- #1

daz71

- 13

- 0

hi all,

I am trying to solve this PDE by separation of variables, it goes like this:

[tex]\frac{\partial u}{\partial t} = \alpha\frac{\partial ^2 u}{\partial z^2} [/tex] for [tex]0\leq z\leq infty[/tex]

the initial condition I have is: t=0; u = uo.

the boundary condtions:

z=0; [tex]\frac{\partial u}{\partial z}\right\rfloor_{z=0} = k\left(u-u_{b}\right) ... ...(1)[/tex]

z= [tex]\infty[/tex]; [tex]\frac{\partial u}{\partial z}\right\rfloor_{z=\infty} = 0......(2)[/tex]

where, uo, [tex]k[/tex],[tex]u_{b}[/tex], and h are constants.

I write:

u(z,t) = F(z)G(t),

with the subsitition I got:

[tex]\frac{1}{G}\frac{\partial G}{\partial t} = \frac{\alpha}{F}\frac{\partial ^2 F}{\partial Z^2}[/tex]

Setting this to some constant: [tex]omega^2[/tex], I can have the two ODEs.

The problem is when I try to use the first boundary condition, bcos of the second term in the parenthesis of the RHS, I seem to have [tex]\frac{u_{b}}{G}[/tex], which I have no clue how to deal with.

pls could some one point to me how i can go about this?

thanks in advance.

I am trying to solve this PDE by separation of variables, it goes like this:

[tex]\frac{\partial u}{\partial t} = \alpha\frac{\partial ^2 u}{\partial z^2} [/tex] for [tex]0\leq z\leq infty[/tex]

the initial condition I have is: t=0; u = uo.

the boundary condtions:

z=0; [tex]\frac{\partial u}{\partial z}\right\rfloor_{z=0} = k\left(u-u_{b}\right) ... ...(1)[/tex]

z= [tex]\infty[/tex]; [tex]\frac{\partial u}{\partial z}\right\rfloor_{z=\infty} = 0......(2)[/tex]

where, uo, [tex]k[/tex],[tex]u_{b}[/tex], and h are constants.

I write:

u(z,t) = F(z)G(t),

with the subsitition I got:

[tex]\frac{1}{G}\frac{\partial G}{\partial t} = \frac{\alpha}{F}\frac{\partial ^2 F}{\partial Z^2}[/tex]

Setting this to some constant: [tex]omega^2[/tex], I can have the two ODEs.

The problem is when I try to use the first boundary condition, bcos of the second term in the parenthesis of the RHS, I seem to have [tex]\frac{u_{b}}{G}[/tex], which I have no clue how to deal with.

pls could some one point to me how i can go about this?

thanks in advance.

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