Physics
One Dimensional Laplace Equation
The one-dimensional Laplace equation is a partial differential equation that describes the distribution of a scalar field in one spatial dimension. It is used to model steady-state phenomena, such as heat conduction or electrostatics, where the field does not change with time. The equation is expressed as the second derivative of the field with respect to the spatial coordinate equal to zero.
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10 Key excerpts on "One Dimensional Laplace Equation"
- 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 .
- 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
eBook - PDFIntroductory 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.
- 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.
eBook - PDF- 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.
eBook - PDF- Virginia W. Noonburg(Author)
- 2019(Publication Date)
- American Mathematical Society(Publisher)
(8.27) If the 2-dimensional heat equation is written in the form ? ?? = ?(? ?? + ? ?? ), and the temperature ?(?, ?) is no longer changing with time, then this equation reduces to (8.27); therefore, Laplace’s equation is often referred to as the steady-state heat equa-tion. It can be easily checked that equation (8.27) is an elliptic partial differential equa-tion by noting that ? = ? = 1 and ? = 0 imply ? 2 − ?? = −1 < 0 . As an example of a physical situation where this equation arises, consider the tem-perature ?(?, ?) in a thin rectangular metal plate which is insulated on the top and bot-tom so that heat cannot flow in the ? -direction. If the temperatures on all four edges of the rectangle are specified appropriately as functions of ? or ? and kept fixed , then given any initial temperature distribution the solution of the heat equation in the inte-rior of the rectangular plate will approach its steady state value; that is, it will approach the solution of (8.27). Solution of Laplace’s Equation by Separation of Variables. The method of sep-aration of variables that we applied to the heat equation and the wave equation can also be used to solve equation (8.27) if it is assumed that three sides of the rectangle are held at temperature 0 ∘ ? . The temperature on the fourth side can be specified by an arbitrary piecewise continuous function. Since Laplace’s equation is linear and ho-mogeneous, we can find four different series solutions, each one satisfying a nonzero condition on a different side, and add them together to get a solution which satisfies arbitrary conditions around the entire boundary of the rectangle. We will assume first that the boundary conditions are as shown in the figure below. If ?(?, ?) = ?(?)?(?) -6 y a b ?(?, ?) ≡ 0 ?(?, 0) = ?(?) ?(0, ?) ≡ 0 ?(?, ?) ≡ 0 x ? ?? + ? ?? = 0 ? 8.5 Laplace’s Equation 319 is substituted into (8.27), and the result is divided by ?? , then ? ″ ? ?? + ?? ″ ?? = 0 ⇒ ? ″ ? = − ? ″ ? = −?.
eBook - PDF- 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.
eBook - PDFDifferential Equations
A first course on ODE and a brief introduction to PDE
- Shair Ahmad, Antonio Ambrosetti(Authors)
- 2019(Publication Date)
- De Gruyter(Publisher)
15 A primer on linear PDE in 2D II: second order equations A second order partial differential equation is an equation in which the higher par-tial derivatives are of second order. We will deal with linear second order PDE in two dimensions, whose general form is au xx + bu xy + cu yy + du x + eu y + fu = h ( x , y ), ( x , y ) ∈ ℝ 2 with a , b , c not all zero. As usual the independent variables x , y can be labeled differ-ently, such as t , x or t , y . The features of this equation essentially depend on the co-efficients a , b , c of the second order partial derivatives. It is convenient to distinguish among three cases: 1. b 2 − 4 ac > 0, elliptic equations; 2. b 2 − 4 ac = 0, parabolic equations; 3. b 2 − 4 ac < 0, hyperbolic equations. We focus on very specific classes of elliptic, parabolic and hyperbolic equations: 1. the Laplace equation u xx + u yy = 0, which is the simplest elliptic linear second order equation; 2. the heat equation u t − u xx = 0, which is the simplest parabolic linear second order equation; 3. the vibrating string equations u tt − c 2 u xx = 0, which is the simplest hyperbolic linear second order equation. 15.1 The Laplace equation The Laplace equation, Δ u = u xx + u yy = 0 , (L) is the prototype of elliptic equations. Its solutions are called harmonic functions . This equation arises in very many situations. For example, a solution of (L) can be the sta-tionary solution of an evolutionary equation, or it can be the electrostatic potential in a region with no charges. As another example, consider a holomorphic function f ( x + iy ) in complex analysis. If f ( x + iy ) = u ( x , y )+ iv ( x , y ) is holomorphic then u , v (are C ∞ and) satisfy the Cauchy–Riemann equations, u x = v y , u y = − v x . It follows that u xx = v yx , u yy = − v xy and hence Δ u = v yx − v xy . Using the Schwarz theorem about mixed partial derivatives we infer that v yx = v xy and hence Δ u = 0.
eBook - PDF- 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.
eBook - PDFMathematical Methods for Physicists
A Concise Introduction
- Tai L. Chow(Author)
- 2000(Publication Date)
- Cambridge University Press(Publisher)
The general solution of a linear non-homo-geneous partial dierential equation is obtained by adding a particular solution of the non-homogeneous equation to the general solution of the homogeneous equation. The homogeneous form of Eq. (10.1) resembles the equation of a general conic: ax 2 bxy cy 2 dx ey f 0 : We thus say that Eq. (10.1) is of elliptic hyperbolic parabolic 9 > = > ; type when B 2 4 AC < 0 B 2 4 AC > 0 B 2 4 AC 0 8 > < > : : For example, according to this classification the two-dimensional Laplace equation @ 2 u @ x 2 @ 2 u @ y 2 0 is of elliptic type ( A C 1 ; B D E F G 0 , and the equation @ 2 u @ x 2 2 @ 2 u @ y 2 0 is a real constant is of hyperbolic type. Similarly, the equation @ 2 u @ x 2 @ u @ y 0 is a real constant is of parabolic type. 388 PARTIAL DIFFERENTIAL EQUATIONS We now list some important linear second-order partial dierential equations that are of physical interest and we have seen already: (1) Laplace’s equation: r 2 u 0 ; 10 : 2 where r 2 is the Laplacian operator. The function u may be the electrostatic potential in a charge-free region. It may be the gravitational potential in a region containing no matter or the velocity potential for an incompressible fluid with no sources or sinks. (2) Poisson’s equation: r 2 u x ; y ; z ; 10 : 3 where the function x ; y ; z is called the source density. For example, if u repre-sents the electrostatic potential in a region containing charges, then is propor-tional to the electrical charge density. Similarly, for the gravitational potential case, is proportional to the mass density in the region. (3) Wave equation: r 2 u 1 v 2 @ 2 u @ t 2 ; 10 : 4 transverse vibrations of a string, longitudinal vibrations of a beam, or propaga-tion of an electromagnetic wave all obey this same type of equation.
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