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Chapter 5 Dynamic Programming and Elliptic Equations 1. The Potential Equation In this chapter, we will present an approach to the numerical solution of some types of elliptic equations based on the theory of dynamic programming. We start with the problem introduced in Chapter 4, the potential equation v 2 u = u,, uyy = 0, (1) + over a region R with the boundary conditions 24 (x,r) = f(x,r) 9 (2) on r the boundary of R. We have seen that (1) is the Euler equation associated with the minimization of the quadratic functional (3) subject t o (2).
Let u = 0 on r so that the functions u and u+ u satisfy the same boundary conditions on r. Then D ( u + u) = D(u) + D(u) + 2 SR (u,u, + u,uJ dR. (1) Applying Green’s theorem, we see that the third term vanishes if u satisfies the potential equation. 1) we have for any nontrivial u. 2) possess a solution, they possess a unique solution, since any solution minimizes D ( u ) . Conversely, it can be shown from first principles that D ( u ) possesses a minimum over the class of functions u such that u,, u,€L2 with u satisfying the specified boundary condition.
See R. Bellman, “Functional Equations in the Theory of Dynamic Programming XVII : Minimum Convolutions and Green’s Functions,” J. Math. Anal. , Vol. 33, 1971, pp. 497499. Section 8. The problem of computing 2‘ with the least number of multiplications is an unsolved problem of mathematics. For some partial results, see H. D. Thesis, Univ. of Southern California, 1970. D. E. Knuth, The Art of Computer Programming 11: Semi-Numerical Algorithms, Addison-Wesley, Reading, Massachusetts, 1969. Section 12.