Two Dimensional Laplace Equation

12 Key excerpts on "Two Dimensional Laplace 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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  A Practical Guide to Boundary Element Methods with the Software Library BEMLIB

  • C. Pozrikidis(Author)
  • 2002(Publication Date)
  • CRC Press

    (Publisher)

  Laplace’s equation in two dimensions 11 (a) C A A c c C (b) n n n n n Figure 2.1.1 A control area in the plane enclosed by (a) a closed line, and (b) a collection of closed lines. Problem P.2.1.1 Kirchhoff transformation Laplace’s equation describes the steady-state distribution of temperature in a do-main occupied by a homogeneous material with uniform thermal conductivity . If the thermal conductivity is not constant but varies with position over the medium, then the temperature distribution is described by the generalized equation (1) which arises by balancing the thermal fluxes expressed by Fick’s law over an in-finitesimal area. Consider heat conduction in a homogeneous material where the thermal conductivity is a function of the local temperature, that is, , where is a given function, and introduce the Kirchhoff transformation (2) where is an arbitrary reference temperature, and is a dummy variable of integra-tion. The function is an implicit function of position through the dependence of on and . Show that (a) the gradients of and are related by , and (b) satisfies Laplace’s equation (2. 1). 12 A Practical Guide to Boundary-Element Methods 2.2 Green’s functions The Green’s functions of Laplace’s equation in two dimensions constitute a special class of harmonic functions that are singular at an arbitrary point . By definition, a Green’s function satisfies the singularly forced Laplace’s equation (2.2.1) where: is the variable “field point.” is the fixed location of the “singular point” or “pole.” , written more explicitly as , is Dirac’s delta function in two dimensions. By construction, Dirac’s delta function in two dimensions is endowed with the fol-lowing properties: 1. vanishes everywhere except at the point , , where it becomes infinite. 2. The integral of the delta function over an area that contains the singular point is equal to unity, (2.2.2) This property illustrates that the delta function in the plane has units of inverse squared length.

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  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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  Introductory Applications of Partial Differential Equations

  With Emphasis on Wave Propagation and Diffusion

  • G. L. Lamb, Jr.(Authors)
  • 2011(Publication Date)
  • Wiley-Interscience

    (Publisher)

  For problems involving only two space dimensions, V 2 u = u xx + u yy and s(r, t) then refers to s(x, y, t). For steady state problems u, = 0 and we have Κ pc ( 3 . 1 . 1 ) V 2 « = --s(r) ( 3 . 1 . 2 ) an equation sometimes referred to as Poisson's equation. 85 86 TWO AND THREE DIMENSIONS Similar extensions can be applied to the wave equation. For two dimensions, in place of waves on a string we may consider waves on a membrane. The expression for the force $F(r, acting on a square element of a membrane is a direct extension of that used for an element of a string, and as will be shown in Section 3.4, we obtain the two-dimensional wave equation in the form V 2 w (r, t) - \ ^ ? = ^ff ( r , 0 (3.1.3) c at Waves in three dimensions may be considered in many contexts, one of which is sound propagation. Again, the only mathematical extension required is that of the three-dimensional Laplacian plus a consideration of the boundary con- ditions that model various physical situations. 3.2 STEADY STATE TEMPERATURE DISTRIBUTION IN RECTANGULAR COORDINATES-LAPLACE'S EQUATION As a simple example of the extension to two dimensions of a one-dimensional problem treated previously, consider a plate in an XY plane as shown in Figure 3.1. The two surfaces of the plate that face in the Ζ direction (i.e., perpendicular to the page) are assumed to be insulated, as was the case for the length of the beam in the one-dimensional problems, and can thus be ignored. The two rims of the plate that face in the Y direction will also be assumed to be perfectly insulated, but since we will now consider temperature variations in the Y di- rection, we must introduce the insulation-type boundary condition explicitly and write u Y (x, 0) = u v (x, W) = 0. In addition, the rim at = 0 is kept at zero temperature while at JC = L we specify the temperature variation in the y w yx .w)=o w •Si V 2 u =0 r~ is.

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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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  Conformal Mapping

  Methods and Applications

  • Roland Schinzinger, Patricio A. A. Laura(Authors)
  • 2012(Publication Date)
  • Dover Publications

    (Publisher)

  Chapter 5

  MATHEMATICAL MODELS

  In conformally mapping a physical domain into the model domain we change the boundaries, the region, and generally also the corresponding partial differential equation. As we select a suitable mapping function and apply it, we must keep in mind the physical process which gives rise to the equation and its boundary conditions in the first place. This is the reason for this chapter.

  First we describe how the Laplace equation is transformed during conformal mapping. Then we show how the Laplace equation arises in some common potential fields. The presentation is kept rudimentary to highlight the analogies among several physical systems. Next the Poisson equation and equations for waves, diffusion, and elastic deformation, along with their boundaries, are conformally transformed. These examples cover the canonical cases of second-order differential equations: the elliptic (Laplace and Poisson), parabolic (diffusion), and hyperbolic (wave) equations. It is shown that while the Laplacian remains invariant under conformal mapping, adjustments must be made with the other equations.

  5.1 POTENTIAL FIELDS AND THE LAPLACE EQUATION

  Consider a region in real space in which each point is given by a set of coordinate values, accompanied by a single physical quantity φ (such as stress, strain, density, temperature, or electric potential) which may change with time, t. The totality of these values for one particular type of physical entity is defined as a physical field. The mathematical treatment of this subject is field theory. [Moon & Spencer, 1961]

  The values which we ascribe to the physical entities of a field as a function of space and time constitute its scalar potential φ(x,y,z,t). Hereafter we shall leave out the third axis “z” or, when it is unavoidable, label it z or ζ so as not to cause confusion with the complex variable z

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  Computational Methods for Physics

  • Joel Franklin(Author)
  • 2013(Publication Date)
  • Cambridge University Press

    (Publisher)

  4 Partial differential equations So far, we have been working with ordinary differential equations, functions of a single variable, satisfying a relation between the function and its derivative(s). But in many physical contexts, what we have is a function of multiple variables. The relevant differential equations then depend on derivatives with respect to each of the variables. The first example one encounters in physics is in E&M, where the electrostatic potential V (x, y, z) is determined by a distribution of source charges ρ (x, y, z) via:  ∂ 2 V ∂x 2 + ∂ 2 V ∂y 2 + ∂ 2 V ∂z 2  = − ρ  0 . (4.1) We use the “Laplace operator”, ∇ 2 , as shorthand for the derivatives appearing on the left, ∇ 2 ≡ ∂ 2 ∂x 2 + ∂ 2 ∂y 2 + ∂ 2 ∂z 2 , so we can write, compactly, ∇ 2 V = − ρ  0 . (4.2) When boundary conditions are provided, this differential equation has a unique solution, and is an example of the “Poisson problem.” We’ll take the generic setup: ∇ 2 f (x, y, z) = s (x, y, z) (4.3) for a “source” s (x, y, z) and f (x, y, z) given on the boundary of some volume, as the definition of the Poisson problem. Its “source-free” form, ∇ 2 f = 0 (with boundary conditions), is referred to as the Laplace problem. In either case, the goal is to find f (x, y, z), given source function s (x, y, z) (possibly zero), in some region , with f (x, y, z) on the boundary of that region, ∂, matching a provided function. 86 4.1 Physical motivation 87 There are many other examples of linear differential operators – e.g. the Helmholtz equation: ( ∇ 2 − μ 2 ) f = s, (4.4) appropriate for fields with “mass” μ (see Section 15.1.2). The operator is linear because it acts on f (and not f 2 or sin(f ), say). The goal is the same: Find f given source s and boundary values. Numerically, we exploit the linearity of these differential operators.

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  Partial Differential Equations

  A First Course

  • Rustum Choksi(Author)
  • 2022(Publication Date)
  • American Mathematical Society

    (Publisher)

  Chapter 4 The Wave Equation in Three and Two Space Dimensions The reader is encouraged to first review Sections A.4–A.6 and A.9 from the Appendix. In this chapter we address the wave equation in space dimensions three and two. In general, the wave equation in ? space dimensions is the second-order PDE for ?( x , ?) , x = (? 1 , ... , ? ? ) ∈ ℝ ? , and ? ∈ ℝ : ? ?? = ? 2 Δ?, (4.1) where Δ? denotes the Laplacian operator ; i.e., Δ? = div ∇? = ? ∑ 𝑖=1 ? ? 𝑖 ? 𝑖 . In the previous chapter we dealt with one space dimension ? = 1 and found that the solution at any point ? and time ? depended on initial data on a spatial inter-val [? − ??, ? + ??] . In other words, the solution involved an integral over the one-dimensional ball centered at ? with radius ?? . When the space dimension is larger than one, we would expect that these single spatial integrals would be replaced by multiple integrals. But of what type? We will soon see that these multiple integrals are surface integrals over spheres in 3D, and bulk integrals over balls (in 2D). It is thus very use-ful to review Section A.5 before embarking any further. We begin with two separate physical derivations of the 3D wave equation. 4.1. • Two Derivations of the 3D Wave Equation In this section, we denote the coordinates of three-dimensional space by (?, ?, ?) . We give two derivations of the 3D wave equation: ? ?? = ? 2 (? ?? + ? ?? + ? ?? ) = ? 2 Δ?. (4.2) 125 126 4. The Wave Equation in Three and Two Space Dimensions The first is for the propagation of electromagnetic waves in a vacuum and is exact in the sense that no approximations are needed for the equation to be valid. The sec-ond one is for acoustics — the propagation of sound in air. Here, following the same path as for the vibrating string, we will derive the wave equation by making an assump-tion that disturbances are relatively small.

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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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  Advanced Engineering Mathematics, SI Edition

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

    (Publisher)

  This can be done by solving the Laplace equation ∇ 2 V = 0 for the potential difference V between the conducting plates of the capacitor. This calculation is a gateway to determining other engineering quantities, such as resistance or capacitance. For instance, once V is determined through the solution of Laplace’s equation, the electric field E = −∇ V is calculated. This subsequently determines the current I = S σ E · dS , which can be used to determine the resistance of the capacitor via Ohm’s Law, R = V / I . Similar calculations can also determine the capacitance for the capacitor. As such, discovery of the potential difference spatial distribution via the Laplace equation is a useful skill for electrical engineers. Section 7.5 Problems In each of Problems 1–4, solve the Dirichlet problem. 1. ∇ 2 u ( x , y , z ) = 0 for 0 < x < 1, 0 < y < 1, 0 < z < 1, u ( 0, y , z ) = u ( 1, y , z ) = 0, u ( x , 0, z ) = u ( x , 1, z ) = 0, u ( x , y , 0 ) = 0, u ( x , y , 1 ) = xy 2. ∇ 2 u ( x , y , z ) = 0 for 0 < x < 2 π , 0 < y < 2 π , 0 < z < 1, u ( x , y , 0 ) = u ( x , y , 1 ) = 0, u ( x , 0, z ) = u ( x , 2 π , z ) = 0, u ( 0, y , z ) = 0, u ( 2 π , y , z ) = z Copyright 2018 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. WCN 02-300 7.6 The Neumann Problem 227 3. ∇ 2 u ( x , y , z ) = 0 for 0 < x < 1, 0 < y < 2 π , 0 < z < π , u ( 0, y , z ) = u ( 1, y , z ) = 0, u ( x , 0, z ) = u ( x , y , 0 ) = 0, u ( x , y , π) = 1, u ( x , 2 π , z ) = xz 2 4. ∇ 2 u ( x , y , z ) = 0 for 0 < x < 1, 0 < y < 2, 0 < z < π , u ( x , 0, z ) = u ( x , 2, z ) = 0, u ( 0, y , z ) = u ( x , y , π) = 0, u ( x , y , 0 ) = x 2 ( 1 − x )( 2 − y ) , u ( 1, y , z ) = sin (π y ) sin ( z ) 7.6 The Neumann Problem Let D be a region in the plane, bounded by a piecewise smooth closed curve C . This means that C has a continuous tangent at all but perhaps a finite number of points.

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  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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  Evolution Equations With A Complex Spatial Variable

  • Ciprian G Gal, Sorin G Gal, Jerome A Goldstein(Authors)
  • 2014(Publication Date)
  • WSPC

    (Publisher)

  Chapter 2

  Heat and Laplace equations of complex spatial variables

   

  In this chapter we study the one-dimensional heat and Laplace equations with real temporal variable and complex spatial variable. All the results presented in this chapter are contained in Gal-Gal-Goldstein [32 ], except for Theorems 2.2.8 and 2.2.9 which are new.

  2.1Introduction

  Let us consider the Cauchy problem, with both temporal and spatial variables real, for the following two classical evolution equations: (i) The case of the one-dimensional heat equation

  where f ∈ BUC(ℝ), which is defined as the space of all bounded uniformly continuous functions on ℝ. It is well-known that its unique bounded solution is given by the semigroup of linear operators (see, e.g., Goldstein [39 ]):

  (ii) The case of the Laplace equation

  where f ∈ BUC(ℝ). This initial boundary value problem has a bounded solution given by the semigroup of linear operators (see, e.g., Goldstein [39 ]):

  As it was already pointed out by Chapter 1 , it is well-known known that when the temporal variable and the spatial variable are both real or the temporal variable is complex and the spatial variable is real, the theories of real or analytic semigroups of linear operators become a powerful of study for the one-dimensional heat equation. Also, there it was pointed out that if both variables, temporal and spatial, are complex, then the Cauchy problem for the heat equation admits as a solution, only a formal power series which, in general, converges nowhere. The purpose of this chapter is, in a sense, complementary: to study the complex versions of the classical heat equation in Section 2.2 and of the Laplace equation in Section 2.3

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