miranker well posed

A Well Posed Problem for the Backward Heat Equation Author(s): W. L. Miranker Source: Proceedings of the American Mathematical Society, Vol. 12, No. 2 (Apr., 1961), pp. 243-247 Published by: American Mathematical Society Stable URL: http://www.jstor.org/stable/2034314 . Accessed: 31/03/2011 20:12 Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at . http://www.jstor.org/action/showPublisher?publisherCode=ams. . Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. American Mathematical Society is collaborating with JSTOR to digitize, preserve and extend access to Proceedings of the American Mathematical Society. http://www.jstor.org A WELL POSED PROBLEM FOR THE BACKWARD HEAT EQUATION W. L. MIRANKER 1. Introduction. A boundary value problem is well posed in the sense of Hadamard if three conditions are met. These are existence, uniqueness, and continuous dependence of the solution on the bound- ary data. In recent years some attention has been given to improperly posed problems [1; 2; 3 ]. The point of view in these matters is directed at approximating in some stable way the solution of a problem which violates the last condition of well-posedness just described. In this paper we will consider the backward heat equation, which leads to a classical improperly posed problem. To see that this is the case, consider the forward heat equation (1.1) ut = u, t > 0, with the initial condition (1.2) u(x, 0) = f(x) = n sin nx. The solution is (1.3) u(x, t) = ne-2n sin nx. Thus the forward heat equation has a sequence of solutions which tend to zero while their corresponding initial values tend to infinity. Thus the backward heat equation which consists of solving (1.1) for t < T say, with data given for t = T has solutions which do not depend continuously on their initial data. The question of existence of a solution to the backward heat equa- tion is also a delicate matter. To see this we observe that the solution of (1.1) and (1.2) may be written as rX (1.4) u(x,t) = f(fa)K(x-aT,t)do- KTf Goo where 1 (1 .5s) K(x, t) =(g) 1K2 exp (-x2/4t). 2(7rt)'/2 Therefore when a solution of the forward heat equation exists it is analytic for all t >0. Therefore one can only prescribe certain analytic Presented to the Society, August 30, 1960; received by the editors April 1, 1960. 243 244 W. L. MIRANKER [April data for the starting values of the backward heat equation. Moreover, since f(a) need not be analytic, the time over which the solution to the backward heat equation exists depends on the initial values. In this paper we will consider the backward heat equation in L2. We will show that in a subspace of L2 consisting of functions whose Fourier Transforms have compact support, the backward heat equa- tion leads to a well posed problem in the sense of Hadamard. We also provide a stable and convergent iteration scheme by means of which the solution may be computed. 2. Statement of results. Letf(x) C L2(- oo, oo), i.e., 11fl 2=flf(X) J 2dx < 00 ,1 and let the Fourier Transform of f(x) be denoted by (2.1) F(co) = Sf(x) = j e-Xf(x)dx. (2)/2J Let Q be a fixed positive number and denote by B the following sub- space of L2: B = {f(x) E L2l F(w)=-O, I l I> Q4.2 For convenience we introduce the following: DEFINITION. A function u(x) is said to have diffused from f(x) in time T if u(x) = u(x, T) with u(x, T) given by (1.4). Our result is given in the following: THEOREM. Let u(x) -B. Then to each T> 0 there exists one and only one f(x) CB such that u(x) has diffused from f(x) in time T. f(x) is given by (2.2) lim llfn -fl - 0 where (2 . 3) fnIfl (x) = ul(x) + (I - KT)fn(X)X (2.3) ~~~fo (x) = 0. Moreover, if u(x) is changed to u(x) +Au(x) then the change Af in f satisfies the relation 1 All integrals are taken from (- oo, oo) unless otherwise specified. 2 The linearity of S implies that B is a linear manifold. That it is closed follows from the Plancherel theorem since fJfn -f 2dx=f - -FJ 2dco assures us that Fn- +F whenever fn- f. i96i] WELL POSED PROBLEM FOR THE BACKWARD HEAT EQUATION 245 (2.4) lI ? CjjAU where C is a constant. PROOF. Since (2.5) SKTf = (27r)-1/2 e-T'F(w), and Fo(w) = 0, it follows from (2.3) that fnGB for all n. Now making use of the fact that S is of an isometry of L2 onto itself, we have from (2.3), If+ -hfnil = 11 (I -KT)(fn -fn-1)1I =-|| (1 - (2r)12 ~e-T ) (Fn - Fn-1)| (2.6) = [f0 1(1 _ )(Fn - Fnj1) do] _ ( e-TQ2) - Fn_.|J -1 |fn -fn-11 I J Since 0 < 1, the sequence f.(x) is a Cauchy sequence of functions in B. Since B is closed, this sequence has a limit fGEB. (2.6) shows that (I-KT) is a continuous operator in B. Thus taking the limit as n- o> o in (2.3) we obtain (2.7) u(x) = KTf(x). If u(x) is replaced by u(x) +Au, it follows from N f = lim E (fn+l -fn) N- o n-0 N = lim a (I- KT)(fJ -f - ) +fi (2.8) N = lim a (I-KT)n(fl--fo) +fl N-c n-1 N - lim E (I - KT)nU N- c n-O that 246 W. L. MIRANKER [April N (2.9) Af = lim , (I-KT)nAU. N-*oo n =O Thus 1, (2 .10) 114fjj ? jj_M- tjj.l This also proves the uniqueness of f. Q.E.D. REMARK. The theorem assures both the convergence of fn to f and the boundedness of Af in the L2 sense. To satisfy ourselves that the instabilities of the backward diffusion process have not been masked by an integral mollifying process, we remark that the convergence of fn to f and the boundedness of Af are assured also in the maximum norm. This we state as the following: COROLLARY. (a) The sequencef. produced in the previous theorem con- verges in the maximum norm, i.e., (2 . 11) lim max fn(x) -f(x) O - (b) The error Af defined in the previous theorem is bounded in the maxi- mum norm, i.e., (2.12) max I Af I ; C| jAuj I. x PROOF. The proof consists of the observation that in B, L2-con- vergence implies Lo-convergence. Viz., if f(x) CB then max = max 1I2 e%@0F(w)dco (2ir)'1 I aK \1/2/ r \1/2 (2.13) X (2mr)a/x ( f d )( fF2d- ) -' (Q)iI21f. REMARK. (2.12) shows that the iteration scheme is stable even if arbitrarily large errors in the maximum sense are made in u, provided only that the corresponding L2 norm of these errors is small. REMARK. If the iteration scheme is implemented, numerically say, there is always the danger that a specific iterate will not be in B be- cause of errors in the actual means of implementation. However, it is clear that even though the successive functions so obtained do not I96I] PERRON-FROBENIUS THEORY AND THE ZEROS OF POLYNOMIALS 247 appear to converge, the projections of the Fourier Transforms of this divergent series onto I w I are converging both in the L2 norm and the maximum norm. BIBLIOGRAPHY 1. J. Douglas, Jr. and T. M. Gallie, An approximate solution of an improper boundary value problem, Duke Math. J. vol. 26 (1959) pp. 339-347. 2. F. John, Numerical solution of the equation of heat conduction for preceding times, Ann. Mat. Pura Appl. ser. IV vol. 40 (1955) pp. 129-142. 3. C. Pucci, Sui problemi di Cauchy non "ben posti," Atti Accad. Naz. Lincei. Rend. Cl. Sci. Fis. Mat. Nat. vol. 18 (1955) pp. 473-477. INTERNATIONAL BUSINESS MACHINES CORPORATION PERRON-FROBENIUS THEORY AND THE ZEROS OF POLYNOMIALS HERBERT S. WILF1 1. Introduction. Our purpose here is to show that many of the classical root location theorems for. polynomial equations, normally derived by the methods of complex analysis, can be obtained easily, and in a purely algebraic manner, from the Perron-Frobenius theo- rems on matrices with nonnegative elements. An important result of this approach is a "minimax" principle which gives precisely the largest root of an equation which dominates the given one. It will be seen that from this principle the above-mentioned location theorems follow, and can be sharpened almost at will. Finally some applica- tions to the theory of orthogonal polynomials will be made, the result again being a minimax principle for the largest zero from which two- sided bounds can be deduced by specialization. 2. Cauchy's Theorem. Let C be an nXn complex matrix, and let C+ be given by (1) (C+) ij c= | 0l ij=1,**, n). A lemma of Wielandt [1] asserts that if C+ is irreducible, y is any eigenvalue of C, and r is the largest real eigenvalue of C+, then y I <_r. Let Presented to the Society, January 24, 1961; received by the editors May 13, 1960. 'This work was supported in part by the National Science Foundation.