Mathematical Methods for Physics and Engineering : A Comprehensive Guide

PDES: GENERAL AND PARTICULAR SOLUTIONS

C

y

x

dx

dy

nˆds

dr

Figure 20.4 A boundary curveCand its tangent and unit normal at a given

point.

For second-order equations we might expect that relevant boundary conditions

would involve specifyingu, or some of its first derivatives, or both, along a

suitable set of boundaries bordering or enclosing the region over which a solution

is sought. Three common types of boundary condition occur and are associated

with the names of Dirichlet, Neumann and Cauchy. They are as follows.

(i)Dirichlet: The value ofuis specified at each point of the boundary.

(ii)Neumann: The value of∂u/∂n,thenormal derivativeofu,isspecifiedat

each point of the boundary. Note that∂u/∂n=∇u·ˆn,wherenˆis the

normal to the boundary at each point.

(iii)Cauchy:Bothuand∂u/∂nare specified at each point of the boundary.

Let us consider for the moment the solution of (20.43) subject to the Cauchy

boundary conditions, i.e.uand∂u/∂nare specified along some boundary curve

Cin thexy-plane defined by the parametric equationsx=x(s),y=y(s),sbeing

the arc length alongC(see figure 20.4). Let us suppose that alongCwe have

u(x, y)=φ(s)and∂u/∂n=ψ(s). At any point onCthe vectordr=dxi+dyjis

a tangent to the curve andnˆds=dyi−dxjis a vector normal to the curve. Thus

onCwe have

∂u

∂s

≡∇u·

dr

ds

=

∂u

∂x

dx

ds

+

∂u

∂y

dy

ds

=

dφ(s)

ds

,

∂u

∂n

≡∇u·nˆ=

∂u

∂x

dy

ds

−

∂u

∂y

dx

ds

=ψ(s).

These two equations may then be solved straightforwardly for the first partial

derivatives∂u/∂xand∂u/∂yalongC. Using the chain rule to write

d

ds

=

dx

ds

∂

∂x

+

dy

ds

∂

∂y

,