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438 chapter 17TABLE 17.1The Relation Between Boundary Conditions and Uniqueness for PDEsBoundary Elliptic Hyperbolic ParabolicCondition (Poisson Equation) (Wave Equation) (Heat Equation)Dirichlet open surface Underspecified Underspecified Unique and stable (1-D)Dirichlet closed surface Unique and stable Overspecified OverspecifiedNeumann open surface Underspecified Underspecified Unique and Stable (1-D)Neumann closed surface Unique and stable Overspecified OverspecifiedCauchy open surface Nonphysical Unique and stable OverspecifiedCauchy closed surface Overspecified Overspecified Overspecifiedinfinite heat bath are physical situations for which the boundary conditions areadequate. If the boundary condition is the value of the solution on a surroundingclosed surface, we have a Dirichlet boundary condition. If the boundary condition isthe value of the normal derivative on the surrounding surface, we have a Neumannboundary condition. If the value of both the solution and its derivative are specifiedon a closed boundary, we have a Cauchy boundary condition. Although havingan adequate boundary condition is necessary for a unique solution, having toomany boundary conditions, for instance, both Neumann and Dirichlet, may be anoverspecification for which no solution exists. 1Solving PDEs numerically differs from solving ODEs in a number of ways. First,because we are able to write all ODEs in a standard form,dy(t)dt= f(y,t), (17.2)with t the single independent variable, we are able to use a standard algorithm, rk4in our case, to solve all such equations. Yet because PDEs have several independentvariables, for example, ρ(x, y, z, t), we would have to apply (17.2) simultaneouslyand independently to each variable, which would be very complicated. Second,since there are more equations to solve with PDEs than with ODEs, we need moreinformation than just the two initial conditions [x(0), ẋ(0)]. In addition, because eachPDE often has its own particular set of boundary conditions, we have to develop aspecial algorithm for each particular problem.1 Although conclusions drawn for exact PDEs may differ from those drawn for the finitedifferenceequations, they are usually the same; in fact, Morse and Feshbach [M&F 53]use the finite-difference form to derive the relations between boundary conditions anduniqueness for each type of equation shown in Table 17.1 [Jack 88].−101COPYRIGHT 2008, PRINCET O N UNIVE R S I T Y P R E S SEVALUATION COPY ONLY. NOT FOR USE IN COURSES.ALLpup_06.04 — 2008/2/15 — Page 438