Maple - PDE substitution

Question

Lets say I have the heat equation:

$$u_t-u_{xx}=0.$$

I want to apply the following substitution to the heat equation in Maple.

$$\zeta=t/x^2$$ $$u(x,t)=x^cF(t/x^2)=x^cF(\zeta)$$

How can i achive this?

The result should be:

$$cF-c^2F+\frac{d F}{d\zeta}-6\zeta\frac{d F}{d\zeta}+4c\zeta\frac{d F}{d\zeta}-4\zeta^2\frac{d^2 F}{d \zeta^2}=0$$

Answer 1

In Maple, you could do it like so,

eq := diff(u(x,t),t) - diff(u(x,t),x,x) = 0:
s1 := t/x^2 = xi(x,t):
s2 := u(x,t) = x^c*F(t/x^2):

eval(eq, [s2, s1]):
normal( simplify( %, {s1} ) ):
simplify( %/x^(c-2), power ):

And, depending on how you want xi and the derivatives represented,

convert( subs(xi(x,t)=xi,%), diff );

And terms may be grouped together,

#simplify(%,size);
collect(%,diff,factor);

                                     /  2        \                          
                    / d        \     | d         |   2                      
(4 c xi - 6 xi + 1) |---- F(xi)| - 4 |----- F(xi)| xi  - F(xi) c (c - 1) = 0
                    \ dxi      /     |    2      |                          
                                     \ dxi       /

Answer 2

I don't see how Maple is involved in this.

Nevertheless, I agree with the expected result, as shown below :

Answer 3

I found a better approach to this problem, that is why I wanted to share this approach:

gc();
restart;
with(PDEtools);

declare(u(x, t), F(z));

heat := diff(u(x, t), t)-(diff(u(x, t), `$`(x, 2))) = 0;
tr := {t = X^2/Z, x = X, u(x, t) = X^c*F(Z)};
dchange(tr, heat, [X, Z, F(Z)]);