published in 1970"

Applied numerical analysis

Abstract: The fifth edition of this book continues teaching numerical analysis and techniques. Suitable for students with mathematics and engineering backgrounds, the breadth of topics (partial differential equations, systems of nonlinear equations, and matrix algebra), provide comprehensive and flexible coverage of numerical analysis.

Advanced Calculus for Applications

Abstract: 1. Ordinary Differential Equations 1.1 Introduction 1.2 Linear Dependence 1.3 Complete Solutions of Linear Equations 1.4 The Linear Differential Equation of First Order 1.5 Linear Differential Equations with Constant Coefficients 1.6 The Equidimensional Linear Differential Equation 1.7 Properties of Linear Operators 1.8 Simultaneous Linear Differential Equations 1.9 particular Solutions by Variation of Parameters 1.10 Reduction of Order 1.11 Determination of Constants 1.12 Special Solvable Types of Nonlinear Equations 2. The Laplace Transform 2.1 An introductory Example 2.2 Definition and Existence of Laplace Transforms 2.3 Properties of Laplace Transforms 2.4 The Inverse Transform 2.5 The Convolution 2.6 Singularity Functions 2.7 Use of Table of Transforms 2.8 Applications to Linear Differential Equations with Constant Coefficients 2.9 The Gamma Function 3. Numerical Methods for Solving Ordinary Differential Equations 3.1 Introduction 3.2 Use of Taylor Series 3.3 The Adams Method 3.4 The Modified Adams Method 3.5 The Runge-Kutta Method 3.6 Picard's Method 3.7 Extrapolation with Differences 4. Series Solutions of Differential Equations: Special Functions 4.1 Properties of Power Series 4.2 Illustrative Examples 4.3 Singular Points of Linear Second-Order Differential Equations 4.4 The Method of Frobenius 4.5 Treatment of Exceptional Cases 4.6 Example of an Exceptional Case 4.7 A Particular Class of Equations 4.8 Bessel Functions 4.9 Properties of Bessel Functions 4.10 Differential Equations Satisfied by Bessel Functions 4.11 Ber and Bei Functions 4.12 Legendre Functions 4.13 The Hypergeometric Function 4.14 Series Solutions Valid for Large Values of x 5. Boundary-Value Problems and Characteristic-Function Representations 5.1 Introduction 5.2 The Rotating String 5.3 The Rotating Shaft 5.4 Buckling of Long Columns Under Axial Loads 5.5 The Method of Stodola and Vianello 5.6 Orthogonality of Characteristic Functions 5.7 Expansion of Arbitrary Functions in Series of Orthogonal Functions 5.8 Boundary-Value Problems Involving Nonhomogeneous Differential Equations 5.9 Convergence of the Method of Stodola and Vianello 5.10 Fourier Sine Series and Cosine Series 5.11 Complete Fourier Series 5.12 Term-by-Term Differentiation of Fourier Series 5.13 Fourier-Bessel Series 5.14 Legendre Series 5.15 The Fourier Integral 6. Vector Analysis 6.1 Elementary Properties of Vectors 6.2 The Scalar Product of Two Vectors 6.3 The Vector Product of Two Vectors 6.4 Multiple Products 6.5 Differentiation of Vectors 6.6 Geometry of a Space Curve 6.7 The Gradient Vector 6.8 The Vector Operator V 6.9 Differentiation Formulas 6.10 Line Integrals 6.11 The Potential Function 6.12 Surface Integrals 6.13 Interpretation of Divergence. The Divergence Theorem 6.14 Green's Theorem 6.15 Interpretation of Curl. Laplace's Equation 6.16 Stokes's Theorem 6.17 Orthogonal Curvilinear Coordinates 6.18 Special Coordinate Systems 6.19 Application to Two-Dimensional Incompressible Fluid Flow 6.20 Compressible Ideal Fluid Flow 7. Topics in Higher-Dimensional Calculus 7.1 Partial Differentiation. Chain Rules 7.2 Implicit Functions. Jacobian Determinants 7.3 Functional Dependence 7.4 Jacobians and Curvilinear Coordinates. Change of Variables in Integrals 7.5 Taylor Series 7.6 Maxima and Minima 7.7 Constraints and Lagrange Multipliers 7.8 Calculus of Variations 7.9 Differentiation of Integrals Involving a Parameter 7.10 Newton's Iterative Method 8. Partial Differential Equations 8.1 Definitions and Examples 8.2 The Quasi-Linear Equation of First Order 8.3 Special Devices. Initial Conditions 8.4 Linear and Quasi-Linear Equations of Second Order 8.5 Special Linear Equations of Second Order, with Constant Coefficients 8.6 Other Linear Equations 8.7 Characteristics of Linear First-Order Equations 8.8 Characteristics of Linear Second-Order Equations 8.9 Singular Curves on Integral Surfaces 8.10 Remarks on Linear Second-Order Initial-Value Problems 8.11 The Characteristics of a Particular Quasi-Linear Problem 9. Solutions of Partial Differential Equations of Mathematical Physics 9.1 Introduction 9.2 Heat Flow 9.3 Steady-State Temperature Distribution in a Rectangular Plate 9.4 Steady-State Temperature Distribution in a Circular Annulus 9.5 Poisson's Integral 9.6 Axisymmetrical Temperature Distribution in a Solid Sphere 9.7 Temperature Distribution in a Rectangular Parallelepiped 9.8 Ideal Fluid Flow about a Sphere 9.9 The Wave Equation. Vibration of a Circular Membrane 9.10 The Heat-Flow Equation. Heat Flow in a Rod 9.11 Duhamel's Superposition Integral 9.12 Traveling Waves 9.13 The Pulsating Cylinder 9.14 Examples of the Use of Fourier Integrals 9.15 Laplace Transform Methods 9.16 Application of the Laplace Transform to the Telegraph Equations for a Long Line 9.17 Nonhomogeneous Conditions. The Method of Variation of Parameters 9.18 Formulation of Problems 9.19 Supersonic Flow of ldeal Compressible Fluid Past an Obstacle 10. Functions of a Complex Variable 10.1 Introduction. The Complex Variable 10.2 Elementary Functions of a Complex Variable 10.3 Other Elementary Functions 10.4 Analytic Functions of a Complex Variable 10.5 Line Integrals of Complex Functions 10.6 Cauchy's Integral Formula 10.7 Taylor Series 10.8 Laurent Series 10.9 Singularities of Analytic Functions 10.10 Singularities at Infinity 10.11 Significance of Singularities 10.12 Residues 10.13 Evaluation of Real Definite Integrals 10.14 Theorems on Limiting Contours 10.15 Indented Contours 10.16 Integrals Involving Branch Points 11. Applications of Analytic Function Theory 11.1 Introduction 11.2 Inversion of Laplace Transforms 11.3 Inversion of Laplace Transforms with Branch Points. The Loop Integral 11.4 Conformal Mapping 11.5 Applications to Two-Dimensional Fluid Flow 11.6 Basic Flows 11.7 Other Applications of Conformal Mapping 11.8 The Schwarz-Christoffel Transformation 11.9 Green's Functions and the Dirichlet Problem 11.10 The Use of Conformal Mapping 11.11 Other Two-Dimensional Green's Functions Answers to Problems Index Contents

Mathematical Methods of Physics

Abstract: 1. Ordinary Differential Equations. 2. Infinite Series. 3. Evaluation of Integrals. 4. Integral Transforms. 5. Further Applications of Complex Variable. 6. Vectors and Matrices. 7. Special Functions. 8. Partial Differential Equations. 9. Eigenfuctions, Eigenvalues, and Green's Functions. 10. Peturbation Theory. 11. Integral Equations. 12. Calculus of Variations. 13. Numerical Methods. 14. Probability and Statistics. 15. Tensor Analysis and Differential Geometry. 16. Introduction to Groups and Group Representations. Appendix: Some Properties of Functions of a Complex Variable. Bibliography. Index.

Shape from shading: a method for obtaining the shape of a smooth opaque object from one view

Abstract: A method will be described for finding the shape of a smooth opaque object from a monocular image, given a knowledge of the surface photometry, the position of the light-source and certain auxiliary information to resolve ambiguities This method is complementary to the use of stereoscopy which relies on matching up sharp detail and will fail on smooth objects Until now the image processing of single views has been restricted to objects which can meaningfully be considered two-dimensional or bounded by plane surfaces It is possible to derive a first-order non-linear partial differential equation in two unknowns relating the intensity at the image points to the shape of the object This equation can be solved by means of an equivalent set of five ordinary differential equations A curve traced out by solving this set of equations for one set of starting values is called a characteristic strip Starting one of these strips from each point on some initial curve will produce the whole solution surface The initial curves can usually be constructed around so-called singular points A number of applications of this method will be discussed including one to lunar topography and one to the scanning electron microscope In both of these cases great simplifications occur in the equations A note on polyhedra follows and a quantitative theory of facial make-up is touched upon An implementation of some of these ideas on the PDP-6 computer with its attached image-dissector camera at the Artificial Intelligence Laboratory will be described, and also a nose-recognition program

Smoothing, filtering, and boundary effects

Abstract: Numerical integrations of finite-difference analogs of systems of nonlinear partial differential equations, such as those arising in atmospheric dynamics, are subject to computational instability from a variety of causes. One type of instability is produced by a spurious, nonlinear growth of high-frequency components that may be introduced by roundoff, truncation, and observational error. This type of instability, first discussed by N. A. Phillips, can be suppressed by a suitable choice of finite-difference method or by the use of a filter that selectively damps the high-frequency components. Though much effort is being devoted to the development of stable finite-difference procedures, and considerable success has been achieved, all such methods involve high-frequency smoothing either implicitly or explicitly. It is therefore important that the effects of such filtering be fully understood. Filtering and smoothing operators are developed for use in conjunction with the numerical integration of nonlinear systems and for other purposes. The general procedure is demonstrated for simple one-dimensional operators and the properties of such operators are thoroughly explored. The development is then expanded to allow for compound operators designed to suit some particular requirement and further extended to more than one dimension. Both real and complex operators are discussed. Reverse smoothers or wave amplifiers are introduced, and some of the problems associated with their use are discussed. A general procedure is outlined for the construction of an ‘ideal’ low-pass filter; that is, a filter that removes the shortest resolvable wave component (the 2-grid-interval wave) but restores all other wave components as close as is desired to their original amplitudes without amplifying or changing the phase of any wave component. Finally, the effects, sometimes disastrous, of finite domains on the properties of the smoothing operators are explored for a variety of common boundary assumptions.

The direct solution of the discrete Poisson equation on irregular regions

Abstract: There are several very fast direct methods which can be used to solve the discrete Poisson equation on rectangular domains. We show that these methods can also be used to treat problems on irregular regions.

Pseudoparabolic Partial Differential Equations

Abstract: This is the publisher’s final pdf The published article is copyrighted by the Society for Industrial and Applied Mathematics and can be found at: http://epubssiamorg/loi/sjmaah

The Direct Solution of the Discrete Poisson Equation on a Rectangle

Abstract: where G is a rectangle, Au = 82u/8x2 + 82u/8y2, and v, w are known functions. For computational purposes, this partial differential equation is frequently replaced by a finite difference analogue. These discrete models for (1) consist of linear systems of equations of very large dimension, and it is widely recognized that the usual direct methods (e.g., Gaussian elimination) are unsatisfactory for such systems [18, ?? 21.2-21.3]. Theoretical investigation has, therefore, been primarily directed toward the development of effective iterative methods for the solution of these problems [64], [66]. In recent years, however, direct methods that take advantage of the special block structure of these linear equations have appeared. For the rectangular regions under consideration, these methods can be considerably faster than iterative methods. The purpose of this survey paper is to provide brief summaries and a list of references for methods which can be used to directly solve the finite difference equations. Some of these methods can be applied to problems in more general domains. However, the extensions generally include only simple rectilinear regions, such as L-shaped or T-shaped domains. This is basically due to the fact that the direct methods require a great degree of regularity in the block structure of the matrix equation. In our discussion, we will indicate whether the methods are easily adaptable to more general regions, and to more general elliptic partial differential equations.

Hopscotch: a Fast Second-order Partial Differential Equation Solver

Abstract: An idea of Gordon for the numerical solution of evolutionary problems is reformulated and shown to be equivalent to a Peaceman-Rachford process A fast computational process is then developed and applied to parabolic and elliptic problems, both linear and non-linear This algorithm is very efficient with regard to computing time, storage requirements and ease of programming Several fairly general conditions are given which ensure convergence for parabolic and elliptic problems

Nonlinear Controllability via Lie Theory

Abstract: Complete system associated with given control, discussing trajectories uniform approximation and nonlinear controllability conditions based on linear partial differential equation

Engineering Mathematics Handbook

Abstract: Algebra. Geometry. Trigonometry. Plane Analytic Geometry. Space Analytic Geometry. Elementary Functions. Differential Calculus. Sequences and Series. Integral Calculus. Vector Analysis. Functions of a Complex Variable. Fourier Series. Higher Transcendent Functions. Ordinary Differential Equations. Partial Differential Equations. Laplace Transforms. Numerical Methods. Probability and Statistics. Table of Indefinite Integrals. Tables of Definite Integrals. Plane Curves and Areas. Space Curves and Surfaces. Appendices: A: Numerical Tables. B: Glossary of Symbols and Mathematics Terms. C: Units of Measurement and Conversions Between International and U. S. Customary Systems. D: Sample Problems, Mathematics Programs, and Modern Handheld Calculators. E: References and Bibliography.

Stability of Spatially Periodic Supercritical Flows in Hydrodynamics

Abstract: Recently, Eckhaus developed a theory for a class of nonlinear stability problems which can be formulated in terms of a scalar partial differential equation with quadratic nonlinearities. It is demonstrated that Eckhaus' work on the development and stability of periodic solutions can be extended to a class of nonlinear matrix partial differential equations. The equations governing axisymmetric viscous flow between concentric rotating cylinders belong to the class of equations considered. When the Taylor number T is slightly above the minimum critical value Tc there exists an interval of possible equilibrium flows (Taylor‐vortex flows) growing out of the instability of Couette flow. It is shown that within the interval of possible Taylor‐vortex flows, there exists a subinterval of stable vortex flows.

On a perturbation theory using Lie transforms

Abstract: Kamel has recently extended to non-Hamiltonian equations a perturbation theory using Lie transforms. We show here how Kamel's extension can be approached from an intrinsic viewpoint, which reformulation leads to a simpler algorithm. Then we complete Kamel's contribution by establishing the rules for inverting the transformation generated by the perturbation theory, and for composing two such transformations.

Pitfalls in computation, or why a math book isn''t enough

Abstract: The floating-point number system is contrasted with the real numbers. The author then illustrates the variety of computational pitfalls a person can fall into who merely translates information gained from pure mathematics courses into computer programs. Examples include summing a Taylor series, solving a quadratic equation, solving linear algebraic systems, solving ordinary and partial differential equations, and finding polynomial zeros. It is concluded that mathematics courses should be taught with a greater awareness of automatic computation.

A fast, approximative method for integrating the stochastic coalescence equation

Abstract: The use of mean values of the drop number density in the coalescence equation is shown to lead to a system of ordinary differential equations that provide good approximative solutions with a minimum of computational effort. The coefficients in this system of equations are functions of the kernel and can be obtained by simple quadrature. Comparisons are made between numerical and analytic solutions for two types of kernels (constant and sum of arguments), using a Gaussian initial distribution of cloud droplets. Systematic deviations of the numerical results from corresponding analytic solutions are shown to depend on the choice of variables, and methods to suppress these deviations are discussed.

Steady Infiltration from Line Sources and Furrows

Abstract: Steady infiltration from an array of equally spaced line sources or furrows at the surface of a semi-infinite soil profile is analyzed. The discussion is based on the assumption that the hydraulic conductivity is an exponential function of the pressure head. It is shown that, under this assumption, the matric flux potential and the stream function for plane flows satisfy the same linear partial differential equation. Explicit expressions for the stream function, the flux, the matric flux potential, the pressure head, and the total head are obtained. Some implications with regard to furrow irrigation are discussed. The solution provides a rational basis for the discussion of leaching under furrow irrigation. Additional

Nonlinear Vibrations of a Beam Under Harmonic Excitation

Abstract: A straight beam with fixed ends, excited by the periodic motion of its supporting base in a direction normal to the beam span, was investigated analytically and experimentally. By using Galerkin’s method (one mode approximation) the governing partial differential equation reduces to the well-known Duffing equation. The harmonic balance method is applied to solve the Duffing equation. Besides the solution of simple harmonic motion (SHM), many other branch solutions, involving superharmonic motion (SPHM) and subharmonic motion (SBHM), are found experimentally and analytically. The stability problem is analyzed by solving a corresponding variational Hill-type equation. The results of the present analysis agree well with the experiments.

On the Control of a Linear Functional Differential Equation with Quadratic Cost

Abstract: Riccati-like linear functional differential equation with quadratic cost, analyzing feedback control solution existence and uniqueness

A finite element solution of the one-dimensional diffusion-convection equation

Abstract: The one-dimensional diffusion-convection equation has been widely used to describe approximately the transient motion of a subset of particles in river flow or porous media flow. An equivalent variational principle to the governing partial differential equations of motion is given, and a finite element solution is developed requiring only approximations in the space domain. The solution is applicable to a wide variety of field problems because it can account for a variety of boundary conditions. Additionally, the solution is not dependent upon constant parameters of motion over the entire domain of interest.

Analytic Approach to the Theory of Phase Transitions

Abstract: An approximation to the first equation of the Kirkwood coupling parameter hierarchy and other model equations for the singlet distribution function are cast into the standard Hammerstein form of nonlinear integral equation. We give a criterion for the existence and uniqueness of solutions of this equation involving the first negative eigenvalue of the kernel, which allows us to establish temperatures and densities where the solution is unique. Multiple solutions of the nonlinear equation are associated with instability of the single phase and thus signal a phase transition. A necessary condition for the existence of other solutions of small norm is given by a bifurcation equation. These new solutions are associated with the freezing transition, and the periodic singlet density of the solid falls naturally out of the theory. The bifurcation equation can be related to the Kirkwood instability criterion, but, in contrast to this, predicts no transition for a system of hard rods when a model kernel is used. T...

PDEL—a language for partial differential equations

Abstract: Conventional computer methods available to solve continuous system problems characterized by partial differential equations are very time-consuming and cumbersome. A convenient, easy to learn and to use, high level problem oriented language to solve and study partial differential equation problems has been designed; a practical translator for the language has also been designed, and a working version of it has been constructed for a significant portion of the language. This Partial Differential Equation Language, PDEL, is outlined, and the highlights of the translator are briefly summarized.

Relativistic Systems of Interacting Particles

Abstract: A multitemporal non-hereditary formulation of relativistic dynamics is given. According to it, a dynamical system of N particles involves some realization of the N-parameter abelian group. The usual concept of flow is replaced by more general tools coming from bitensor calculus. The basic axioms of the theory are expressed in terms of partial differential equations and a non-trivial solution is exhibited.