SINGULAR PERTURBATION FOR NONLINEAR BOUNDARY-VALUE PROBLEMS

Asymptotic solutions of a class of nonlinear boundary-value problems are studied. The problem is a model arising in nuclear energy distribution. For large values of the parameter, the differential equations are of the singularperturbation type and approximations are constructed by the method of matched asymptotic expansions.

where r2(x) and f(x) are positive functions, n is a positive integer > 2, a > 0 and b > 0, by the method of matched asymptotic expansions.
The problem arises in connection with the distribution of the energy released r 2 in a nuclear power reactor as a result of a power excursion; (x) is the space- dependent perturbation in the neutron multiplication of a reactor, and (I.I) and (1.2) give the distribution of the energy release from the start of the perturbation till the neutron population again becomes zero, see Ergen (3).The case of zero boundary conditions and constant coefficients has been investigated in Canosa and Cole (4).It would be assumed that y is positive and bounded.An upper bound for the solution is given by y'(x) < ximum of 63 Note that if both r 2 and f are constant, then (2.2) implies that y cannot have any relative minimum.For definitness, it would be assumed in general that y'(0) > 0 and y'(1) < 0 But the results can be easily modified for the other possibilities.

SINGULAR PERTURBATION PROBLEM.
Consider the asymptotic case    where al (e) / 0 as e / 0.
Let f( ) be expanded as f(g ) f(0) + al (g) f R + In matching with the outer solution, the constant in (4.5) can be obtained.
If f' (0) 0, then (4.5) becomes 2 2 f (0') (go) is a polynomial of degree (n-i) in go" Similar can be obtained for the boundary layer at x i, with g0(R) replaced by say h0() where (I x)/e 1/2 f(0) by f(1) and f'(0) by f'(1) 5. EXPLICIT ASYMPTOTIC SOLUTIONS FOR SPECIAL CASES AND DISCUSSION OF RESULTS.
There are two special cases in which explicit asymptotic solutions can be obtained.For n 2, the first term in the outer expansion is given by I Y0 t() Transforming back to the origional variable y, we see that, away from the boundaries, r 2 the solution is given asymptotically by f(x) Equation (4.6) becomes 2 /2 f(0)3 (go f('10)') /go + 2 f(0)l or d go y2 f(0) Near the boundary x I, the first term of the boundary-layer solution is given by h (5.B) Equations (5.2) and (5.3) show the exponential decay of the boundary solutions into the outer solution, and the symmetry of the solution about the domain center if f has such symmetry and a b.The first term outer solution and (5.2) reduce to the ones given in Canose and Cole (4) when the coefficient f(x) m 1 and the boundary conditions are zero.
When n 3, the first term of the outer solution is Y0 /f(x) and so away from the boundaries, the solution is given asymptotically by Equation (4.6) now becomes v/f (x) In this case, we see from (5.5) and (5.6), the exponential growth of the boundary- layer solutions into the outer solution, and again the symmetry of the solution about the domaln center If f is symmetric and a b.The first term outer solutlon and (5.5) reduce to the ones given in Canosa and Cole (4) when the coefficient f(x) 1 and the boundary conditions are zero.

2 .
UPPER BOUND FOR THE MAXIMUM OF THE SOLUTION.Let the maximum value of the solution occur at x c d2 Y + p(x)Y f(x) yn 0(3.5)    dx Equation (3.5) is a singular perturbation equation, the asymptotic expansions of which and (1.2) will be studied in the remaining sections.
Outer SolutionAssuming the solution in the form of an asymptotic series ) + e Yl(X) + and substituting it into (4.1), the functions Y. (x) can be determined recursively.The first two terms are given by I f(e1/2 )yn 0 (4.2)The boundary-layer solution has the form Y(, e) go () + al (e) gl () +(4.3)