Mathematical Methods for Physics and Engineering : A Comprehensive Guide

20.9 HINTS AND ANSWERS 20.25 The Klein–Gordon equation (which issatisfied by the quantum-mechanical wave- function Φ(r) of a rel ...

PDES: GENERAL AND PARTICULAR SOLUTIONS 20.23 (a) Parabolic, open, Dirichletu(x,0) given, Neumann∂u/∂x=0atx=±L/ 2 for allt; (b) e ...

21 Partial differential equations: separation of variables and other methods In the previous chapter we demonstrated the methods ...

PDES: SEPARATION OF VARIABLES AND OTHER METHODS When seeking PDE solutions of the form (21.1), we are requiring not that there i ...

21.1 SEPARATION OF VARIABLES: THE GENERAL METHOD Since there is only one equation to be satisfied and four constants involved, t ...

PDES: SEPARATION OF VARIABLES AND OTHER METHODS This gives a particular solution of the original PDE (21.3) u(x, y, z, t)=exp(il ...

21.2 SUPERPOSITION OF SEPARATED SOLUTIONS In order to satisfy the boundary conditionu→0ast→∞,λ^2 κmust be>0. Sinceκ is real a ...

PDES: SEPARATION OF VARIABLES AND OTHER METHODS superposing solutions corresponding to different allowed values of the separatio ...

21.2 SUPERPOSITION OF SEPARATED SOLUTIONS y b 0 x u=0 u=0 u=f(y) u→ 0 Figure 21.1 A semi-infinite metal plate whose edges are ke ...

PDES: SEPARATION OF VARIABLES AND OTHER METHODS f(y) −b 0 b y Figure 21.2 The continuation off(y) for a Fourier sine series. The ...

21.2 SUPERPOSITION OF SEPARATED SOLUTIONS solutions for differentnwe then obtain u(x, y)= ∑∞ n=1 Bnsinh[nπ(a−x)/b]sin(nπy/b), (2 ...

PDES: SEPARATION OF VARIABLES AND OTHER METHODS y y y b b b f(y) f(y) 0 0 0 0 0 0 0 0 a a a x x x g(x) g(x) (a) (b) (c) Figure 2 ...

21.2 SUPERPOSITION OF SEPARATED SOLUTIONS
A bar of lengthLis initially at a temperature of 0 ◦C. One end of the bar(x=0)is held ...

PDES: SEPARATION OF VARIABLES AND OTHER METHODS −L L f(x) 0 x −HL/k Figure 21.4 The appropriate continuation for a Fourier serie ...

21.3 SEPARATION OF VARIABLES IN POLAR COORDINATES 21.3 Separation of variables in polar coordinates So far we have considered th ...

PDES: SEPARATION OF VARIABLES AND OTHER METHODS Thus, writingu(ρ, φ)=P(ρ)Φ(φ) and using the expression (21.23), Laplace’s equati ...

21.3 SEPARATION OF VARIABLES IN POLAR COORDINATES whereA, B, C, Dare arbitrary constants andnis any integer. We have not yet, ho ...

PDES: SEPARATION OF VARIABLES AND OTHER METHODS Laplace’s equation in cylindrical polars Passing to three dimensions, we now con ...

21.3 SEPARATION OF VARIABLES IN POLAR COORDINATES As in the two-dimensional case, single-valuedness ofurequires thatmis an integ ...

PDES: SEPARATION OF VARIABLES AND OTHER METHODS z x a y u=0 u=0 u=T 0 Figure 21.5 A uniform metal cylinder whose curved surface ...