Verification of solution to Heat Equation

[rexasaurus]

1. verify that u(t,x,y)=e^-λtsin(αt)cos(βt) (for arbitrary α, β and with λ=α²+β²) satisfies the 2-D Heat Equation.



2. u_t=Δu



3. I began with:
Δu=u_xx+u_yy.
note the equation does not contain variable "x"
so u_xx=0 i.e. Δu=u_yy

u_y=e^-λtsin(αt){-βsin(βt)}
u_yy=e^-λtsin(αt){-β²cos(βt)}

next I found u_t
u_t=cos(βy) {e^-λtαcos(αt)+sin(αt)-λe^-λt}

I have tried to reduce both equations but don't see how they are equal. I also have tried using the λ=α²+β² to re-write the eq. Any suggestions?

 

[LCKurtz]

1. verify that (for arbitrary α, β and with λ=α²+β²) satisfies the 2-D Heat Equation.
2. u_t=Δu
3. I began with:
Δu=u_xx+u_yy.
note the equation does not contain variable "x"
so u_xx=0 i.e. Δu=u_yy

u_y=e^-λtsin(αt){-βsin(βt)}
u_yy=e^-λtsin(αt){-β²cos(βt)}

next I found u_t
u_t=cos(βy) {e^-λtαcos(αt)+sin(αt)-λe^-λt}

I have tried to reduce both equations but don't see how they are equal. I also have tried using the λ=α²+β² to re-write the eq. Any suggestions?

Homework Statement

Your equation is transcribed incorrectly. Your original equation should be:

u(t,x,y)=e^-λtsin(αx)cos(βy)

 

[rexasaurus]

The original equation was posted incorrectly.

The actual question states:
u(t,x,y)=e^-λtsin(αt)cos(βy)

My apologies

 

[LawrenceC]

The heat equation in 2D is the Fourier equation. I think the answer they want you to check out is:

u(x,y,t) = e**(-lambda*t) * sin(alpha*x) * cos(beta*y)

Take the partial derivatives and plug them back into the original equation and you'll see that the equation is satisfied if lambda = alpha**2 + beta**2. The original equation is the Laplacian of u equals the partial of u with respect to time.

 

[rexasaurus]

I checked the problem statement

The question actually states:
u(t,x,y)=e^-λtsin(αt)cos(βy)

Could the instructor possibly have made a typo??

When I set u_t=u_yy (from simplifying Δu) I was able to substitute: λ=²+β²

My last line reads:
αcos(αt)+β²sin(αt)=(α²+β²)sin(αt)

Any thoughts?

 

[LCKurtz]

I checked the problem statement

The question actually states:
u(t,x,y)=e^-λtsin(αt)cos(βy)

Could the instructor possibly have made a typo??

Of course he could have. That t in the sine term should be x. Then it all works.

 

[rexasaurus]

the last substitution should read λ=α²+β²

 

[rexasaurus]

Thx for your help.