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John1987
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Last week, the professor gave a difficult PDE to solve as a bonus exercise, describing the motion of a conveyor belt. From experience, he knew that only 5% of the students (applied physics) is able to solve this problem. I got stuck and I really hope to get some help on this forum. This is the problem:
Consider the conveyor belt equation:
[tex]u_{tt}+2*V*u_{tx}+(V^{2}-c^{2})*u_{xx}[/tex]=0, 0<x<L and t>0. (eq. 1)
Furthermore, V,L and c are all constants and u= u(x,t). Boundary conditions: u(0,t)=u(L,t)=0
The question: construct nontrivial solutions of this problem in the form X(x)*T(t). He gave as a hint: "substitute X(x)*T(t) into the PDE and after dividing by X(x)*T(t), differentiate with respect to t or x.
2. Solution attempt
Well, this is how I started:
From the boundary conditions it follows that X(0)=X(L) = 0, since otherwise the solution would be trivial.
Substitution of u(x,t)= X(x)*T(t) into the PDE gives:
[tex]X(x)*T''(t)+2*V*X'(x)*T'(t)+(V^{2}-c^{2})*X''(x)*T(t)=0[/tex] (eq. 2)
Dividing by X(x)*T(t) yields:
[tex]\frac{T''(t)}{T(t)}+\frac{2*V*X'(x)*T'(t)}{X(x)*T(t)}+(V^{2}-c^{2})*\frac{X''(x)}{X(x)}=0[/tex] (eq. 3)
Now, I follow the professor's hint and differentiate this equation with respect to x. This gives:
[tex]2*V*\frac{T'(t)}{T(t)}*(X(x)*X''(x)-X'(x)*X'(x))+(V^{2}-c^{2})*(X(x)*X'''(x)-X''(x)*X'(x))=0[/tex] (eq. 4)
In this last step, I already multiplied both sides of the equation by [tex]X(x)^{2}[/tex] which appeared after applying the quotient rule for fraction differentiation. This last expression doesn't seem to help me so I also differentiate the original equation w.r.t. t. This gives:
[tex]T(t)*T'''(t)-T''(t)*T'(t)+2*V*\frac{X'(x)}{X(x)}*(T(t)*T''(t)-T'(t)*T'(t))=0[/tex] (eq. 5)
Here, I already multiplied out the factor [tex]T(t)^{2}[/tex] that came from applying the quotient differentiation rule.
This is the point where I get stuck: following the hints, I ended up with two nasty equations, neither one I can solve and it also seems very hard to use one of the two equations for solving the other.
Does anyone know how to proceed?
Thanks a lot,
John
Homework Statement
Consider the conveyor belt equation:
[tex]u_{tt}+2*V*u_{tx}+(V^{2}-c^{2})*u_{xx}[/tex]=0, 0<x<L and t>0. (eq. 1)
Furthermore, V,L and c are all constants and u= u(x,t). Boundary conditions: u(0,t)=u(L,t)=0
The question: construct nontrivial solutions of this problem in the form X(x)*T(t). He gave as a hint: "substitute X(x)*T(t) into the PDE and after dividing by X(x)*T(t), differentiate with respect to t or x.
2. Solution attempt
Well, this is how I started:
From the boundary conditions it follows that X(0)=X(L) = 0, since otherwise the solution would be trivial.
Substitution of u(x,t)= X(x)*T(t) into the PDE gives:
[tex]X(x)*T''(t)+2*V*X'(x)*T'(t)+(V^{2}-c^{2})*X''(x)*T(t)=0[/tex] (eq. 2)
Dividing by X(x)*T(t) yields:
[tex]\frac{T''(t)}{T(t)}+\frac{2*V*X'(x)*T'(t)}{X(x)*T(t)}+(V^{2}-c^{2})*\frac{X''(x)}{X(x)}=0[/tex] (eq. 3)
Now, I follow the professor's hint and differentiate this equation with respect to x. This gives:
[tex]2*V*\frac{T'(t)}{T(t)}*(X(x)*X''(x)-X'(x)*X'(x))+(V^{2}-c^{2})*(X(x)*X'''(x)-X''(x)*X'(x))=0[/tex] (eq. 4)
In this last step, I already multiplied both sides of the equation by [tex]X(x)^{2}[/tex] which appeared after applying the quotient rule for fraction differentiation. This last expression doesn't seem to help me so I also differentiate the original equation w.r.t. t. This gives:
[tex]T(t)*T'''(t)-T''(t)*T'(t)+2*V*\frac{X'(x)}{X(x)}*(T(t)*T''(t)-T'(t)*T'(t))=0[/tex] (eq. 5)
Here, I already multiplied out the factor [tex]T(t)^{2}[/tex] that came from applying the quotient differentiation rule.
This is the point where I get stuck: following the hints, I ended up with two nasty equations, neither one I can solve and it also seems very hard to use one of the two equations for solving the other.
Does anyone know how to proceed?
Thanks a lot,
John
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