Separation of Variables

6 Key excerpts on "Separation of Variables"

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  An Introduction to Computational Physics

  • Tao Pang(Author)
  • 2006(Publication Date)
  • Cambridge University Press

    (Publisher)

  Even in the numerical study of partial differential equations, due to the limitation of computing resources, the Separation of Variables can serve the purpose of simplifying a problem to the point where it can be solved with the available computing resources. Later in the chapter, we will introduce several numerical schemes used mainly in solving linear partial differential equations. 7.2 Separation of Variables Before we introduce numerical schemes for partial differential equations, we now discuss an analytic method, that is, the Separation of Variables , for solving partial differential equations. In many cases, using a combination of the Separation of Variables and a numerical scheme increases the speed of computing and the accu-racy of the solution. The combination of analytic and numerical methods becomes critical in cases where the memory and speed of the available computing resources are limited. This section is far from being a complete discussion; interested read-ers should consult a standard textbook, such as Courant and Hilbert (1989). Here we will just illustrate how the method works. Each variable (time or one of the spatial coordinates) is isolated from the rest, and the solutions of the resulting ordinary differential equations for all the variables are obtained before they are combined into the general solution of the original partial differential 7.2 Separation of Variables 199 equation. Boundary and initial conditions are then used to determine the unknown parameters left in the solution, which usually appear as the coefficients or eigenvalues. Consider first the standing waves on a light, uniform string that has both ends ( x = 0 and x = L ) fixed. The wave equation is ∂ 2 u ( x , t ) ∂ t 2 − c 2 ∂ 2 u ( x , t ) ∂ x 2 = 0 , (7.8) where u ( x , t ) is the displacement of the string at the location x and time t , and c = √ T /ρ is the phase speed of the wave, with T being the tension on the string and ρ the linear mass density of the string.

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  Mathematical Methods for Mathematicians, Physical Scientists and Engineers

  • Jeremy Dunning-Davies(Author)
  • 2003(Publication Date)
  • Woodhead Publishing

    (Publisher)

  Clearly the solution is unique and is stable also since it depends contin-uously on the values of the initial conditions. 12.5 Separation of Variables As has been mentioned already, the general solutions of partial differential equations are of little use owing to the difficulty of choosing the arbitrary functions so as to satisfy the boundary conditions. For some linear partial Sec. 12.5] Separation of Variables 311 differential equations, this difficulty may be avoided in various ways. One such method is based on the principle of superposition which was encountered in Chapter 8. A set of solutions of a given linear homogeneous partial differen-tial equation may be found by using the method of Separation of Variables. Here the dependent variable is assumed to be a product of functions, each depending on just one of the independent variables. It may be possible, at this stage, to find a solution to the original equation, which satisfies all the given boundary conditions, by choosing an appropriate linear combination of the solutions found by the Separation of Variables technique. This complete method, a combination of Separation of Variables and the superposition of solutions, is far less complicated than it sounds and is probably introduced most easily by means of an example: Obtain a solution of the one-dimensional wave equation dx 2 ~ c 2 dt 2 ( l ) which satisfies the Cauchy boundary conditions u(0,<) = uil,t) = 0 for t ^ 0 (ii) uixfi) = fix) for 0 « χ « I (iii) (θι//3«, = 0 = gix) for 0 « χ « I (iv) where / and g are given functions and / is a given constant. Assume a separable solution of the form uix,t) = XixWit) where AT is a function of χ only, and Γ is a function of t only. Then (i) becomes 1 d 2 X 1 d 2 r X dx 2 ~ c 2 T dt 2 ' The left-hand side of this equation is a function of χ only, and the right-hand side is independent of x. Hence, the equation may hold only if both sides have the same constant value, say k.

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  Differential Equations with Maple V®

  • Martha L Abell, James P. Braselton(Authors)
  • 2014(Publication Date)
  • Academic Press

    (Publisher)

  In the previous chapters, we have seen that many physical and mathematical situations are de-scribed by ordinary differential equations. We have also seen that others are described by partial differential equations. In this chapter, we investigate one way to solve some partial differential equations, the method of Separation of Variables, which was mostly developed by Fourier in his study of the heat equation. 12.1 I N T R O D U C T I O N T O P A R T I A L D I F F E R E N T I A L E Q U A T I O N S A N D S E P A R A T I O N O F V A R I A B L E S We begin our study of partial differential equations with an introduction of some of the termi-nology associated with the topic. A linear second-order partial differential equation in the two independent variables χ and y has the form Auxx H-Buxy + Cuyy + + Euy + Fw = G{x, Xf), where A, B, C, D, E, and F are constants and the solution is u{x, y). If G(x, y) = 0 for all χ and y, then we say that the equation is homogeneous. Otherwise, the equation is nonhomogeneous. EXAMPLE: Classify the foUowing partial differential equations: (a) Ηχχ + Uyy = u and (b) Uxx + UUx = X, SOLUTION: (a) This equation satisfies the form of the linear second-order partial differential equation with A = C = 1, F = -1 , and B = D = Ε = 0. Because G(x,y) = 0, the equation is homogeneous, (b) This equation is nonlinear because the coefficient of Ux is not a constant. It is also nonhomogeneous because G{x, y) = x. 5 Í 9 Partial Differential Equations 590 Differential Equations with Maple V EXAMPLE: Use Separation of Variables to solve xux = Wy. SOLUTION: If u{x, y) = X{x)Y{y), then Ux = Χ Ύ and Uy = XY'. The equation then becomes xX'Y = XY', which can be written as the separated equation xx^ X Y' Notice that the left-hand side of the equation is a function of x, while the right-hand side is a xX' Y' function of y. Hence, the only way that this situation can be true is for — and γ both to be constant.

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  Partial Differential Equations

  A First Course

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

    (Publisher)

  We will most often use ? to denote the unknown function, i.e., the dependent variable. So, for example, in purely spatial variables we would be dealing with ?(?, ?), ?(?, ?, ?) , or more generally ?( x ) . When time is also relevant, we will deal with ?(?, ?), ?(?, ?, ?), ?(?, ?, ?, ?) , or more generally ?( x , ?) . 1.2. • What Are Partial Differential Equations and Why Are They Ubiquitous? “ The application of calculus to modern science is largely an exercise in the formulation, solution, and interpretation of partial differential equations . ... Even at the cutting edge of modern physics, partial dif-ferential equations still provide the mathematical infrastructure. ” -Steven Strogatz a a Strogatz is an applied mathematician well known for his outstanding undergraduate book on the qualitative theory of ODEs: Nonlinear Dynamics and Chaos , Westview Press. This quote is taken from his popular science book Infinite Powers: How Calculus Reveals the Secrets of the Universe , Houghton Mifflin Harcourt Publishing Company, 2019. We begin with a precise definition of a partial differential equation. Definition of a Partial Differential Equation (PDE) Definition 1.2.1. A partial differential equation (PDE) is an equation which relates an unknown function ? and its partial derivatives together with inde-pendent variables. In general, it can be written as ?( independent variables , ?, partial derivatives of ?) = 0, for some function ? which captures the structure of the PDE. For example, in two independent variables a PDE involving only first-order partial derivatives is described by ?(?, ?, ?, ? ? , ? ? ) = 0, (1.1) where ? ∶ ℝ 5 → ℝ . Laplace’s equation is a particular PDE which in two independent variables is associated with ?(? ?? , ? ?? ) = ? ?? + ? ?? = 0. Note that often there is no explicit appearance of the independent variables, and the function ? need not depend on all possible derivatives.

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  Partial Differential Equations

  An Introduction

  • Walter A. Strauss(Author)
  • 2012(Publication Date)
  • Wiley

    (Publisher)

  1 WHERE PDEs COME FROM After thinking about the meaning of a partial differential equation, we will flex our mathematical muscles by solving a few of them. Then we will see how naturally they arise in the physical sciences. The physics will motivate the formulation of boundary conditions and initial conditions. 1.1 WHAT IS A PARTIAL DIFFERENTIAL EQUATION? The key defining property of a partial differential equation (PDE) is that there is more than one independent variable x , y , . . . . There is a dependent variable that is an unknown function of these variables u ( x , y , . . . ). We will often denote its derivatives by subscripts; thus ∂ u /∂ x = u x , and so on. A PDE is an identity that relates the independent variables, the dependent variable u , and the partial derivatives of u . It can be written as F ( x , y , u ( x , y ) , u x ( x , y ) , u y ( x , y )) = F ( x , y , u , u x , u y ) = 0 . (1) This is the most general PDE in two independent variables of first order. The order of an equation is the highest derivative that appears. The most general second -order PDE in two independent variables is F ( x , y , u , u x , u y , u xx , u xy , u yy ) = 0 . (2) A solution of a PDE is a function u ( x , y , . . . ) that satisfies the equation identically, at least in some region of the x , y , . . . variables. When solving an ordinary differential equation (ODE), one sometimes reverses the roles of the independent and the dependent variables—for in-stance, for the separable ODE du dx = u 3 . For PDEs, the distinction between the independent variables and the dependent variable (the unknown) is always maintained. 1 2 CHAPTER 1 WHERE PDEs COME FROM Some examples of PDEs (all of which occur in physical theory) are: 1. u x + u y = 0 (transport) 2. u x + yu y = 0 (transport) 3. u x + uu y = 0 (shock wave) 4. u xx + u yy = 0 (Laplace’s equation) 5. u tt − u xx + u 3 = 0 (wave with interaction) 6. u t + uu x + u xxx = 0 (dispersive wave) 7.

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  Mathematical Methods for Physicists

  A Concise Introduction

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

    (Publisher)

  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 . (3) Laplace transform method: We first obtain the Laplace transform of the partial di€erential equation and the associated boundary conditions with respect to one of the independent variables, and then solve the resulting equation for the Laplace transform of the required solution which can be found by taking the inverse Laplace transform. Solutions of Laplace's equation: Separation of Variables (1) Laplace’s equation in two dimensions x ; y † : If the potential is a function of only two rectangular coordinates, Laplace’s equation reads @ 2 @ x 2 ‡ @ 2 @ y 2 ˆ 0 : It is possible to obtain the general solution to this equation by means of a trans-formation to a new set of independent variables: ˆ x ‡ iy ; ˆ x iy ; 392 PARTIAL DIFFERENTIAL EQUATIONS where I is the unit imaginary number. In terms of these we have @ @ x ˆ @ @ @ @ x ‡ @ @ @ @ x ˆ @ @ ‡ @ @ ; @ 2 @ x 2 ˆ @ @ x @ @ ‡ @ @ ˆ @ @ @ @ ‡ @ @ @ @ x ‡ @ @ @ @ ‡ @ @ @ @ x ˆ @ 2 @ 2 ‡ 2 @ @ @ @ ‡ @ 2 @ 2 : Similarly, we have @ 2 @ y 2 ˆ @ 2 @ 2 ‡ 2 @ @ @ @ @ 2 @ 2 and Laplace’s equation now reads r 2 ˆ 4 @ 2 @@ ˆ 0 : Clearly, a very general solution to this equation is ˆ f 1 † ‡ f 2 † ˆ f 1 x ‡ iy † ‡ f 2 x iy † ; where f 1 and f 2 are arbitrary functions which are twice di€erentiable. However, it is a somewhat dicult matter to choose the functions f 1 and f 2 such that the equation is, for example, satisfied inside a square region defined by the lines x ˆ 0 ; x ˆ a ; y ˆ 0 ; y ˆ b and such that takes prescribed values on the boundary of this region. For many problems the method of Separation of Variables is more satisfactory.

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