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Change of variables - Linear PDEs

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Hello, I'm really struggling with this. I'm finding the way my lecturer has described what is going on to be really confusing and I'm not sure what is going on.

I've tried to use the example to help me, but (I guess using the chain rule somehow?) I don't get how we've arrived at:

lambda^2*x^2 + lambda*(x^3)/t.

Are we change variables from lambda*x to X? or x to X? I'm so lost.

Thank you.
Reply 1
Avatar for Bameron
Bameron
OP
9
Oh I think I've understood a bit of the example, we've subbed in lambda*x for x and lambda^2*t for t and subbed them into v(x,t).
Original post by Bameron

Hello, I'm really struggling with this. I'm finding the way my lecturer has described what is going on to be really confusing and I'm not sure what is going on.

I've tried to use the example to help me, but (I guess using the chain rule somehow?) I don't get how we've arrived at:

lambda^2*x^2 + lambda*(x^3)/t.

Are we change variables from lambda*x to X? or x to X? I'm so lost.

Thank you.

Look, you have that v(x,t)=x2+x3tv(x,t) = x^2 + \dfrac{x^3}{t}.

Let X=λxX= \lambda x and T=λ2tT = \lambda^2 t.

Then the function V(x,t)V(x,t) is defined to be v(X,T)=X2+X3T=λ2x2+λx3tv(X,T) = X^2 + \dfrac{X^3}{T} = \lambda^2 x^2 + \dfrac{\lambda x^3}{t}.


There are two ways for you to compute VxV_x you should ensure that both ways give the same result;

Directly: Vx=∂∂x(λ2x2+λx3t)=2λ2x+3λx2tV_x = \dfrac{\partial}{\partial x} \left(\lambda^2 x^2 + \dfrac{\lambda x^3}{t}\right) = 2\lambda^2 x + \dfrac{3\lambda x^2}{t}

Or via the chain rule without the need to express VV explicitly in terms of x,tx,t:

Vx=∂∂xv(X,T)=∂X∂x∂∂Xv(X,T)=(λ)⋅(2X+3X2T)V_x = \dfrac{\partial}{\partial x} v(X,T) = \dfrac{\partial X}{\partial x}\dfrac{\partial}{\partial X} v(X,T) = (\lambda) \cdot \left(2X + \dfrac{3X^2}{T} \right)

and then sub in X=λx,T=λ2tX=\lambda x, T=\lambda^2 t into this last result.

You should notice that it's the same as the approach above.
(edited 3 years ago)
Reply 3
Avatar for Bameron
Bameron
OP
9
Thank you for clarifying RDK, I get it now.

Excuse me for being nosy, but how are you so knowledgeable about maths? Are you a current/former student? A teacher/ lecturer? Still have a keen interest in maths and helping people? Or is this a good way of remembering how to do all these things by helping people on student room?
Original post by Bameron
Thank you for clarifying RDK, I get it now.

Excuse me for being nosy, but how are you so knowledgeable about maths? Are you a current/former student? A teacher/ lecturer? Still have a keen interest in maths and helping people? Or is this a good way of remembering how to do all these things by helping people on student room?

I'm finishing my MMath degree at this moment so I'd say I have several years of experience with maths!

But yeah, you're also correct on your last point :smile:

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