If ux t defined for all x R and t 0 is a solution to the h

If u(x, t) defined for all x ? R and t > 0 is a solution to the heat equation u_t = u_xx, then for any constant ? > 0, the function w(x, t) = u(?x, ?² t) is a solution too

The function w(x, t) is called a parabolic scaling of u(x, t). We are interested in finding solutions u(x, t) that are p-homogeneous for parabolic scalings. That is, we seek solutions u(x, t) such that u(?x, ?² t) = ? ^pu(x, t).

Question 2 in the picture.

Solution

In order to prove that w(x,t) is a solution of above heat equation, we need to show that it satisfies heat equation.

Gamma = g

Want to check: wt = wxx

Wt = dw(x,t)/dt = d/dt of u(gx, g2t) = g2*ut

Wxx = dw/dx of dw/dx = dw/dx of g*ux = g2* uxx

Ut =Uxx hence g2*Ut=g2*Uxx implies Wt=Wxx. Hence W(x,t)is also a solution.