Deathslayer7

2013-10-19, 01:47 AM

I'm not sure what the question is asking for.

Given:

Geometry: a square box with the top and right side equal to T1

k[T_xx+T_yy]=pcT_t

where T=Temp and t=time and T_xx (read: partial squared T with respect to partial x squared) is the 2nd derivative with respect to those variables.

Given the boundary conditions:

T_x(0,y,t)=0

T(L,y,t)=T1

T_y(x,0,t)=0

T(x,L,t)=T1

and the initial condition:

T(x,y,0)=t0(x,y)

a) Determine a(x0 and b(y) that will give a homogeneous problem for u(x,y,t). State the differential equation, boundary conditions and initial condition for u(x,y,t). Done

b) Under what conditions will you be able to solve for u(x,y,t) by assuming that u(x,y,t)=v(x,t)*w(y,t)?

c) Determine the problems that must be solved for v(x,t) and w(y,t)

Solution:

The professor forces us to normalize the equation before even starting the problem and I thank him for that. That being said, part a) is done. I can do that part. But I do not understand what part b) is asking for.

I have the following for a) after normalizing.

a(x)=0

b(y)=0

u_xx+u_yy=u_t

u_x(0,y,t)=0

u(L,y,t)=0

u_y(x,o,t)=0

u(x,L,t)=0

u(x,y,0)=1

So what exactly is part b) asking for? I don't understand. Note this is a Heat Conduction class, not a math class. This is the first time I ever encountered separation of variables and I am teaching myself because the teacher isn't that great at it. I got the knack of it down, but the wording is what trips me up.

Please note: I know I can normally go straight to separation of variables, but the problem specifically asks for this step.

Given:

Geometry: a square box with the top and right side equal to T1

k[T_xx+T_yy]=pcT_t

where T=Temp and t=time and T_xx (read: partial squared T with respect to partial x squared) is the 2nd derivative with respect to those variables.

Given the boundary conditions:

T_x(0,y,t)=0

T(L,y,t)=T1

T_y(x,0,t)=0

T(x,L,t)=T1

and the initial condition:

T(x,y,0)=t0(x,y)

a) Determine a(x0 and b(y) that will give a homogeneous problem for u(x,y,t). State the differential equation, boundary conditions and initial condition for u(x,y,t). Done

b) Under what conditions will you be able to solve for u(x,y,t) by assuming that u(x,y,t)=v(x,t)*w(y,t)?

c) Determine the problems that must be solved for v(x,t) and w(y,t)

Solution:

The professor forces us to normalize the equation before even starting the problem and I thank him for that. That being said, part a) is done. I can do that part. But I do not understand what part b) is asking for.

I have the following for a) after normalizing.

a(x)=0

b(y)=0

u_xx+u_yy=u_t

u_x(0,y,t)=0

u(L,y,t)=0

u_y(x,o,t)=0

u(x,L,t)=0

u(x,y,0)=1

So what exactly is part b) asking for? I don't understand. Note this is a Heat Conduction class, not a math class. This is the first time I ever encountered separation of variables and I am teaching myself because the teacher isn't that great at it. I got the knack of it down, but the wording is what trips me up.

Please note: I know I can normally go straight to separation of variables, but the problem specifically asks for this step.