Mathematical Methods for Physics and Engineering : A Comprehensive Guide

21.5 Inhomogeneous problems – Green’s functions the infinite range inx, rather than of the transform method itself. In fact the ...

PDES: SEPARATION OF VARIABLES AND OTHER METHODS we haveL=∇^2 , whereas for Helmholtz’s equationL=∇^2 +k^2. Note that we have not ...

21.5 INHOMOGENEOUS PROBLEMS – GREEN’S FUNCTIONS if the boundary conditions onu(r) are homogeneous then a solution to (21.76) tha ...

PDES: SEPARATION OF VARIABLES AND OTHER METHODS conditions in the new variables are homogeneous although the equation itself is ...

21.5 INHOMOGENEOUS PROBLEMS – GREEN’S FUNCTIONS ˆn nˆ nˆ V V S S 1 S 2 (a) (b) Figure 21.11 Surfaces used for solving Poisson’s ...

PDES: SEPARATION OF VARIABLES AND OTHER METHODS written as u(r 0 )= ∫ V G(r,r 0 )ρ(r)dV(r)+ ∫ S [ u(r) ∂G(r,r 0 ) ∂n −G(r,r 0 ) ...

21.5 INHOMOGENEOUS PROBLEMS – GREEN’S FUNCTIONS the corresponding homogeneous equation (i.e. Laplace’s equation) insideVbut is a ...

PDES: SEPARATION OF VARIABLES AND OTHER METHODS for example, the electrostatic potentialu(r) due to some distribution of electri ...

21.5 INHOMOGENEOUS PROBLEMS – GREEN’S FUNCTIONS (ii) Since all the image sources lie outsideV, the fundamental solution cor- res ...

PDES: SEPARATION OF VARIABLES AND OTHER METHODS y z x V r^0 r 1 + − Figure 21.12 The arrangement of images forsolving Laplace’s ...

21.5 INHOMOGENEOUS PROBLEMS – GREEN’S FUNCTIONS example, in solving Poisson’s equation in two dimensions in the half-spacex> ...

PDES: SEPARATION OF VARIABLES AND OTHER METHODS −λ −λ +λ +λ x 0 y 0 r 0 r 2 r 1 r 3 C x y V Figure 21.13 The arrangement of imag ...

21.5 INHOMOGENEOUS PROBLEMS – GREEN’S FUNCTIONS y z a −a B A V x −a |r 0 | r 1 +1 r 0 Figure 21.14 The arrangement of images for ...

PDES: SEPARATION OF VARIABLES AND OTHER METHODS Therefore, in order thatG=0at|r|=a, the strength of the image charge must be −a/ ...

21.5 INHOMOGENEOUS PROBLEMS – GREEN’S FUNCTIONS Using the fact thatr 1 =(a^2 /|r 0 |^2 )r 0 and the geometrical result (21.99), ...

PDES: SEPARATION OF VARIABLES AND OTHER METHODS and, in a similar way, we might wish to choose∂G(r,r 0 )/∂n= 0 in the solution o ...

21.6 EXERCISES Using plane polar coordinates, the radial (i.e. normal) derivative of this function is given by ∂G(r,r 0 ) ∂ρ = r ...

PDES: SEPARATION OF VARIABLES AND OTHER METHODS 21.2 A cube, made of material whose conductivity isk, has as its six faces the p ...

21.6 EXERCISES (a) EvaluatedPm(μ)/dμandd^2 Pm(μ)/dμ^2 , using the forms given in (21.47), and substitute them into (21.45). (b ...

PDES: SEPARATION OF VARIABLES AND OTHER METHODS x= 0, and free at the other,x=L, show that the angular frequency of vibration ωs ...