A VARIATIONAL FORMALISM FOR THE EIGENVALUES OF FOURTH ORDER BOUNDARY VALUE PROBLEMS

This paper describes a variational approach for computing eigenalues of a two point boundary value problem associated with coupled second order equations to which a fourth order linear differential equation is reduced. An attractive feature of this approach is the technique of enforcing the boundary conditions by the variational func- tional. Consequently, the expansion functions need not satisfy any of them.


I. INTRODUCTION.
In a number of papers finite difference methods have been used to solve the fourth order linear differential equation: y(4) + p(x) q(x) ]y 0 (I.I) subject to one of the following pairs of homogeneous boundary conditions: In (I.i), the functions p(x),q(x) eC[a,b] and they satisfy the conditions p(x) 0, q(x) > 0, x E [a,b]. (1.3) Such boundary value problems occur frequently in applied mathematics, modern physics and engineering,see [1,2,3,4].Chawla and Katti [5] have developed a finite difference method of order 2 for computing approximate values of for a boundary value problem (l.l)-(l.2a).For the same problem, a fourth order method is developed by Chawla [6] which leads to a generalized seven- band symmetric matrix eigenvalue problem.More recently, Usmani [7] has presented finite differnce methods for (1.1)-(1.2b)and (I.I)-(1.2c)which lead to generlized five-band and seven-band symmetric matrix eigenvalue problem.
In the present paper we follow a different approach.We reduce the fourth order A.A. HAJJ equation (I.I) to two coupled second order equations as follows: The problem (I.I) can now be written as f" + [p(x) q(x)]y 0 (l.5a) y" f 0 (1.Sb) The associated boundary conditions (1.2) can be written in this case as: (1.6c) In the nert section we propose a variational priciple for the solution of (1.5) and (1.6) with the following attractive features: I.The proposed functional is a general one in the sense that it solves (1.5) and any pair of associated boundary conditions (1.6).
If.The boundary conditions are enforced via suitable terms in the functional and hence the expansion (trial) functions need not satisfy any of them.III.The variational technique employed leads to stable calculations and to a high con- vergence rate.
2. A FUNCTIONAL EMBODYING ALL THE BOUNDARY CONDITIONS.
In this section we produce the functional: b h(u,v) which incorporates the boundary conditions (1.6).The parameters a, a2, and 0.3 are set equal to either or 0 depending on which pair of the boundary conditions (1.6) is taken with (1.5).
Theoreml.The functional (2.1) is stationary at the solution of (1.5)-(1.6a),where for this pair of boundary conditions a is set equal to I; a=a3=O.
Upon substituting (2.4) in (2.3) and using (1.5a), we get G[u,v 1,Xv] -26X qy dx 0 llence, die 0 to OII6Yll-In an identical manner, it can be show, that 6X 0 to ol16fll.Thence, the equation G[u,v,X] 0 does not change to the first order in 6y and 6f.Fhis establishes the val- idity of X(u,v) as a functional for this problem.
The proof of theorem2 and Theorem3 Parallel that of theoreml and, therefore, are omitted.
Inserting (3.1) into the functional (2.1) and finding the stationary value of the func- tional leads to the 2x2 block matrix equation In order not to introduce artificial singularities in the matrix D and for stability reasons (see mikhlin [8]), we choose in (3.1): where for convenience, we number the basis functions from -2 to N-3 and where the Ti(x are chebyshev polynomials of the first kind; o is a linear map of x onto [-I 4. EFFICIENT CALCULATIONS. To calculate the elements of the matrices D and @ in (3.3), we need tile following: (1-x 2)k TE dx @ji @ij @-I,-I -2/3, 0__i,__2 O, 0__2,__2 -2, 01 -(,/4) [8 (3/2)   ",-i+l 3/2) Proof.For i,j _-> O, @.. can be written in the form 'l (l-x2)5/2Ti(x)T|(x) dx Oij -I (l_x2) (4.9) 'rhe result then follows from (4.1) and (4.5) afld the orthogonality relations of the Chebyshev polynomials; and similarly for the first two rows and columns of @.
In the same manner, the elements of the matrix D in (3.3) can be related to B s+)" using the identity 2 The elements of the matrices and require a slightly differet treatment due to the presence of the function p(x) in ij and q(x) in ij" and similarly for the function q(x).We assume that Fast Fourier Transform techniques are used to approximate a(O), y 0,1 N-1 via the scheme 21/2 a(0) (2/) [p(x)Ty(x)/(1-x dx (4.12) -1 n m m3 (2/n) p(cos---)cos(----_,, n='N (4.13) m=0 and that we approximate a(0)=" 0, y N wllenever these coefficients appear.It is not difficult to show that the coefficients exhjr and axh+2jr (and similarly for the elements of in terms of the expans+/-on coefficients of q(x)), in a manner that parallels that of theorem3 and details are omitted.
To test the formalism itroduced in this paper numerically, we take the second order version of problem (I.I) by considering the solution of the following regular Sturm-Liov- ille problem: The proof of the validity of this functional parallels that of theoreml and, therefo is omitted.Next, let N y(x) YN(X) Y. zihi(x), x e [a,b]   (5.3) i=I Substituting YN(X) for w(x) in (5.2) and finding the stationary value of the functional leads to the symmetric matrix eigenvalue problem: dimensional program an excellent approximation to the eigenvalue using inverse iterations.
Searching for the eigenvalue closest to and using a zero starting value with the num- ber of expansion functions N=7, it took the program only three iterations to produce an approximated eigenvalue with an error of order 10-8.This shows that the variational principle derived here givem nn attractive extension of the global variational method t the eigenvalue problems.Tb? technique avoids the need to search for trial functions that must satisf the boundary conditions since searching for such trial functions has proven, in many cases, to be technically complicated.

2 )B
-, k >= 2 can be related to I)-by the following:Similarly the half-integers can easily be related to B s) by the following: m= 4m 2 2m+E + B 2m El which follows from (4.1), (4.5) and the well known chebyshev expansion: At this stage, the elements of the matrices as follows: Assume, without loss of generality, that [a,b] [-I,I], then Theorem3.The elements @.. are given by 13 llere we require the expansion (1-x2) k p(x) 'klT(x)(4.11) [r(x)y']' + (p(x) q(x)'r(x)w' + q(x)w2)dx + 2[w(b)r(b)w'(b) w(a)r(a)w'(a)] and S are given by: b R [ [h'r(x)h' + h q(x)h dx b)r(b)h'(b) -h (a)r(a)h'(a)] To apply the technique presented in this paper, we consider the numerical solution of problem (5.1)where we choose r(x)= |, p(x)=1/2, q(x)=0 and [a,b] [O,I].In this case problem (5.1) has a theoretical eigenvalue =2.With basis (3.4), we obtain from a one !x, h >= are related to akul'o" by