- #1
masnet
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Hello all!
I appreciate it if you can share any thoughts that you may have regarding how to solve the following PDE:
[tex]\frac{\partial U(z,t)}{\partial t}+(1-z)\frac{\partial U(z,t)}{\partial z}=(\frac{1}{z}-1)\left(U(z,t)-U(0,t)\right)[/tex]
Initial condition:[tex]U(z,0)=z^{K}[/tex]
[tex]U(0,t)[/tex] arises due to a boundary condition, but it is unknown and should be derived by solving the PDE itself (details are irrelevant).
A typical strategy would be using http://www-solar.mcs.st-and.ac.uk/~alan/MT2003/PDE/node8.html" to tackle this. But [tex]U(0,t)[/tex] can't be dealt with.
Specifically, I am stuck when integrating both sides of the following subsidiary equation due to [tex]U(0,t)[/tex]:
[tex]\frac{dU(z,t)}{U(z,t)-U(0,t)}=(\frac{1}{z}-1)dt[/tex]
I greatly appreciate it if you could share any thoughts you might have regarding how I can proceed to solve this through this method or any other method.
Thanks!
I appreciate it if you can share any thoughts that you may have regarding how to solve the following PDE:
[tex]\frac{\partial U(z,t)}{\partial t}+(1-z)\frac{\partial U(z,t)}{\partial z}=(\frac{1}{z}-1)\left(U(z,t)-U(0,t)\right)[/tex]
Initial condition:[tex]U(z,0)=z^{K}[/tex]
[tex]U(0,t)[/tex] arises due to a boundary condition, but it is unknown and should be derived by solving the PDE itself (details are irrelevant).
A typical strategy would be using http://www-solar.mcs.st-and.ac.uk/~alan/MT2003/PDE/node8.html" to tackle this. But [tex]U(0,t)[/tex] can't be dealt with.
Specifically, I am stuck when integrating both sides of the following subsidiary equation due to [tex]U(0,t)[/tex]:
[tex]\frac{dU(z,t)}{U(z,t)-U(0,t)}=(\frac{1}{z}-1)dt[/tex]
I greatly appreciate it if you could share any thoughts you might have regarding how I can proceed to solve this through this method or any other method.
Thanks!
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