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Math. Anal. Appl. Vol. 5, 1962, pp. 499-501. Chapter 4 The P o t e n t i a l E q u a t i o n 1. Introduction In this chapter we wish to discuss some aspects of the potential equation, u,, + uyy = 0 , * = L7(X,Y), k Y )5 R, (1) (-X,Y) E r, (2) where I- is the boundary of the region R in Fig. 1. We want to consider its connection with the minimization of the quadratic functional ,. a ~ ( u= ) 22 J R (u,’ + uY2)d R , FIGURE I (3) 23 2. The Euler-Lagrange Equation the Dirichlet functional, and a number of problems associated i n this fashion.
Quadratic Case The results simplify greatly when h ( x , y ) is quadratic in x and y and g(x, y ) is linear. Consider, for example, the minimization of PT J ( x ) = J [(x’,x’) 0 + ( x , A X ) ] dt , (1) where A is positive definite and x(0) = c. It is clear that f ( c , T ) = minJ(x) X = (2) (c,R(T)c), where R ( T ) depends only on T. 6) yields fT = min z [(z,z ) + (c, A c ) + ( z , gradf’)] . (3) The minimization with respect to z is readily accomplished, yielding z = -gradf/2, fT = (4) (c, A c) - [(gradj; grad f I/ 41 .
See R . Bellman, and J. Math. Anal. , Vol. 34, 1971, pp. 235-238. Section 16. This approach is taken by R . Varga t o analyze a number of standard techniques. See R. Varga, Matrix lteratice Analysis, Prentice-Hall, Englewood Cliffs, New Jersey, 1962. A. Ralston and H. S. Wilf, Marheniatical Methodsf~irDigital Contprrters, Vols. I and 2, Wiley, New York, 1965 and 1967. Secfion 17. John Todd points out that AD1 only works if certain matrices commute and we are in standard academic situations. In practice, however, the situation is often favorable.