SPLINE SOLUTIONS FOR NONLINEAR TWO POINT BOUNDARY VALUE PROBLEMS

Necessary formulas are developed for obtaining cubic, quartic, quintic, and sextic spline solutions of nonlinear boundary value problems. These methods enable us to approximate the solution of the boundary value problems, as well as their successive derivatives smoothly. Numerical evidence is included to demonstrate the relative performance of these four techniques.

n n n 0(1)N + i, and let s(x) be a spline function of degree m on [a,b].Thus, in each subinterval Ix i, Xi+l], s(x) is a polynomial of degree at most m and cm-i s (x) e [a,b].We shall designate this polynomial by j=m j P.(x) aij (xx i) i 0(1)N, x e ix i, Xi+l].
In this paper we shall present some methods for the continuous approximation of the solution of the two point real nonlinear boundary value problem y (x) f(x, y(x)), a < x -< b y(a) A y(b) B 0 (1.2) by the use of spline functions of orders up to six.The function f(x, y(x)) is a continuous function of two variables with f continuous and nonnegative in Y the strip S defined by S: a < x < b, < y < It is well-known that the boundary value problem (1.2) with these conditions has a unique solution (Henrici ill, . 347).
2. CUBIC SPLINE SOLUTION (m 3) The possiblities of using spline functions for obtaining smooth approximations of the solution of boundary value problems were first briefly discussed by Ahlberg et al. [2].Following this,Bickley [3] and Albasiny et al. [4] have demonstrated the use of cubic spline function for obtaining an approximate solution of (1.2) when f(x, y(x)) p(x) y'(x) + q(x) y(x) + r(x).

N
The approximate values of y(x) and its derivative at the points other than knots are obtained by evaluating or differentiating the corresponding cubic spline polynomial.
3. SOLUTION OF NONLINEAR EQUATIONS (i. 3) The method which we shall use to obtain the solution of the system (1.3) is a generalization of Newton's method which we summarize very briefly for the sake of completeness (Henrici [I], p.355).Let the nonlinear equations (1.3) in N unknown Yi be written in the form i(Yl' Y2 Yn 0, i I(1)N or,in vector form, R.A. USHANI (3.2) where -- (i), Y (yl) are N dimensional vectors.Let M(Y) (mlj) denote the matrix with elements i, j I(1)N.
Then the Newton's method for the solution of (3.1) is written in the form y(p+l) y(p) [M(y(p))]-i (y(p)), (p 0,i, ). (3.3) In our case ,the elements mij are given as follows: mij 2 + (4h2/6)gi i -1 + (h2/6)gj, i J 1 0 l-Jl> where gi fy(xi' yi ).The vector (Y) is usually referred to as residual vector.The criterion for stopping the iterations defined by (3.4) is that the residual vector (Y (p)) be such that II(Y(P))II l(YCe)) < , (p 0,, ), where e is a preassigned small positive quantity.Also, the system of nonlinear equations (3.2) has a unique solution Y, to which the successive approxtions Y(P) defined by (3.4)  where the function Q(x) providing the initial approximation is such that Q4 max [QiV(x)[, R max iQ"(x) f(x, Q(x))[, and x x y) (x,y) fyy' In deriving (3.11),we use the theory of monotone matrices as given by Henrici ([I], p. 360).If the initial approximation vector y(0 be such that the quantity R is small, then it follows from (3.11) that the solution of the system (1.3) obtained by Newton's method will converge to the solution of (1.2) for all sufficiently small values of h.

QUARTIC SPLINE SOLUTION
We now consider (i.i) for m 4. With the analogy of section 2 on cubic spline, we can determine the five coefficients of (i.i) in terms of Yi' Yi+l' Mi' Mi+l and D i where we now write P'.
Continuity of the first and third derivative at x x i gives the relations [that is P' (xi) P'i (xi) and Pi-I"' (xi) e'"i (xi) 4hai_l, + ai_l, 1 ai, 1 which on using (4.1) reduce to ( and We equate the expressions on the right side of the equality sign in (4.3) and (4.4) respectively and obtain which collapses,on simplification, to the recurrence relation which is the same as the well-known Noumerov's formula.As before, we first determine the unknowns Yi' i I(1)N by solving the system of nonlinear algebraic equations by Newton's iterative method explained in the previous section and then compute Di, i l(1)N+l, using (4.3)And now the knowledge of Yi' Di' Mi' i 0(1)N+I enables us to produce the coefficients of quartlc spline as given by (4.1).An approximation of the third derivative at knots is given by Yi 6ai, l, i 0(1)N 24haN, 0 + 6aN,I -12(YN+ I yN)/h 3 + 12DN/h 2 (4.7) + 3(+ + )/h, i N+I.

CONVERGENCE
Define e i y(x i) y+/- and E (e n), T (t n) are N-dimensional vectors; then the error equation in any of the four methods is obtained in a standard manner in the form where M is a tridlagonal matrix for cubic and quartic spline functions and M is a five band matrix for quintic and sextlc spllne solutions of the boundary value problem (1.2).It is easily seen that the truncation error associated with (i.3) is 0(h4) and IM-III -< (b a)2/(8h2) where the elements of M are given by (3.5).Thus we easily deduce that for a cubic spline solution of (1.2) and, from this, it follows that as h / 0 (i.e.N / ) the cubic spline solution Yi / Y(Xi)" Thus cubic spline solution based on (1.3) is a second order convergent process.We can similarly prove that the quartlc spllne solution based on (4.5) is a fourth order convergent process.
In order to prove the convergence of the quintic solution based on (5.5) and (5.6), we observe that the truncation error is given by h6y(6)(l)/48' x0 < i < x3 t i h6y (6)(xi)/120 + 0(h7), i The elements of the corresponding matrix M are given below m12 2 + (14h2g2/12), mN,N_1 2 + (14h2gN_i/12).The error analysis depends on the properties of the matrix M and M where M is a five band matrix obtained from M by setting each gt O, so that M (mi) and.
(7.10) Following a similar t.echniqu% we can establish that for m sextic spline solution lEll < h6.

NUMERICAL ILLUSTRATIONS
We solve two nonlinear boundary value problems of the form (1.2). y'' 0 5(x + y + 1) 3 y(0) y() 0, with y(x) 2/(2 x) x-i.The function Q(x) is chosen to satisfy the system Q" Here Q(x) [sinh x + sinh (i x)]/sinh 1 i.The numerical calculations are made using double precision arithmetic in order to keep the rounding errors to a minimum.The numerical results are briefly summarized in Tables i-3.

CONCLUDING REMARKS
Our numerical results on test problems indicate that results based on quintic and sextic spline are only marginally better than those obtained by quartic spline solution.Moreover, in order to obtain nonlinear equations equal to the number of unknowns in quintic and sextic spline solution, an ad hoc procedure is used near the boundaries of the interval.Also, the matrices that arise are five band matrices whereas in case of cubic and quartic spline solution the bandwidth of the matrices that arise is three, which makes them slightly simpler to implement.
Since formulas (1.3) and (4.5) satisfy the conditions of Theorem 7.4 (Henrici [I]), Richardson's h 2 extrapolation method can be used to push the accuracy of these formulas to 0(h4) and 0(h6) respectively, whereas in case of quintic and sextic spline solut ions we can only use h-extrapolation technique to improve the numerical solution.The latter techniques also suffer from a disadvantage that they require approximate formulas for S O and SN+I [see (5.7)].Similarly, in sextic spline solution, in order to compute Di, i I(1)N+I, we must provide D O [see (6.11)].
Finally, in the opinion of this author, one should rely on quartic spline solution for a smooth approximation of the solution and its successive derivates for the nonlinear boundary value problem of the type (1.2).
i fi fi f(xi'

Table I
Observed max.error IEII for (s.x) in Yi based on