Showing papers by "George F. Carrier published in 1976"
Book•
01 Jan 1976
TL;DR: The Diffusion Equation as discussed by the authors is a generalization of the Wave Equation, which is used in the Laplace Transform Methods (LTLM) and Green's Functions.
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.
154 citations
01 Jan 1976
TL;DR: In this article, the Laplacian equations are used to divert the gradient of a function and the mathematical operations that divert a function's gradient are called LaPlacian operations.
Abstract: This chapter explains the potential equation. In mathematics, Poission's equation is a partial differential equation of elliptic type with broad utility in electrostatics, mechanical engineering, and theoretical physics. The mathematical operations that divert the gradient of a function are called the LaPlacian equations. The chapter considers the functions φ(x, y) that satisfy Poisson's equation. The temperature φ is a function of x and y only. The real world rarely has discontinuities—like apparent sharp corners or sudden transitions from, say, a conductor at one potential to a conductor at another; therefore, potential become blurred when examined microscopically. However, it is often convenient to simplify the exact boundary conditions, in which there are narrow transition regions, by permitting discontinuities in boundary data.
11 citations
01 Jan 1976
TL;DR: In this article, a boundary layer method is used to analyze the behavior of the solution near a caustic, i.e., near a ray envelope, and the asymptotic analysis of wave motion.
Abstract: A boundary value problem contains a small parameter for which the perturbation methods are inadequate. This chapter discusses the use of a boundary layer method to analyze the behavior of the solution near a caustic, that is, near a ray envelope. In an ordinary differential equation, the formal use of two or more naturally occurring scales in a partial differential equation problem can lead to a perturbation expansion that is valid over a larger range of an independent variable. Some singular perturbation problems in linear differential equations can be solved using the exponential method. The chapter presents the asymptotic analysis of wave motion.
3 citations
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TL;DR: In this article, the authors explain the wave equation with respect to the second derivative of the diffusion equation, where a second derivative is added to the first derivative to alter the solution behavior.
Abstract: This chapter explains the wave equation. A simple form of this equation is u tt = c 2 u xx , where c(x) is some positive function of x. In physical aplications, x frequently represents position and t time. In comparison with the diffusion equation, there is a second derivative with respect to t instead of a first derivative; this change markedly alters the solution behavior. To better understand wave equation, a tightly stretched string of negligible thickness that in an undisturbed state is coincident with the x-axis should be considered. In response to external disturbances, the string can deflect in the (x, y) plane. The displacement of a string particle in the y direction can be denoted by u(x, t).
1 citations