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Since there is no time dependence in any of the problems mentioned previously, there are no initial conditions to be satisfied by the solutions of Eq. (1) or (2). They must, however, satisfy certain boundary conditions on the bounding curve or surface of the region in which the differential equation is to be solved. Since Laplace’s equation is of second order, it might be plausible to expect that two boundary conditions would be required to determine the solution completely. This, however, is not the case. Recall that in the heat conduction problem for the finite bar (Sections 10.5 and 10.6) it was necessary to prescribe one condition at each end of the bar, that is, one condition at each point of the boundary. If we generalize this observation to multidimensional problems, it is then natural to prescribe one condition on the function u at each point on the boundary of the region in which a solution of Eq. (1) or (2) is sought. The most common boundary condition occurs when the value of u is specified at each boundary point; in terms of the heat conduction problem this corresponds to prescribing the temperature on the boundary. In some problems the value of the derivative, or rate of change, of u in the direction normal to the boundary is specified instead; the condition on the boundary of a thermally insulated body, for example, is of this type. It is entirely possible for more complicated boundary conditions to occur; for example, u might be prescribed on part of the boundary, and its normal derivative specified on the remainder. The problem of finding a solution of Laplace’s equation that takes on given boundary values is known as a Dirichlet problem, in honor of P. G. L. Dirichlet.9 In contrast, if the values of the normal derivative are prescribed on the boundary, the problem is said to be a Neumann problem, in honor of K. G. Neumann.10 The Dirichlet and Neumann problems are also known as the first and second boundary value problems of potential theory, respectively.
Physically, it is plausible to expect that the types of boundary conditions just mentioned will be sufficient to determine the solution completely. Indeed, it is possible to establish the existence and uniqueness of the solution of Laplace’s equation under the
9Peter Gustav Dirichlet (1805-1859) was a professor at Berlin and, after the death of Gauss, at Gottingen. In 1829 he gave the first set of conditions sufficient to guarantee the convergence of a Fourier series. The definition of function usually used today in elementary calculus is essentially the one given by Dirichlet in 1837. While he is best known for his work in analysis and differential equations, Dirichlet was also one of the leading number theorists ofthe nineteenth century.
10Karl Gottfried Neumann (1832-1925), professor at Leipzig, made contributions to differential equations, integral equations, and complex variables.
Chapter 10. Partial Differential Equations and Fourier Series
boundary conditions mentioned, provided the shape of the boundary and the functions appearing in the boundary conditions satisfy certain very mild requirements. However, the proofs of these theorems, and even their accurate statement, are beyond the scope of the present book. Our only concern will be solving some typical problems by means of separation of variables and Fourier series.
While the problems chosen as examples are capable of interesting physical interpretations (in terms of electrostatic potentials or steady-state temperature distributions, for instance), our purpose here is primarily to point out some of the features that may occur during their mathematical solution. It is also worth noting again that more complicated problems can sometimes be solved by expressing the solution as the sum of solutions of several simpler problems (see Problems 3 and 4).
Dirichlet Problem for a Rectangle. Consider the mathematical problem of finding the function u satisfying Laplace’s equation (1),
Uxx + Uyy = 0
in the rectangle 0 < x < a, 0 < y < b, and also satisfying the boundary conditions
u(x, 0) = 0, u(x, b) = 0, 0 < x < a,
u(0, y) = 0, u(a, y) = f (y), 0 < y < b,
where f is a given function on 0 < y < b (see Figure 10.8.1).
To solve this problem we wish to construct a fundamental set of solutions satisfying the partial differential equation and the homogeneous boundary conditions; then we will superpose these solutions so as to satisfy the remaining boundary condition. Let us assume that
u(x, y) = X(x)Y(y) (5)
and substitute for u in Eq. (1). This yields
X = - Y = ’
where X is the separation constant. Thus we obtain the two ordinary differential equations
X" - XX = 0, (6)
Y" + XY = 0. (7)
FIGURE 10.8.1 Dirichlet problem for a rectangle.
10.8 Laplace’s Equation
If we now substitute for u from Eq. (5) in each of the homogeneous boundary conditions, we find that
X (0) = 0 (8)
Y (0) = 0, Y (b) = 0. (9)
We will first determine the solution of the differential equation (7) subject to the boundary conditions (9). However, this problem is essentially identical to one encountered previously in Sections 10.1, 10.5, and 10.7. We conclude that there are nontrivial solutions if and only if X is an eigenvalue, namely,