Three Dimensional Laplace Equation

9 Key excerpts on "Three Dimensional Laplace Equation"

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  Models and Modeling

  An Introduction for Earth and Environmental Scientists

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

    (Publisher)

  Furthermore, Laplace’s equation predicts that every point in the problem domain is the average of the neighboring points, and this interpretation of the equation was born out by the form of the finite difference operator for the second derivative. Strictly speaking, the so‐called 1D Laplace equation examined in this chapter is not a true Laplace equation, because Laplace’s equation is a PDE, the solutions of which are harmonic functions. Despite this, the simplified version used in this chapter provides a good illustration of many of the features of Laplace’s equation. From a practical standpoint, we have seen that some of the procedures commonly followed in, for example, environmental consulting companies, derive from the assumption that Laplace’s equation adequately describes the subsurface head distribution. This may or may not be a good assumption for a given aquifer, but it is important to evaluate the assumption before relying on it in a field setting. As was noted in the introductory section to this chapter, the Laplace equation is one of the most important equations of mathematical physics. Time spent understanding this fundamental equation will be repaid manyfold; this is particularly true because Laplace’s equation is a special case of many of the equations we will be looking at in subsequent chapters. 4.10 Problems Nondimensionalize the following field equations: (4.54) (4.55) (4.56) (4.57) (4.58) (4.59) (4.60) (4.61) (4.62) (4.63) with. HINT: You will need to define a dimensionless constant in this problem. Perhaps you could call it θ 2 or similar. If masl, masl, km, and m/s, what is the flux (in m/s) across the Elysian Fields? What is the total discharge from the World Encircling Sea to the River Acheron? You may assume the aquifer thickness is 500 m. Explain the value you estimated for the total discharge across the Elysian Fields

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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 Course of Mathematics for Engineers and Scientists

  • Brian H. Chirgwin, Charles Plumpton(Authors)
  • 2016(Publication Date)
  • Pergamon

    (Publisher)

  CHAPTER II THE SOLUTION OF SOME DIFFERENTIAL EQUATIONS 2:1 Laplace's equation in two and three dimensions Laplace's equation, V 2 / = 0, introduced in Chapter I takes the forms d 2 f , d 2 f dx 2 dy 2 d 2 f d 2 f , d 2 f = 0, (2.1) 0 (2.2) dx 2 dy 2 dz 2 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, / = sinax coshay, f = e kz sinlx cosmy, (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 117 118 A COURSE OF MATHEMATICS 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 gradf 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). v/-i{£(^U«(i4'U-.£UiiU ,4,/ r (dr dr] δθ r dd J dr 2 r dr r 2 δθ 2 (2.3) To separate the variables we write / = R(r)0 (0), where R is a function of r only and Θ a function of 0 only, and divide through by the product R Θ .

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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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  Introductory Applications of Partial Differential Equations

  With Emphasis on Wave Propagation and Diffusion

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

    (Publisher)

  Two and Three Dimensions The one-dimensional problems analyzed previously are, of course, idealiza- tions. They are useful because they display in a simple way many of the physical concepts that are described more realistically but in a more cumber- some mathematical framework by problems in two and three dimensions. For instance, consideration of temperature variation over the beam cross section in any of the previous one-dimensional diffusion problems introduces one of the main limitations in treating partial differential equations by analytical methods. It is found that only certain shapes for the cross-sectional area are amenable to exact solution. In this chapter we introduce some of the extensions of the previous developments that are associated with this continuation into additional space dimensions. 3.1 INTRODUCTION Although the solutions of partial differential equations in more than one di- mension may be more difficult to obtain, the equations themselves are readily extended to more space dimensions by merely introducing vector concepts. For diffusion problems, the heat flow J = —Ku x is replaced by the heat flow vector J = -K(iu x + ju y + ku z ) = - Κ V u where i, j , and k are unit vectors in the JC, y, and directions, respectively, and V = i d/dx + j d/dy + k d/dz. In place of the conservation law pc du/dt + dJ/dx = s(x, t) there is the corre- sponding vector extension pcu, + V · J = s(r, t) where s(r, t) is an abbre- viation for s(x, y, z, t). The diffusion equation now becomes pcu, - V · (K VK) = s(r, t). For constant K, we obtain where V 2 u = V · Vu = u xx + u yy + M z z . 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.

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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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  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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  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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  Spherical and Plane Integral Operators for PDEs

  Construction, Analysis, and Applications

  • Karl K. Sabelfeld, Irina A. Shalimova(Authors)
  • 2013(Publication Date)
  • De Gruyter

    (Publisher)

  13.3 3D Laplace equation | 279 is a zero mean homogeneous random field on the boundary { y = 0 } with the correlation function B g ( x ) which is bounded in dimension n = 2 , or tends to zero as | x | → ∞ if n > 2 . Then B u ( x , y ) = B u ( x 2 − x 1 , y 1 + y 2 ) , the correlation function of the solution, is a harmonic function in R n + , and is related to B g by the Poisson-type formula: B u ( x 2 − x 1 , y 1 + y 2 ) = Γ ( n / 2 ) π n / 2 ∂D + ( y 1 + y 2 ) B g ( x ) dS ( x ) [( x − ( x 2 − x 1 )) 2 + ( y 1 + y 2 ) 2 ] n / 2 . (13.34) The proof is obtained by the same Fourier transform technique as we used above. Remark 13.3. We remark that exactly as in the case of a disk as discussed in Re-mark 13.2 to Theorem 13.3, the same convolution relation (13.34) is true for the cross-correlation functions, we need to write it only for the kernel K p : B u 1 u 2 = K p ∗ B g 1 g 2 . Note that in the n -dimensional case, K p has the form of the kernel given in (13.34). In practice, it is often important to know the statistical structure of the gradient of the solution. Let us denote by B u x i ( x , y ) , i = 1 , . . . , n − 1, and B u y ( x , y ) the correlation functions of the partial derivatives of the solution u . They also obviously depend only on x = x 2 − x 1 and y = y 1 + y 2 by the assumption that g is homogeneous. Direct evaluation gives B u x i = − ∂ 2 B u ∂x i , i = 1 , . . . , n − 1 , B u y = ∂ 2 B u ∂ 2 y . Note that since the correlation function B u is harmonic, this implies the following re-markable property: B u y = ∑ n − 1 i = 1 B u x i . So in dimension 2, B u y = B u x . 13.3 3D Laplace equation For a ball in 3D, all considerations are quite similar, where the eigenfunctions involved are the spherical harmonics.

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