Quadrature approaches to the solution of two point boundary value problems

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1 Quadrature approaches to the solution of two point boundary value problems Seth F. Oppenheimer Mohsen Razzaghi Department of Mathematics and Statistics Mississippi State University Drawer MA MSU, MS Abstract In a split level numerical analysis course or introductory applied mathematics course the topics of Gaussian Quadrature and the solution of two point boundary value problems are often covered. It seems to the students that these topics are unrelated. However, by using Green s functions, one can develop very nice easily implemented numerical schemes to approimate the solution to two point boundary value problems via Gaussian quadrature. 1 Introduction The purpose of this note is to present a technique for the numerical solution of two point boundary value problems which, while well known in the research community [4], is not taught at the undergraduate level. This is unfortunate as the technique is easy to implement and remarkably accurate. In a first numerical analysis course, Gaussian quadrature is usually introduced. However, beyond approimating the integrals of some known functions, little is done with it. We will demonstrate a very nice, nontrivial application for Gaussian quadrature to the numerical solution of differential equations. For completeness, we will outline all of the necessary background topics. In section 2, we review orthogonal polynomials and Gaussian quadrature. In section 3, we will state the general two point boundary value problem we wish to solve and give an eample of transforming such a problem into an equivalent Fredholm integral equation via the use of a Green s function. Finally, in the last section we will show how to use Gaussian quadrature to transform the integral representation of the differential equation into a system of algebraic equations that will yield an approimate solution of the problem. Two eamples will be given. 1

2 2 Orthogonal polynomials and Gaussian quadrature In this section we will develop the set of orthogonal polynomials on the interval [- 1,1] called the Legendre polynomials and their application to the approimation of definite integrals, Gaussian quadrature. Treatments of both of these subjects may be found in standard numerical analysis books such as [2] and [5]. An inepensive monograph on orthogonal polynomials is [6] and the more general application of the techniques we are discussing in this paper may be found in [3]. We consider the space of continuous functions on [, 1], C ([, 1]). We know from the Stone-Weierstrass theorem that the set of polynomials is dense in C ([, 1]), that is given a continuous function f and a positive number ε, there is a polynomial p so that for all in [, 1] p () f () < ε. The question is, given an f, howdowefind p? The answer is interpolation. That is, we choose N +1 points 0,..., N and find the unique polynomial of degree N +1 that agrees with f at those points. That is, we want the polynomial to take on the value f ( i ) at i. However, for some functions and some choices of points, increasing the number of interpolation points will not improve the polynomial approimation. To find sets of interpolation points so that as the number of interpolating points increases the difference between f and the interpolating polynomial decreases requires a detour into a discussion of orthogonal polynomials. Given a positive integrable function ρ on [, 1], called a weight function, we can define an inner product on C [(, 1)] by (f,g) = f()g()ρ()d This allows us to define orthogonal functions by saying that f and g are orthogonal if (f,g) =0. Since a polynomial of finite degree has only a finite set of roots, it is easy to see that any set 1,, 2,..., Mª is linearly independent in C [(, 1)]. We can therefore use the Gram-Schmidt orthogonalization process q 0 () = 1 q k () = k X k,q i () (q i (),q i ()) q i (),k=2,..., M i=0 to obtain a set of orthogonal polynomials {q 0,q 1,..., q M } which spans the same subspace in C [(, 1)] as 1,, 2,..., Mª. When ρ () 1, we obtain the Legendre polynomials, L 0,L 1,..., L M This set of orthogonal polynomials has many nice properties. One of these nice properties is the fact that the Legendre polynomials satisfy the following recursion 2

3 relation L 0 () = 1,L 1 () = L m+1 () = 2m +1 m +1 L m () m m +1 L m (). This allows us to easily generate the Legendre polynomials. The property of the Legendre polynomials we are most interested in, concerns polynomial interpolation. Let I N map C ([, 1]) onto the polynomials of degree N or less wherei N (f) is the interpolating polynomial for f where the interpolating points are the zeros of the N th Legendre polynomial. Then we have the following. lim I N (f)() f () 2 d =0. N In other words, under rather weak conditions, the sequence of interpolating polynomials will converge mean square to the function f. Our approach to approimating will be to approimate it by f () d I N+1 (f)() d an easily computed integral of a polynomial. (The increase in interpolation points is for notational convenience below.) We can make the computation of the integral still easier by bypassing the computation of the interpolating polynomial I N+1 (f)(). We do that with the theorem below. The quadrature method below [3] is equivalent to simply integrating I N+1 (f)(). Theorem 1 Let 0,..., N be the roots of the (N +1)-th orthogonal polynomial, p N+1, on (a, b) with respect to the weight function ρ and let w 0,..., w N be the solution of the linear system Then ( j ) k w j = Z b a k ρ () d, k =0,...N w j > 0 for j =0,..., N and p ( j ) w j = Z b a p()ρ () d, for all p P 2N+1. The w j are called weights. We will call the roots, j, the collocation points. 3

4 We will find the roots and weights for N =7which we will use below. In this case the approimate roots, calculated on Mathematica, are , , , , , , , and We set up the 8 8 linear system in Matlab and solved for the weights Just to see how well this works, observe that i w i e d = e 1 e and Not too bad. In general we have, 7X ep ( j ) w j lim I N+1 (f)() d = f () d N 3 Two point boundary value problems We will consider the following nonlinear two point boundary value problem L[u] = p 0 () u 00 + p 1 ()u 0 + p 2 () u = g(, u), <<1 B l [u] = α l u() + β l u 0 () = 0 (NLBVP) B r [u] = α r u(1) + β r u 0 (1) = 0 First we will eamine the following simpler boundary value problem. L[u] = p 0 () u 00 + p 1 ()u 0 + p 2 () u = f(), <<1 (BVP) B l [u] = α l u() + β l u 0 () = 0 B r [u] = α r u(1) + β r u 0 (1) = 0 4

5 where p 0 is nonzero and continuously differentiable. It is known, [1], that if the operator L does not have 0 as an eigenvalue, then (BVP) has a unique solution for each continuous f. Thechallengeisfinding the promised solution. There are a wide variety of methods for approimating a solution to (BVP) such as shooting methods and finite difference methods. However, classically, one sought a Green s function and found the solution to (BVP) as an integral. Here we give the integral equation form. u() = K[f] (IEF) = G (, s) f(s)ds. We will now do an eample in which we find the Green s function G. Green s function shall be used in our numerical eamples later on. Consider the following problem This u 00 = f (EX1) u 0 () = 0 u (1) = 0 We may interpret u physically as the temperature in a bar with an insulated left hand end, a right hand end held a fied temperature of 0, and a distributed heat source of intensity f. We will now do various manipulations to put (EX1) into integral equation form (IEF). First we integrate from to to obtain u 0 () u 0 () = Using the boundary condition at yields We now integrate from to 1. u 0 () = u (1) u () = Z Using the boundary condition at 1 yields u () = We now integrate by parts setting w = Z f (t) dt Z s Z s Z s and dv = ds f (t) dt f (t) dtds f (t) dtds f (t) dt 5

6 Then " Z s u () = s = = = f (t) dt 1 f (t) dt f (s) ds + G (, s) f (s) ds Z Z # sf (s) ds f (t) dt f (s) ds + sf (s) ds sf (s) ds where ½ 1 s< G (, s) = s 1 s>. We are now ready to return to (NLBVP). The idea at this point is very simple. If u is a solution to (NLBVP), then setting f () =g (, u) will yield that u also satisfies (BPV ) and consequently or u() = u() = G (, s) f(s)ds. G (, s) g (s, u (s)) ds, (NLIEF) which is a Fredholm integral equation of the first kind. This is a nonlinear integral equation. Since we will be working in situations where the solution to (NLBVP) is unique, a solution to (NLIEF) is the unique solution to (NLBVP). In the net section we put together what we have done so far and give an algorithm for the numerical solution of (NLIEF). 4 The numerical approach Let 0,..., N be the roots of the (N +1)-th Legendre polynomial and w 0,..., w N be the corresponding weights so that for a continuous function f Z b a f()d f ( j ) w j We consider (NLIEF) and apply the approimation u() = G (, s) g (s, u (s)) ds G (, j ) g ( j,u( j )) w j. 6

7 Since this approimation is valid for every in [, 1], itistruefor i, i = 0, 1,...N. Thus u( i ) G ( i, j ) g ( j,u( j )) w j,i=0, 1,...N (AIEF) Let us denote the approimation of u( i ) by u i and define u i by requiring (AIEF) to hold eactly for the approimation. That is u i = G ( i, j ) g ( j,u j ) w j,i=0, 1,...N (AE) This a system of N +1 algebraic equations in the N +1 unknowns u 0,u 1,..., u N. For our first eample, we will study a linear problem. u 00 = g(, u) u 0 () = 0 u(1) = 0. where g(u, ) =u + e e The eact solution to this problem is u() =e + 1 e2. e We will use N =7, for eight points. The system turns out to be an 8 8 linear system which we can solve by our favorite method. The results may be 7

8 seen in Figure 1 and Table 1. 0 Figure 1. -The true solution, * the approimate solution y i u( i ) u i error Table 1. Comparison of the true solution with the quadrature approimation For our second eample, we will merely change g to a nonlinear function of u. g(, u) =u In this case the solution is u () = We will use N =7, for eight points. The system turns out to be an 8 8 nonlinear system. The results may be seen in Figure 2 and Table 2. The 8

9 solution to (AE) in this case was found using the built in Matlab function fmins. However, in a numerical analysis class, this could be used as an opportunity to apply a quasi-newton method. 0 Figure 2. -The true solution, *the approimate solution y i u( i ) u i error Table 2. Comparison of the true solution with the quadrature approimation While not as nice as the results for the linear problem, it is still very good for eight points. To see this, let us compare the results obtained for a finite 9

10 difference approimation. The results may be seen in Table 3 and Figure 3. Figure 3. -The true solution, +the approimate solution via finite difference, * via quadrature y i u( i ) u i error Table 3. Comparison of the true solution with the finite difference approimation 5 References References [1] G. Birkhoff and G. C. Rota, Ordinary Differential Equations Second Edition, Blaisdell Publishing Company,

11 [2] R. L. Burden and J. D. Faires, Numerical Analysis, Sith Edition, Brooks/Cole, 1997 [3] C. Canuto, M.Y. Hussaini, A. Quarteroni and T.A. Zang, Spectral Methods in Fluid Dynamic, Springer-Verlag, 1987 [4] G.N.ElnagarandM.Razzaghi,A Pseudospectral Method for Hammerstein Equations, Journal of Mathematical Analysis and Applications 199, (1996). [5] F. B. Hildebrand, Introduction to Numerical Analysis, Dover, [6] G. Sasone, Orthogonal Functions, Dover,

Recall that any inner product space V has an associated norm defined by

Hilbert Spaces Recall that any inner product space V has an associated norm defined by v = v v. Thus an inner product space can be viewed as a special kind of normed vector space. In particular every inner