Laplace's Equation

10 Key excerpts on "Laplace's Equation"

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  eBook - ePub

  Models and Modeling

  An Introduction for Earth and Environmental Scientists

  • Jerry P. Fairley(Author)
  • 2017(Publication Date)
  • Wiley-Blackwell

    (Publisher)

  CHAPTER 4 The Laplace equation Chapter summary The Laplace equation may very well be the single most important equation in mathematical physics. In groundwater studies, it describes the steady flow of groundwater in a domain with no sources or sinks, where all the driving potential is located on the boundaries of the domain. Although it may sound somewhat limited in application, it is a commonly employed conceptual representation of groundwater flow; furthermore, the Laplace equation is a limiting case of several other important field equations we will see in future chapters. In this chapter, we will apply the Laplace equation to understand the configuration of the water table below an idealized region, located somewhere between Heaven and Hell. 4.1 Laplace’s equation In previous chapters, we have been concerned with modeling processes that vary over time. Spatial variability did not enter into our considerations; to the extent that spatial variability was a part of our problems, we glossed over it, using averaged or effective properties to represent the model space. Beginning with this chapter, we will turn our attention to fields that vary in space—for example, the distribution of head in an aquifer, or the distribution of temperature in a steel plate. In this chapter, we will confine ourselves to steady‐state processes lacking source/sink terms. These types of problems belong to the class of systems described by the Laplace equation. The Marquis Pierre‐Simon de Laplace (1749–1827) was a famous French mathematician, and author of the Méchanique Céleste. Laplace has been credited with making important contributions to many areas of mathematics and physics, including potential theory, the theory of probability, and celestial mechanics, as well as being possibly the first person to propose what we would call a “black hole,” by suggesting that a sufficiently massive star might have a gravitational field so great that even light could not escape from its surface

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  Differential Equations with Boundary-Value Problems, International Metric Edition

  • Dennis Zill(Author)
  • 2017(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  544 15.1 Laplace’s Equation 15.2 Heat Equation 15.3 Wave Equation C H A P T E R 15 I N R E V I E W W e saw in Section 9.5 that one way of approximating a solution of a second-order boundary-value problem was to work with a finite difference equation replacement of the linear ordinary differential equation. The difference equation was constructed by replacing the ordinary derivatives d 2 y y dx 2 and dy y dx by difference quotients. We will see in this chapter that the same idea carries over to boundary-value problems involving linear partial differential equations. 15 Sdecoret/Shutterstock.com Numerical Solutions of Partial Differential Equations Copyright 2018 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. WCN 02-300 15.1 LAPLACE’S EQUATION 545 INTRODUCTION In Section 12.1 we saw that linear second-order PDEs in two independent variables are classified as elliptic , parabolic , and hyperbolic . Roughly, elliptic PDEs involve partial derivatives with respect to spatial variables only, and as a consequence solutions of such equations are determined by boundary conditions alone. Parabolic and hyperbolic equations involve partial derivatives with respect to both spatial and time variables, so solutions of such equations generally are determined from boundary and initial conditions. A solution of an elliptic PDE (such as Laplace’s equation) can describe a physical system whose state is in equilibrium (steady-state); a solution of a parabolic PDE (such as the heat equation) can describe a diffusional state, whereas a hyperbolic PDE (such as the wave equation) can describe a vibrational state. In this section we begin our discussion with approximation methods that are appropriate for elliptic equations. Our focus will be on the simplest but probably the most important PDE of the elliptic type: Laplace’s equation.

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  A Course of Mathematics for Engineerings and Scientists

  Volume 4

  • Brian H. Chirgwin, Charles Plumpton(Authors)
  • 2013(Publication Date)
  • Pergamon

    (Publisher)

  C H A P T E R I I T H E S O L U T I O N OF S O M E D I F F E R E N T I A L E Q U A T I O N S 2:1 Laplace's Equation in two and three dimensions Laplace's Equation, V 2 / = 0, introduced in Chapter I takes the forms in cartesian coordinates of two and three dimensions respectively. La-place's equation occurs frequently in physics, and the operator V 2 occurs in the equation of heat conduction, the wave equation, and in other contexts. In Vol. II Chapter II we introduced the method of solution of a partial differential equation by separation of variables and illustrated there how the boundary conditions often suggest the ap-propriate coordinate systems to be used and the form of a separable solution. For example, / = sinao: coshat/, / = e* z sinZ# vosmy, (k 2 — I 2 + m 2 ) are, respectively, solutions of eqns. (2.1) and (2.2) in which the variables are separated. However, while cartesian coordinates are suitable for systems with rectangular boundaries other coordinate systems are usually more suit-able for problems involving boundaries of other shapes. In this section we consider the separation of Laplace's and similar equations in various systems of coordinates. We do not consider all possible cases, but indicate the type of solution which is found and leave the reader to 117 (2.1) (2.2) k > 0; Θ = a 2 6>, θ k = 0; Θ = o, Θ k < 0; 6> = -α 2 Θ , Θ 118 A C O U R S E O F M A T H E M A T I C S extend the method to other systems of coordinates and other partial differential equations. To express V 2 / in curvilinear coordinates we write V 2 / = div grad f and use the formulae of § 1:10. (1) Two dimensional polars (r, θ) We quote the general formula (1.99) but omit the third component (c. f. three-dimensional cylindrical polars below). (2.3) To separate the variables we write / = R(r)Q (θ) , where R is a function of r only and Θ a function of θ only, and divide through by the product R Θ .

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  Available until 16 Feb |Learn more

  Advanced Engineering Mathematics with MATLAB

  • Dean G. Duffy(Author)
  • 2021(Publication Date)
  • Chapman and Hall/CRC

    (Publisher)

  Chapter 10 Laplace’s Equation In the previous chapter we solved the one-dimensional heat equation. Quite often we found that the transient solution died away, leaving a steady state. The partial differential equation that describes the steady state for two-dimensional heat conduction is Laplace’s equation ∂ 2 u ∂ x 2 + ∂ 2 u ∂ y 2 = 0. (10.0.1) In general, this equation governs physical processes where equilibrium has been reached. It also serves as the prototype for a wider class of elliptic equations a (x, t) ∂ 2 u ∂ x 2 + b (x, t) ∂ 2 u ∂ x ∂ t + c (x, t) ∂ 2 u ∂ x 2 = f (x, t, u, ∂ u ∂ x, ∂ u ∂ t), (10.0.2) where b 2 < 4 ac. Unlike the heat and wave equations, there are no initial conditions and the boundary conditions completely specify the solution. In this chapter we present some of the common techniques for solving this equation. 10.1 DERIVATION OF LAPLACE’S EQUATION Imagine a thin, flat plate of heat-conducting material between two sheets of insulation. Sufficient time has passed so that the temperature depends only on the spatial coordinates x and y. Let us now apply the law of conservation of energy (in rate form) to a small rectangle with sides Δ x and Δ y. If q x (x, y) and q y (x, y) denote the heat flow rates in the x- and y -direction, respectively, conservation of energy requires that the heat flow into the slab equals the heat flow out of the slab if there is no storage or generation of heat. Now r a t e i n = q x (x, y + Δ y / 2) Δ y + q y (x + Δ x / 2, y) Δ x, (10.1.1) and r a t e o u t = q x (x + Δ x, y + Δ y / 2) Δ y + q y (x + Δ x / 2, y + Δ y)[--=. PLGO-SEPARATOR=--]Δ x. (10.1.2) If the plate has unit thickness, [ q x (x, y + Δ y / 2) − q x (x + Δ x, y + Δ y / 2) ] Δ y + [ q y (x + Δ x / 2, y) − q y (x + Δ x / 2, y + Δ y) ] Δ x = 0. (10.1.3) Upon dividing through by. Δ x Δ y, we obtain two differences quotients on the left side of Equation 10.1.3

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  Mathematical Methods in the Physical Sciences

  • Mary L. Boas(Author)
  • 2011(Publication Date)
  • Wiley

    (Publisher)

  C H A P T E R 13 Partial Differential Equations 1. INTRODUCTION Many of the problems of mathematical physics involve the solution of partial dif-ferential equations. The same partial differential equation may apply to a variety of physical problems; thus the mathematical methods which you will learn in this chapter apply to many more problems than those we shall discuss in the illustrative examples. Let us outline the partial differential equations we shall consider, and the kinds of physical problems which lead to each of them. Laplace’s equation ∇ 2 u = 0 (1.1) The function u may be the gravitational potential in a region containing no mass, the electrostatic potential in a charge-free region, the steady-state temperature (that is, temperature not changing with time) in a region containing no sources of heat, or the velocity potential for an incompressible fluid with no vortices and no sources or sinks. Poisson’s equation ∇ 2 u = f ( x, y, z ) (1.2) The function u may represent the same physical quantities listed for Laplace’s equation, but in a region containing mass, electric charge, or sources of heat or fluid, respectively, for the various cases. The function f ( x, y, z ) is called the source density; for example, in electricity it is proportional to the density of the electric charge. The diffusion or heat flow equation ∇ 2 u = 1 α 2 ∂u ∂t (1.3) Here u may be the non-steady-state temperature (that is, temperature varying with time) in a region with no heat sources; or it may be the concentration of a diffusing substance (for example, a chemical, or particles such as neutrons). The quantity α 2 is a constant known as the diffusivity. Wave equation ∇ 2 u = 1 v 2 ∂ 2 u ∂t 2 (1.4) 619

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  eBook - ePub

  Classical Electromagnetic Radiation

  • Jerry Marion(Author)
  • 2012(Publication Date)
  • Academic Press

    (Publisher)

  CHAPTER 3

  The Equations of Laplace and Poisson

  Publisher Summary

  The Laplacian operator occurs in many different types of physical problems probably the most important of which is that of wave propagation. This chapter discusses some of the mathematical functions that arise in the solution of wave equations that are the same as those that result from the solution of Laplace’s equation. It discusses some of the important properties of harmonic functions—that is, functions that satisfy Laplace’s equations. The chapter discusses the solutions of Laplace’s equation in rectangular coordinates, in spherical coordinates, and in cylindrical coordinates. It describes the parallel-plate diode that is a simple example of Poisson’s equation.

  3.1 Introduction

  In Chapter 1 we found that the general problem of the electrostatic field is described by Poisson’s equation, Eq. (1.9) :

  (3.1a)

  In regions not containing charge, this reduces to Laplace’s equation:

  (3.1b)

  The Laplacian operator occurs in many different types of physical problems, * probably the most important of which is that of wave propagation. Although we are interested in this book primarily in electromagnetic wave phenomena rather than in electrostatics, some of the mathematical functions which arise in the solution of wave equations are the same as those that result from the solution of Laplace’s equation. It is somewhat easier to introduce these harmonic functions (Legendre functions, spherical harmonics, and Bessel functions) in connection with electrostatic problems.

  We shall study such problems in some detail in order to become familiar with the functions that will be of use later in discussions of radiation phenomena. This will be the extent of the treatment of electrostatics; we will not discuss the method of images nor the use of conjugate functions in the solution of problems in electrostatics. The interested reader is referred to the list of Suggested References for sources of such material.

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  Advanced Engineering Mathematics, SI Edition

  • Peter O'Neil(Author)
  • 2017(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  Copyright 2018 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. WCN 02-300 234 CHAPTER 7 Laplace’s Equation Math in Context—Common Capacitor Designs (a) (b) + + + + + + + + + + + + – + – + – + – + – + – d V a – Q A + Q E V b Schematic of Parallel Plate Capacitor One of the most common capacitor designs uses two (or more) parallel plates. This corresponds to the Dirichlet and Neumann problems on the rectangle discussed in this chapter. Another common configuration is arranging the two conducting plates as concentric cylinders with the dielectric in the middle. The one-dimensional cylin-drical case for unbounded mediums can be solved by a trivial double integration to yield a logarithmic poten-tial function. However, solutions in higher dimensions are quite complicated and are expressed in terms of special Bessel functions. Much effort has been expended to find methods to determine the potential function in an electronic com-ponent. Engineers have developed general solutions for the Laplace equation in over two dozen coordinate systems, including rectangular, spherical, oblate, and other geometries. For more complicated geometries, which do not entertain exact analytical solu-tions, numerical methods are used to solve the Laplace equation for V . Section 7.6 Problems In each of Problems 1–3, solve the Neumann problem. 1. ∇ 2 u ( x , y ) = 0 for 0 < x < 1, 0 < y < 1, ∂ u ∂ y ( x , 0 ) = 4 cos (π x ) , ∂ u ∂ y ( x , 1 ) = 0 for 0 ≤ x ≤ 1, ∂ u ∂ x ( 0, y ) = ∂ u ∂ x ( 1, y ) = 0 for 0 ≤ y ≤ 1 2. ∇ 2 u ( x , y ) = 0 for 0 < x < 1, 0 < y < π , ∂ u ∂ y ( x , 0 ) = ∂ u ∂ y ( x , π) = 0 for 0 ≤ x ≤ 1, ∂ u ∂ x ( 0, y ) = y − π 2 , ∂ u ∂ x (π , y ) = cos ( y ) for 0 ≤ y ≤ π 3. ∇ 2 u ( x , y ) = 0 for 0 < x < π , 0 < y < π , ∂ u ∂ y ( x , 0 ) = cos ( 3 x ) for 0 ≤ x ≤ π ∂ u ∂ y ( x , π) = 6 x − 3 π for 0 ≤ x ≤ π ∂ u ∂ x ( 0, y ) = ∂ u ∂ x (π , y ) = 0 for 0 ≤ y ≤ π 4.

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  Mathematical Methods for Physicists

  A Concise Introduction

  • Tai L. Chow(Author)
  • 2000(Publication Date)
  • Cambridge University Press

    (Publisher)

  Thus, we have Gauss’ law in di€erential form: r E ˆ 4 C : Substituting this into the equation r 2 ˆ r E , we get r 2 ˆ 4 C ; which is Poisson’s equation. In the Gaussian system of units, C ˆ 1; in the SI system of units, C ˆ 1 = 4 0 , where the constant 0 is known as the permittivity of free space. If we use SI units, then r 2 ˆ = 0 : 391 LINEAR SECOND-ORDER PDEs In the particular case of zero charge density it reduces to Laplace’s equation, r 2 ˆ 0 : In the following sections, we shall consider a number of problems to illustrate some useful methods of solving linear partial di€erential equations. There are many methods by which homogeneous linear equations with constant coecients can be solved. The following are commonly used in the applications. (1) General solutions: In this method we first find the general solution and then that particular solution which satisfies the boundary conditions. It is always satisfying from the point of view of a mathematician to be able to find general solutions of partial di€erential equations; however, general solutions are dicult to find and such solutions are sometimes of little value when given boundary conditions are to be imposed on the solution. To overcome this diculty it is best to find a less general type of solution which is satisfied by the type of boundary conditions to be imposed. This is the method of separation of variables. (2) Separation of variables: The method of separation of variables makes use of the principle of superposition in building up a linear combination of individual solutions to form a solution satisfying the boundary conditions. The basic approach of this method in attempting to solve a di€erential equation (in, say, two dependent variables x and y ) is to write the dependent variable u x ; y † as a product of functions of the separate variables u x ; y † ˆ X x † Y y † . In many cases the partial di€erential equation reduces to ordinary di€erential equations for X and Y .

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  Mathematical Analysis tools for engineering

  • Franco Tomarelli(Author)
  • 2019(Publication Date)
  • Società Editrice Esculapio

    (Publisher)

  In order to get familiarity with terminology, in the sequel of present Sub-section 2.1.1 we introduce a wide list of PDEs which play a crucial role in Engineering and Mathematical Physics. Starting form Section 2.2, we intro-duce some toools for solving meaningful associated problems. 2.1 Partial Di ↵ erential Equations 73 Examples of homogeneous linear partial di ↵ erential equations (here the physical units are normalized): 1. -Δ u = 0 Laplace’s equation 2. -Δ u + u = 0 elliptic equation 3. -Δ u = λ u Helmoltz’s (or eigenvalue) equation 4. u t -Δ u = 0 heat (or di ↵ usion) equation 5. u tt -Δ u = 0 wave equation 6. u t -i Δ u = 0 Schr¨ odinger’s equation 7. u t + b · Du = 0 linear transport equation 8. u t -Δ u + b · Du = 0 drift-di ↵ usion equation 9. u tt + u t -u xx = 0 telegraph equation 10. u tt + u xxxx = 0 beam equation . If the units are not normalized and the medium is neither homogeneous nor isotropic, then -Δ u has to be replaced in the equations above by -div x ⇣ A ( x ) D x ⌘ = -n X i,j =1 @ @ x i ✓ A i,j ( x ) @ u @ x j ◆ , (2.2) where A is an n ⇥ n matrix describing the property of the medium. The Laplace operator -Δ is a special case of (2.2), when A i,j ( x ) = I . Examples of nonlinear partial di ↵ erential equations: 1. u t + u u x = 0 (inviscid) Burger’s equation 2. u t + div F ( u ) = 0 (scalar) conservation law 3. u t -Δ ( u γ ) = 0 porous medium equation (or nonlinear di ↵ usion equation) 4. u t -Δ u = f ( u ) (scalar) reaction-di ↵ usion equation 5. u tt -Δ u = f ( u ) nonlinear wave equation 6. det ( D 2 u ) = f Monge-Amp` ere equation 7. u t + u u x + u xxx = 0 Korteweg-deVries (KdV) equation 74 2 Partial Di ↵ erential Equations Next we introduce some terminology and notation. Definition 2.1. We classify the partial di ↵ erential equations, in the case of a single PDE with a scalar unknown u , as follows.

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  Introduction to Modeling and Simulation

  A Systems Approach

  • Mark W. Spong(Author)
  • 2023(Publication Date)
  • Wiley

    (Publisher)

  Under these conditions, it is now straight- forward to determine that C = 0 and λ = n L . Therefore, the solution is u(x, t) = ∞ ∑ n=1 B n sin ( n L x ) e − n 2  2  L 2 t , where B n = 2 L ∫ L 0 f (x) sin ( n L x ) dx. Example 12.5 (Adapted from Duffy [20]) Suppose we take L =  and f (x) = x( − x). Then a straightforward calculation shows B n = 2  ∫  0 x( − x) sin(nx)dx = 2 ∫  0 x sin(nx)dx − 2  ∫  0 x 2 sin(nx)dx = 4 1 − (−1) n n 3  , and so the solution becomes u(x, t) = 8  ∞ ∑ k=1 1 (2k − 1) 3 sin((2k − 1)x) exp −(2k − 1) 2 a 2 t. Figure 12.8 shows the temperature profile along the bar over time with a = 1. 12.4 Laplace’s Equation 313 Figure 12.8 The temperature u(x, t ) is a thin bar with both ends held at zero temperature and initial temperature distribution given by x( − x). The temperature converges exponentially to zero over time. 2.5 2 1.5 1 0.5 0 0 0.5 1 1.5 2 2.5 3 3.5 12.4 Laplace’s Equation The third equation we consider is Laplace’s equation  2 u x 2 +  2 u y 2 = 0. Laplace’s equation arises in numerous applications. First, we can see that it repre- sents the steady-state heat equation in two spatial dimensions, i.e., with u t = 0. Laplace’s equation also arises in problems in electrostatics, gravitation, and fluid dynamics, as well as in image processing. Solutions of Laplace’s equation are known as harmonic functions. The general theory of solutions to Laplace’s equation is known as potential theory, and is a vast subject in its own right [42]. Laplace’s Equation on a Rectangle Let us consider the following problem. Find a function u(x, y) satisfying Laplace’s equation on a rectangle with sides of length L and K with the boundary conditions u(x, 0) = f 1 (x), u(x, K) = f 2 (x), 0 ≤ x ≤ L u(0, y) = g 1 (y), u(L, y) = g 2 (y), 0 ≤ y ≤ K, as illustrated in Figure 12.9.

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