Mathematical Methods for Physics and Engineering - Matematica.NET

PDES: GENERAL AND PARTICULAR SOLUTIONSyCdrdydxˆn dsxFigure 20.4point.A boundary curve C and its tangent and unit normal at a givenFor second-order equations we might expect that relevant boundary conditionswould involve specifying u, or some of its first derivatives, or both, along asuitable set of boundaries bordering or enclosing the region over which a solutionis sought. Three common types of boundary condition occur and are associatedwith the names of Dirichlet, Neumann and Cauchy. They are as follows.(i) Dirichlet: The value of u is specified at each point of the boundary.(ii) Neumann: The value of ∂u/∂n, thenormal derivative of u, isspecifiedateach point of the boundary. Note that ∂u/∂n = ∇u · ˆn, whereˆn is thenormal to the boundary at each point.(iii) Cauchy: Bothu and ∂u/∂n are specified at each point of the boundary.Let us consider for the moment the solution of (20.43) subject to the Cauchyboundary conditions, i.e. u and ∂u/∂n are specified along some boundary curveC in the xy-plane defined by the parametric equations x = x(s), y = y(s), s beingthe arc length along C (see figure 20.4). Let us suppose that along C we haveu(x, y) =φ(s) and∂u/∂n = ψ(s). At any point on C the vector dr = dx i + dy j isa tangent to the curve and ˆn ds = dy i − dx j is a vector normal to the curve. Thuson C we have∂u dr≡∇u ·∂s ds = ∂u dx∂x ds + ∂u dy∂y ds = dφ(s) ,ds∂u∂n≡∇u · ˆn =∂u∂xdyds − ∂u∂ydxds = ψ(s).These two equations may then be solved straightforwardly for the first partialderivatives ∂u/∂x and ∂u/∂y along C. Using the chain rule to writedds = dx ∂ds ∂x + dy ∂ds ∂y ,702