How can I transform, step by step, the partial differential equation (pde) u.diff(z,z)k/gw == mvu.diff(t) into u.diff(Z,Z)=u.diff(Tv) where Z=z/H, Tv = cvt/H^2 and cv = k/(gwmv)? As you can see from my attempt below I can isolate cv; after that I have no idea how to do the change of variables. I can do this example by hand: let Z=z/H --> du/dz = du/dZ * dZ/dz --> du/dz = du/dZ *(1/H) ...etc. But Other examples may not be so easy (and I don't want to do it by hand!).
var('u,z,t,Z,H,Tv,k,gw,mv,cv')
u = function('u',z,t)
eq1 = u.diff(z,z)*k/gw == mv * u.diff(t)
print(eq1)
mycoeff=eq1.left_hand_side().coeff(u.diff(z,z),1)
print(mycoeff)
eq2 = eq1.divide_both_sides(mycoeff)
print(eq2)
mycoeff = eq2.right_hand_side().coeff(u.diff(t),1)
eq3 = cv == 1/mycoeff
print(eq3)
eq4 = solve(eq3,eq3.right_hand_side().default_variable(), solution_dict=true)
print(eq4)
eq5 = eq2.subs(eq4[0])
print(eq5)
which gives me the following output:
k*D[0, 0](u)(z, t)/gw == mv*D[1](u)(z, t)
k/gw
D[0, 0](u)(z, t) == gw*mv*D[1](u)(z, t)/k
cv == k/(gw*mv)
[{gw: k/(cv*mv)}]
D[0, 0](u)(z, t) == D[1](u)(z, t)/cv
