BAIL 2006 Book of Abstracts - Institut für Numerische und ...

A.I. ZADORIN: Numerical Method for the Blasius Equation on an Infinite Interval
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where g(t), β(t) are solutions of auxiliary singular Cauchy problems and can be found as series
in (2t) −0.5 with a given accuracy. We can use equation (5) as the exact boundary condition for
a finite interval. Then we return to the variable x and get the exact restriction of problem (3)
to a finite interval as follows:
εu ′′ (x) + [a(x) + x]u ′ (x) = f(x),
u(0) = A,
ε
L u′ (L) + g(L)u(L) = β(L).
Return to problem (2). Consider the iterative method
v ′ n(x) = wn−1(x), vn(0) = 0,
w ′′
n(x) + [vn(x) + x]w ′ n(x) = 0, wn(0) = −1, lim
x→∞ wn(x) = 0.
It is proved that method (7) has the property of convergence. At each iteration, we can transform
problem (7) to a problem for a finite interval, as it was done for a linear problem. One can use
a difference scheme to solve the problem obtained on a finite interval. Theoretical results are
confirmed by results of numerical experiments.
References
[1] G.I. Shishkin, “Grid approximation of the solution to the Blasius Equation and of its Derivatives”,
Computational Mathematics and Mathematical Physics, 41 (1), 37–54 (2001).
[2] A.A. Abramov, N.B. Konyukhova, “Transfer of admissible boundary conditions from a singular
point of linear ordinary differential equations”, Sov. J. Numer.Anal. Math. Modelling,
1, (4), 245–265 (1986).
[3] A.I. Zadorin, “The transfer of the boundary condition from the infinity for the numerical
solution of second order equations with a small parameter” (in Russian), Siberian Journal
of Numerical Mathematics, 2 (1), 21–36 (1999).
Speaker: ZADORIN, A.I. 62 BAIL 2006
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