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## Main Question or Discussion Point

Hey all,

I'm wondering if someone can help me understand how to apply the boundary conditions to the diffusion equation in one dimension. Diffusion equation is:

[tex]\frac{\partial u}{\partial t}[/tex]=D*[tex]\frac{(\partial)^{2}u}{\partial x^{2}}[/tex]

The initial condition is:

[tex]u(x,0)=0[/tex]

And the boundary conditions are:

[tex]\frac{\partial u(0,t)}{\partial x}[/tex]=[tex]\frac{\partial u(L,t)}{\partial x}[/tex]=0

I've been trying to solve this by seperation of variables, and letting [tex]u(x,t)=T(t)X(x)[/tex]I get the two equations:

[tex]\frac{dT}{dt}+DT=a^{2}[/tex]

and

[tex]\frac{d^{2}X}{dx^{2}}+X=a^{2}[/tex]

Then for my solutions I get:

[tex]T(t)=C_{1}e^{-a^{2}Dt}[/tex]

and

[tex]X(x)=C_{2}sin(ax)+C_{3}cos(ax)[/tex]

so then,

[tex]u(x,t)=T(t)X(x)=C_{1}e^{-a^{2}Dt}(C_{2}sin(ax)+C_{3}cos(ax))[/tex]

To apply my boundary/initial conditions, I then differentiate wrt x, and obtain the three simultaneous equations:

[tex]C_{1}(C_{2}sin(ax)+C_{3}cos(ax))=0[/tex]

[tex]C_{2}C_{1}e^{-a^{2}Dt}=0[/tex]

[tex]C_{1}e^{-a^{2}Dt}(C_{2}cos(aL)-C_{3}sin(aL))=0[/tex]

When I try to solve this, I find that the only possible solutions are [tex]C_{1}=C_{2}=C_{3}=0[/tex] but that can't be right.

What am I missing?

I'm wondering if someone can help me understand how to apply the boundary conditions to the diffusion equation in one dimension. Diffusion equation is:

[tex]\frac{\partial u}{\partial t}[/tex]=D*[tex]\frac{(\partial)^{2}u}{\partial x^{2}}[/tex]

The initial condition is:

[tex]u(x,0)=0[/tex]

And the boundary conditions are:

[tex]\frac{\partial u(0,t)}{\partial x}[/tex]=[tex]\frac{\partial u(L,t)}{\partial x}[/tex]=0

I've been trying to solve this by seperation of variables, and letting [tex]u(x,t)=T(t)X(x)[/tex]I get the two equations:

[tex]\frac{dT}{dt}+DT=a^{2}[/tex]

and

[tex]\frac{d^{2}X}{dx^{2}}+X=a^{2}[/tex]

Then for my solutions I get:

[tex]T(t)=C_{1}e^{-a^{2}Dt}[/tex]

and

[tex]X(x)=C_{2}sin(ax)+C_{3}cos(ax)[/tex]

so then,

[tex]u(x,t)=T(t)X(x)=C_{1}e^{-a^{2}Dt}(C_{2}sin(ax)+C_{3}cos(ax))[/tex]

To apply my boundary/initial conditions, I then differentiate wrt x, and obtain the three simultaneous equations:

[tex]C_{1}(C_{2}sin(ax)+C_{3}cos(ax))=0[/tex]

[tex]C_{2}C_{1}e^{-a^{2}Dt}=0[/tex]

[tex]C_{1}e^{-a^{2}Dt}(C_{2}cos(aL)-C_{3}sin(aL))=0[/tex]

When I try to solve this, I find that the only possible solutions are [tex]C_{1}=C_{2}=C_{3}=0[/tex] but that can't be right.

What am I missing?