aaa Partial Differential Equations in Physics

1. Auflage 2012
ISBN: 978-0-323-15496-3
Verlag: Elsevier Science & Techn.
Format: EPUB
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 346 Seiten

Partial Differential Equations in Physics

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CHAPTER II Introduction to Partial Differential Equations
Publisher Summary
This chapter provides an introduction to partial differential equations. It describes how the simplest partial differential equations arise and additionally provides ample examples. The chapter gives the definition of the Laplace operator. It also gives the expression of the field action approach. The chapter provides the Fourier law that states that for an isotropic medium, the flow of heat is in the direction of decreasing temperature and is proportional to the rate of this decrease. It further presents the case of diffusion to which Fourier’s law is adapted. The chapter illustrates hyperbolic, elliptic, and parabolic differential equations, and focuses on their differences. It additionally illustrates Green’s theorem and Green’s function for linear and elliptic differential equations. The chapter provides the definition of a unit source and of the principle solution. The analytic character of the solution of an elliptical differential equation is discussed in the chapter. The definition of Green’s function for self-adjoint differential equation is also given in the chapter. The chapter further illustrates Riemann’s integration of the hyperbolic differential equation. It focuses on Green’s theorem in heat conduction and explains that heat expands with infinite velocity. §7 How the Simplest Partial Differential Equations Arise
The potential equation (1) or (1a) is known in the theory of gravitation as the expression of the field-action approach, as opposed to the action-at-a-distance approach of Newton. The Laplace operator is defined as (2) The same equations (1) and (1a) are fundamental for eloctrostatic and magnetic fields, (1) in empty space, (1a) in the presence of a source of density the factor 4p in (1a) has been put in parentheses since it can be removed by a proper choice of units. Equation (1) appears also in the hydrodynamics of incompressible and irrotational fluids, u standing for the velocity potential. We also mention the two-dimensional potential equation (3) as the basis of Riemannian function theory, which we may characterize as the “field theory” of the analytic functions f (x + iy). Equally well known is the wave equation (4) It is fundamental in acoustics (c = velocity of sound). It is also fundamental in the electrodynamics of variable fields (c = velocity of light), and therefore in optics. In the special theory of relativity one may write (4) as the four-dimensional potential equation (5) by introducing the fourth coordinate x4 (or x0) = ict in addition to the three spatial coordinates x1,x2,x3. For an oscillating membrane we have (4) with two spatial dimensions, for an oscillating string we have one spatial dimension. In the latter case we write (6) (6a) setting, for the time being, y = ct (not y = ict). Neither membrane nor string has a proper elasticity; the constant c is computed from the tension imposed from outside and from the density per unit of area or of length. In the general theory of elasticity one has, as a special case, the differential equation for the transverse vibrations of a thin disc (7) for reasons of dimensionality c here does not stand for the velocity of sound in the elastic material, as it does in acoustics, but is computed from the elasticity, density, and thickness of the disc. Analogously, the differential equation of an oscillating elastic rod is (8) This will be derived in exercise II.1, where the resulting characteristic frequencies will be compared with the acoustic frequencies of open and of covered pipes. As a third type we add to the differential equations of states of equilibrium ((1) to (3)), and of oscillating processes ((4) to (8)), those of equalization processes. As their chief representative we shall here consider heat conduction (equalization of energy differences). We remark, however, that diffusion (equalization of differences of material densities), fluid friction (equalization of impulse differences), and pure electric conduction (equalization of differences of potential), follow the same pattern. Let G be a vector of the magnitude and direction of the heat flow and let the initial point P be surrounded by an element of volume dt. Then div G dt is the outflow of heat energy from dt per unit of time. A decrease per unit of time in the amount of heat in dt, which we shall denote by —?Q/?t, corresponds to this. We then have (9) Our heat conductor is here considered to be a rigid body so that we can neglect expansion; heat content is then the same as energy content. Now every increase dQ in heat causes an increase in the temperature of dt, every decrease — dQ in heat causes a decrease in temperature. Denoting the temperature by u, we have (10) c being the specific heat (for a rigid body we need not distinguish between cv and cp). The factor dm is due to the fact that c is related to the unit of mass. From (9) and (10) we get (11) We now apply Fourier’s law, which determines the relation between G and u. It states that for an isotropic medium (12) the flow of heat is in the direction of decreasing temperature and is proportional to the rate of this decrease. The factor of proportionality x is called the heat conductivity. Introducing (12) in (11) we get the differential equation of heat conduction (13) k is called the temperature conductivity. Fourier’s law was adapted to the case of diffusion by the physiologist Fick. Here u stands for the concentration of dissolved matter in the solvent, G for the material flow of the dissolved matter, and k for the diffusion coefficient. In the case of inner friction of an incompressible fluid, k stands for the kinematic viscosity, and (13) is the Navier-Stokes equation for laminar flow (i.e., flow in a fixed direction). Owing to the tensor character of this process equation (12) has no general validity here. The analogue of Fourier’s law in the electric case is Ohm’s law. Here u stands for the potential, G for the specific electric current (the current per unit of area of the conductor), and k for the specific resistance of the conductor. Equation (13) is of the type of Maxwell’s equations in the case of pure Ohm conduction. Schrödinger’s equation of wave mechanics belongs formally to the same scheme, in particular in the force-free case, to which we restrict ourselves here: (14) However, owing to the fact that the real constant, k, of (13) is replaced here by the imaginary constant i/2m, equation (14) describes an oscillation rather than an equalization process. We see this in the passage to the case of periodicity in time, if we set. (14a) Then (14) becomes (15) This is the same form as we would obtain from the wave equation (4) if we set u = ?. exp (—i?t) and let C = ?2/c2 The so-called case of linear heat conduction, with the thermal state depending on only one variable x, will be treated in detail in the following chapter. In order to compare its differential equation with (3) and (6a), we write it in the form: (16) Looking back on this sketchy survey one notices a family resemblance among the differential equations of physics. This stems from the invariance under rotation and translation, which must be demanded for the case of isotropic and homogeneous media. The differential operator of second order implied by this invariance is just the Laplace ?. In the case of space-time invariance of relativity this is replaced by the corresponding four-dimensional of (15). For the case of an anisotropic medium, ? must be replaced by a sum of all second derivatives with factors determined from the crystal constants. For the case of an inhomogeneous medium these factors will also be functions of position. We shall deal with such generalized differential expressions in the beginning of the next section. The fact that we are dealing...