initial guess for a b.v.p.

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Gualtiero
Gualtiero 2011-2-5
I am working on the solution of the following 4th order ode:
y'''' -2*(pi^2)*y''-(1/eps)*y'+(pi^4)*y=0
b.c.: y(0)=y(1)=0; y''(1)=0; y'''(1)=0
The solution can be worked using bvp4c. The initial guess for y, y', y'' and y''' for bvpinit can be worked out as different polynomials. However, in the command
solinit = bvpinit(linspace(0,1,100),@mat4init);
when I change the number of mesh-points (in the example: 100) the solution changes drastically its shape. Is there any way to solve this problem?

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回答（1 个）

Walter Roberson
Walter Roberson 2011-2-5
I don't know anything about bvp4c, but it appears the solution is
y(t) = Sum(exp(RootOf(Pi^4*eps-_Z-2*Pi^2*_Z^2*eps+_Z^4*eps, index = _a)*t)*_C[_a], _a = 1 .. 4)
That is, there is a quartic, x^4*eps - 2*pi^2*x^2 + pi^4*eps . There are four solutions to that quartic, some of them imaginary. Find those solutions, take exp() of each one and multiply it by an arbitrary constant, and then add all the results together.
If eps is 2^(-52) then the sum involves quantities as high as 3E71719
I am inexperienced in solving differential equations and it is possible that I got the notation wrong when I pumped it through Maple. The Maple was
dsolve({diff(diff(diff(diff(y(t), t), t), t), t)-2*Pi^2*(diff(diff(y(t), t), t))-(diff(y(t), t))/eps+Pi^4*y(t), y(0) = 0, y(1) = 0, ((D@@2)(y))(1) = 0, ((D@@4)(y))(1) = 0})
It looks like the right command when it is typeset.

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