Two New Finite Difference Methods for Computing Eigenvalues of a Fourth Order Linear Boundary Value Problem

This paper describes some new finite difference methods of order 2 and 4 for computing eigenvalues of a two-point boundary value problem associated with a fourth order differential equation of the form (py")" + (q r)y 0. Numerical results for two typical eigenvalue problems are tabulated to demonstrate practical usefulness of our methods.

p(x) d2y + [q(x) -%r(x)]y 0, (1.2) Such boundary value problems occur in applied mathematics, engineering and modern physics, (see ref. [I-4}. in the differential equation (1.1) the functions p(x), q(x), r C[a,b] and satisfy the conditions p(x) > 0, q(x) 0 and r(x) > 0, x [a,b]. (1.3) We cannot compute the exact values of the eigenvalues for which the boundary value problem (i.I) (1.2) has a nontrivial eigensolution y(x) for arbitrary chocies of the functions p(x), q(x) and r(x).We resort to numerical methods for computing approximate values of .The most eononly used technique for approximating A for WhiCh e-sysCem (i.I) (1.2) has a nontrivial eigenfunction y(x) is by finite difference methods.
Recently, the author [2] has analysed some new finite different methods of order 2 and 4 for computing eigenvalues of a two point boundary value problem involving the differential equation (i.I) with p(x) m associated with one of the following pairs of homogeneous boundary conditions: (a) y(a) y(b) y'(a) y'(b) 0 (b) the same boundary conditions as (1.2) (1.4) (c) y(a) y'(a) y"(b) y'"(b) 0. Chawla and Katti [3] have developed a numerical finite difference method of order 2 for approximating the lowest eigenvalue I of the system (I.i) (l.4(a)) with p(x) m i.A fourth order method was later developed by Chawla [4] for the numerical treatment of the same problem.This latter method leads to a generalized seven-band symmetric matrix eigenvalue problem.
Let I be any eigenvalue of the system (i.I) (1.2) and let y(x) 0 be the corresponding eigenfunction.Then on multiplying (1.1) by y(x) and integrating the resulting equation from a to b, we find after integration by parts and on using The purpose of this brief report is to present two new finite difference methods for computing approximate values of for the system (I.i)(1.2).These methods lead to general{zed five-band and nine-band symmetrlx matrlx eigenvalue problems and provide 0(h2) and 0(h4) -convergent approximations for the eigenvalues.

A SECOND ORDER METHOD
For a positive integer N > 5, let h= (b a)/(N + i) and x. a + ih, i 0(1)N + i.We shall designate Yi Y(Xi) Pi P(Xi) qi q(xi) and r i r(xi).
Note that the differential system (i.
The preceding system can be conveniently written in matrix form where Y (yi), V (v i) T 1 (pi) are N-dimensional colunm vectors with Pi Y(4)(Oi) P diag (i i) and J (jmn) is a tridiagonal matrix so that It can be verified that the matrix A JPJ + h4Q is a five-band symmetric matrix.Now, in (2.6), neglect truncation error F, replace Y by , then our method for computing approximations A for A of the system (I.i) (1.2) can be expressed as a generalized seven-band symmetric matrix eigenvalue problem A AN4R (2.8) In fact the matrix JPJ is a positive definite matrix and hence for any step-size h > 0, the approximations A for A by (2.8) are real and positive for all p(x) > 0 and r(x) > 0. That our method provides 0(h2) convergent approximations A for l can be established following Grigorieff [5].We omit the proof of convergence for brevity.
3. A FOURTH ORDER METHOD Following Shoosmith [6] the boundary value problems 2.1(a) and 2.1(b) are discretized by the finite difference scheme Similarly, for the system 2.1(b), we obtain the linear equations M 12h (3.4) The elimination of V from (3.2) and (3.3) gives our method for computing A for k of (I.I) (1.2) in the form (MPM + 144h4Q)TY 144Ah4RTI (3.5)where the matrix MPM is a nine-band positive definite matrix and hence for any step-size h > 0, the approximations A for by (3.5) are real and positive for all p(x), r(x) > 0. As before, it can be proved from the results of Grigorieff [5] that our present method provided 0(h4) convergent approximations A for k.

NUMERICAL RESULTS
In order to illustrate our methods of order 2 and 4 for the approximation of k satisfying (I.I) (1.2), we consider the eigenvalue problems: )4 [(I + x2)y"] + 1 ),(I + x ]y 0 (4.1) (i + x2) y(O) y(1) y"(O) y"(1) 0 The smallest eigenvalue i   ACKNOWLEDGEMENT.This work was supported in part by a grant from the Natural Sciences and Engineering Research Council of Canada.The author also acknowledges the assistance of Mr. Manzoor Hussain for making numerical calculations presented in Tables I and II.