I need help solving this second order differential equation?

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Robin
Robin 2011-8-7
I was given a second order function that models longitudinal displacements in a longitudinally loaded elastic bar.
a(x)*u''(x) + a'(x)*u'(x)=f(x), 0 <= x <=1
I know that the left end is at x=0 and the right end is at x=1. So that means:
u(0)=u(1)=0 (does not bend at the ends)
I know a(x) represents both the elastic properties and the cross-sectional area of the bar and u(x) is the displacement at point x.
for the first part I am told that a(x)=1+x and f(x)= 5*sin(2*pi*x)^2 (this is the case where the force is applied symetrically and is strongest at x=.25 and x=.75)
I need to use dsolve to solve for the problem and plot the result on the interval [0,1]
This is what I tried to do:
ode1= 'D2y*(1+x)+ Dy= 5*sin(2*pi*x)^2';
sol=dsolve(ode1,'y(0)=0','y(1)=0');
ezplot(sol)
Howver the graph I got from this just does not look right. Can someone please help me out?
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Robin
Robin 2011-8-7
ok so I tried to replace what I had for that long solution with mfun, but I get an error:
>> y=(5*x)/2 - (5*log(x + 1))/2 + (5*i*mfun('Ei',((-4)*pi*i)) - 5*i*mfun('Ei',(4*pi*i)))/(16*pi) - (5*i*mfun('Ei',((-4)*pi*i*(x + 1))))/(16*pi) + (5*i*mfun('Ei',(4*pi*i*(x + 1))))/(16*pi) - (log(x + 1)*(40*pi - 40*pi*log(2) + 5*i*mfun('Ei',((-4)*pi*i)) - 5*i*mfun('Ei',(4*pi*i)) - 5*i*mfun('Ei',((-8)*pi*i)) + 5*i*mfun('Ei',(8*pi*i))))/(16*pi*log(2))
??? Error using ==> str2num at 33
Requires string or character array input.
Error in ==> mfun>computeX at 72
x = str2num(x); %#ok
Error in ==> mfun at 43
[x,siz,nans] = computeX(varargin{:});
Walter Roberson
Walter Roberson 2011-8-7
My mistake earlier: Ei is the Exponential Integral, not the Elliptic Integral. The hypergeometric conversion I showed is still valid, though.
I do see any immediate reason why you would be getting the str2num errors, but I can make the suggestion that you optimize using
t = mfun('Ei', pi*i*[-4, 4, -8, 8, -4*(x+1), 4*(x+1)]);
and then replace the individual mfun calls with t(1), t(2), t(5, t(6), t(1), t(2), t(3), and t(4) respectively.

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回答（1 个）

Neels
Neels 2011-8-7
My suggestion would be to rearrange the equation as
u''(x) =f(x)/a(x)- u'(x)/a(x)
z = f(x)/a(x)-y/a(x) , where y is first derivative of u(x) and z is first derivative of y. Then solve it using ode45, giving the initial values. I think matlab computes faster using first order derivatives
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Robin
Robin 2011-8-7
I did try to rearrange the equation every way I could think of, but I still get the same nasty result with "Ei" as part of the answer. We have to solve it symbolically, using dsolve, the problem specifically says to do that :(

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