Mathematical Methods for Physics and Engineering : A Comprehensive Guide

21.3 SEPARATION OF VARIABLES IN POLAR COORDINATES and the coefficientsAnmay be expressed as An= 2 T 0 a^2 J^21 (kna) [ aJ 1 (kna ...

PDES: SEPARATION OF VARIABLES AND OTHER METHODS The first term depends only onrand the second and third terms (taken together) d ...

21.3 SEPARATION OF VARIABLES IN POLAR COORDINATES Having settled the form of Φ(φ), we are left only with the equation satisfied ...

PDES: SEPARATION OF VARIABLES AND OTHER METHODS variable solution of Laplace’s equation in spherical polars. It is u(r, θ, φ)=(A ...

21.3 SEPARATION OF VARIABLES IN POLAR COORDINATES + + + + + + + + + + + + − − − − − − − − − − − − θ a Figure 21.6 Induced charge ...

PDES: SEPARATION OF VARIABLES AND OTHER METHODS x z y a −a φ θ v=0 v=v 0 r Figure 21.7 A hollow split conducting sphere with its ...

21.3 SEPARATION OF VARIABLES IN POLAR COORDINATES In the above example, on the equator of the sphere (i.e. atr=aandθ=π/2) the po ...

PDES: SEPARATION OF VARIABLES AND OTHER METHODS We may treat the diffusion equation κ∇^2 u= ∂u ∂t in a similar way. Separating t ...

21.3 SEPARATION OF VARIABLES IN POLAR COORDINATES and is not to be confused with a spherical harmonic (these are written with a ...

PDES: SEPARATION OF VARIABLES AND OTHER METHODS ω=2. 40 c/a ω=3.^83 c/a ω=5. 14 c/a ω=5.^52 c/a a Figure 21.8 The modes of oscil ...

21.3 SEPARATION OF VARIABLES IN POLAR COORDINATES Helmholtz’s equation in spherical polars is thus F(r, θ, φ)=r−^1 /^2 [AJ+1/ 2 ...

PDES: SEPARATION OF VARIABLES AND OTHER METHODS y P a θ −a O r z x Figure 21.9 The polar axisOzis taken as normal to the plane o ...

21.3 SEPARATION OF VARIABLES IN POLAR COORDINATES surprising that the form of the expression foruchanges there. Let us therefore ...

PDES: SEPARATION OF VARIABLES AND OTHER METHODS Comparing this expression forr=z,θ= 0 with thez<aexpansion of (21.60), which ...

21.3 SEPARATION OF VARIABLES IN POLAR COORDINATES where the coefficientsR(r)andF(r) in the Legendre polynomial expansions are ...

PDES: SEPARATION OF VARIABLES AND OTHER METHODS This can be rearranged to give r^2 R′′ 1 +2rR 1 ′− 2 R 1 =− Ar^3  0 , where the ...

21.4 INTEGRAL TRANSFORM METHODS 21.4 Integral transform methods In the method of separation of variables our aim was to keep the ...

PDES: SEPARATION OF VARIABLES AND OTHER METHODS solution is immediate: ̄u(x, s)=Aexp (√ s κ x ) +Bexp ( − √ s κ x ) , where the ...

21.4 INTEGRAL TRANSFORM METHODS
An infinite metal bar has an initial temperature distributionf(x)along its length. Find the tem ...

PDES: SEPARATION OF VARIABLES AND OTHER METHODS x u x=a t 1 t 2 t 3 Figure 21.10 Diffusion of heat from a point source in a meta ...