Heat equation and taylor's approximation

[orangesun]

Homework Statement

storage of heat, T at time, t (measured in days) at a depth x (measured in metres)
T(x,t)=T0 + T1 e$$^{-\lambda} x$$ sin (wt - $$\lambda$$x)
where w = 2pi/365 and $$\lambda$$ is a positive constant

show that $$\delta$$T/$$\delta$$t = k $$\delta$$^2 T / $$\delta$$x^2Derive the second order Taylor polynomial approximation Q(x, t) for T(x, t)
about the point (-0,1)

Homework Equations

The Attempt at a Solution

I'm sorry i have absolutely no idea how to even begin this question, i would really like it if someone could nudge me in the right direction for the method at least.
thanks heaps

 

[lanedance]

for the first part i would start by attempting the required derivatives to see if the equality is true

 

[orangesun]

I understand that, but I just don't know how you would even begin tackling this question.
How would you even be able to take the derivative of that function?

 

[lanedance]

well you have T(x,t)

it should be stratightforward differentiation using chain rule to find
$$\frac{\partial T(x,t)}{\partial t}$$
$$\frac{\partial^2 T(x,t)}{\partial x^2}$$

as they are partial derivatives when you differentiate the other variables are kept constant

have a go and i'll help you

 

[orangesun]

thanks heaps for that lane dance! I didnt know where to start but I managed to get an answer...for both dT/dt and dT/dx and just derive dt/dx again to get d^2T/dx^2

I just need a bit of help now with the second part, how would you use the taylor approximations for that

 

[lanedance]

so what is the equation for a 2nd order taylor polynomial expansion of a 2 variable equation?

 

[orangesun]

I know what it is for a 1 variable equation, but for a 2 variable equation I am a bit stumped.
is it where you have to replace Q(x,y) with x=a+h y=b+k ?

 

[lanedance]

here's the first thing that pops up on a quick google on "multivariable taylor series"
http://www.fepress.org/files/math_primer_fe_taylor.pdf