Question

Transform the parabolic equations
4Uxx + 12Uxy + 9Uyy − 2Ux + U = 0

A. uξξ −1/3uη +1/9u = −
B. uξξ −1/3uη +1/9u = 0
C. uξξ − uηu=0
D. uξξ−1/5uηξ +1/2u=0

Answer 1

Answer on Question #79426 – Math – Differential Equations

Question

Transform the parabolic equations

$$4 \mathrm{U} \mathrm{x} \mathrm{x} + 12 \mathrm{U} \mathrm{x} \mathrm{y} + 9 \mathrm{U} \mathrm{y} \mathrm{y} - 2 \mathrm{U} \mathrm{x} + \mathrm{U} = 0$$

A. $\mathrm{u} \xi \xi - 1/3 \mathrm{u} \eta + 1/9 \mathrm{u} = -$

B. $\mathrm{u} \xi \xi - 1/3 \mathrm{u} \eta + 1/9 \mathrm{u} = 0$

C. $\mathrm{u} \xi \xi - \mathrm{u} \eta \mathrm{u} = 0$

D. $\mathrm{u} \xi \xi - 1/5 \mathrm{u} \eta \xi + 1/2 \mathrm{u} = 0$

Solution

$$4 U_{xx} + 12 U_{xy} + 9 U_{yy} - 2 U_x + U = 0$$ $$a_{11} = 4, a_{12} = 6, a_{22} = 9$$ $$\Delta = a_{12}^2 - a_{11} a_{22} = 36 - 36 = 0$$ $$\frac{dy}{dx} = \frac{a_{12} \pm \sqrt{\Delta}}{a_{11}} = \frac{6}{4} = \frac{3}{2} \Rightarrow y = \frac{3}{2} x + C \Rightarrow C = y - \frac{3}{2} x$$

Let

$$\xi = y - \frac{3}{2} x$$ $$\eta = y + \frac{3}{2} x$$ $$U(x, y) = u(\xi, \eta)$$

Then

$$U_x = u_\xi \xi_x + u_\eta \eta_x = -\frac{3}{2} u_\xi + \frac{3}{2} u_\eta$$ $$U_y = u_\xi \xi_y + u_\eta \eta_y = u_\xi + u_\eta$$ $$U_{xx} = -\frac{3}{2} \left( u_{\xi \xi} \xi_x + u_{\xi \eta} \eta_x \right) + \frac{3}{2} \left( u_{\eta \xi} \xi_x + u_{\eta \eta} \eta_x \right) = \frac{9}{4} u_{\xi \xi} - \frac{9}{2} u_{\eta \xi} + \frac{9}{4} u_{\eta \eta}$$ $$U_{yy} = u_{\xi \xi} \xi_y + u_{\xi \eta} \eta_y + u_{\eta \xi} \xi_y + u_{\eta \eta} \eta_y = u_{\xi \xi} + 2 u_{\xi \eta} + u_{\eta \eta}$$ $$U_{yx} = u_{\xi \xi} \xi_x + u_{\xi \eta} \eta_x + u_{\eta \xi} \xi_x + u_{\eta \eta} \eta_x = -\frac{3}{2} u_{\xi \xi} + \frac{3}{2} u_{\eta \eta}$$

Then

$$\begin{array}{l} 4 U _ {x x} + 1 2 U _ {x y} + 9 U _ {y y} - 2 U _ {x} + U = 0 \Rightarrow 4 \left(\frac {9}{4} u _ {\xi \xi} - \frac {9}{2} u _ {\eta \xi} + \frac {9}{4} u _ {\eta \eta}\right) + 1 2 \left(- \frac {3}{2} u _ {\xi \xi} + \frac {3}{2} u _ {\eta \eta}\right) + \\ + 9 \left(u _ {\xi \xi} + 2 u _ {\xi \eta} + u _ {\eta \eta}\right) - 2 \left(- \frac {3}{2} u _ {\xi} + \frac {3}{2} u _ {\eta}\right) + u = 0 \Rightarrow \\ \Rightarrow u _ {\xi \xi} (9 - 1 8 + 9) + u _ {\eta \eta} (9 + 1 8 + 9) + u _ {\eta \xi} (- 1 8 + 1 8) + 3 u _ {\xi} - 3 u _ {\eta} + u = 0 \Rightarrow \\ \Rightarrow 3 6 u _ {\eta \eta} = 3 u _ {\eta} - 3 u _ {\xi} - u \Rightarrow u _ {\eta \eta} = \frac {3 u _ {\eta} - 3 u _ {\xi} - u}{3 6} \\ \end{array}$$

Answer provided by https://www.AssignmentExpert.com