Consider the general transformation of independent variables

Question

Assignment:

Consider the general transformation of the independent variables x and y of the equation

Au_xx + Bu_xy + C_yy + Du_x + Eu_y + Fu = G............(1) to new variables μ, ν where

μ= μ (x,y), ν = ν (x,y)

such that the functions and are continuously differentiable and the Jacobian

j = ∂ (μ,ν) / ∂(x,y)= |μ_x μ_y / ν_x ν_y|= (μ_x ν_y - μ_y ν_x) = 0

in the domain Ω where the equation (1) holds.

Using the chain rule of partial differentiation, to show that the differential equation transforms into the differential equation

A‾u_xx + B¯u_xy + C¯u_yy + D¯u_x + E¯u__y + F¯ = ¯G

Where,

A¯ = Aμ2_x+ Bμ_xμ_y + Cμ2_y

B¯ = 2Aμ_xν_x + B (μ_xν_y + μ_y ν_x) + 2Cμ_yν_y

C¯ = Aν²_{X + B}ν_x ν_y + Cν²_y

D¯ = Aμ_xx + Bμ_xy + Cμ_yy + Dμ_x + Eμ_y

E¯ = Aν_xx + Bν_xy + Cν_yy + Dν_x + Eν_y

F¯ = F, G¯= G