User:Egm6322.s09.bit.gk/r3

Solution:

The complete form of the equations 1.2.6 to 1.2.8 from Lapidus and Pinder:

Let,

$$  \xi =\phi (x,y)$$

and

$$ \eta =\psi (x,y) $$

$$\therefore $$ by chain rule we have

$$\frac{\partial u}{\partial x}=\frac{\partial u}{\partial \xi}\frac{\partial \phi}{\partial x} + \frac{\partial u}{\partial \eta}\frac{\partial \psi}{\partial x}$$

and

$$\frac{\partial u}{\partial y}=\frac{\partial u}{\partial \xi}\frac{\partial \phi}{\partial y} + \frac{\partial u}{\partial \eta}\frac{\partial \psi}{\partial y}$$

EXPRESSION FOR $$\frac{\partial^2 u}{\partial x^2}$$
We Know that

$$\frac{\partial^2 u}{\partial x^2}=\frac{\partial }{\partial x}\left (\frac{\partial u}{\partial x} \right )$$ $$ \therefore$$ from chain rule we have

$$ \frac{\partial^2 u}{\partial x^2}=\frac{\partial }{\partial \xi}\left (\frac{\partial u}{\partial x} \right )\frac{\partial \phi}{\partial x} + \frac{\partial }{\partial \eta}\left (\frac{\partial u}{\partial x} \right )\frac{\partial \psi}{\partial x}$$

$$\frac{\partial^2 u}{\partial x^2}=\frac{\partial }{\partial \xi}\left (\frac{\partial u}{\partial \xi}\frac{\partial \phi}{\partial x} + \frac{\partial u}{\partial \eta}\frac{\partial \psi}{\partial x} \right )\frac{\partial \phi}{\partial x} + \frac{\partial }{\partial \eta}\left (\frac{\partial u}{\partial \xi}\frac{\partial \phi}{\partial x} + \frac{\partial u}{\partial \eta}\frac{\partial \psi}{\partial x} \right )\frac{\partial \psi}{\partial x} $$

Let,

$$\frac{\partial }{\partial \xi}\left (\frac{\partial u}{\partial \xi}\frac{\partial \phi}{\partial x} + \frac{\partial u}{\partial \eta}\frac{\partial \psi}{\partial x} \right )\frac{\partial \phi}{\partial x} =I$$

and

$$\frac{\partial }{\partial \eta}\left (\frac{\partial u}{\partial \xi}\frac{\partial \phi}{\partial x} + \frac{\partial u}{\partial \eta}\frac{\partial \psi}{\partial x} \right )\frac{\partial \psi}{\partial x}=II

$$

Differentiating $$I$$ We have

$$I=\frac{\partial }{\partial \xi}\left(\frac{\partial u}{\partial \xi}\right)\left(\frac{\partial \phi}{\partial x}\right)^2 + \frac{\partial }{\partial \xi}\left(\frac{\partial \phi}{\partial x}\right )\frac{\partial u}{\partial \xi}\frac{\partial \phi}{\partial x}$$

$$ + \frac{\partial }{\partial \xi}\left(\frac{\partial u}{\partial \eta}\right)\frac{\partial \psi}{\partial x}\frac{\partial \phi}{\partial x} + \frac{\partial u}{\partial \eta}\frac{\partial }{\partial \xi}\left(\frac{\partial \psi}{\partial x}\right)\frac{\partial \phi}{\partial x} $$

and

$$II=\frac{\partial }{\partial \eta}\left(\frac{\partial u}{\partial \xi}\right)\frac{\partial \phi}{\partial x}\frac{\partial \psi}{\partial x} + \frac{\partial }{\partial \eta}\left(\frac{\partial \phi}{\partial x}\right )\frac{\partial u}{\partial \xi}\frac{\partial \psi}{\partial x} $$

$$+\frac{\partial }{\partial \eta}\left(\frac{\partial u}{\partial \eta}\right)\left(\frac{\partial \psi}{\partial x}\right)^2 + \frac{\partial u}{\partial \eta}\frac{\partial }{\partial \eta}\left(\frac{\partial \psi}{\partial x}\right)\frac{\partial \psi}{\partial x}

$$

$$\Rightarrow I=\frac{\partial^2 u}{\partial \xi^2}\left(\frac{\partial \phi}{\partial x}\right)^2 + \frac{\partial }{\partial \xi}\left(\frac{\partial \phi}{\partial x}\right )\frac{\partial u}{\partial \xi}\frac{\partial \phi}{\partial x} + \frac{\partial^2 u}{\partial \xi{\partial \eta}}\frac{\partial \psi}{\partial x}\frac{\partial \phi}{\partial x} + \frac{\partial u}{\partial \eta}\frac{\partial }{\partial \xi}\left(\frac{\partial \psi}{\partial x}\right)\frac{\partial \phi}{\partial x}

$$

$$\Rightarrow II=\frac{\partial^2 u}{\partial \xi{\partial \eta}}\frac{\partial \psi}{\partial x}\frac{\partial \phi}{\partial x} + \frac{\partial }{\partial \eta}\left(\frac{\partial \phi}{\partial x}\right )\frac{\partial u}{\partial \xi}\frac{\partial \psi}{\partial x} + \frac{\partial^2 u}{\partial \eta^2}\left(\frac{\partial \psi}{\partial x}\right)^2 + \frac{\partial u}{\partial \eta}\frac{\partial }{\partial \eta}\left(\frac{\partial \psi}{\partial x}\right)\frac{\partial \psi}{\partial x} $$

The complete expansion of $$u_{xx}=$$

$$ I+II=\frac{\partial^2 u}{\partial x^2}=\frac{\partial^2  u}{\partial \xi^2}\left(\frac{\partial \phi}{\partial x}\right)^2+2\frac{\partial^2 u}{\partial \xi{\partial \eta}}\frac{\partial \psi}{\partial x}\frac{\partial \phi}{\partial x}+\frac{\partial^2 u}{\partial \eta^2}\left(\frac{\partial \psi}{\partial x}\right)^2$$

$$+\frac{\partial }{\partial \xi}\left(\frac{\partial \phi}{\partial x}\right )\frac{\partial u}{\partial \xi}\frac{\partial \phi}{\partial x} +\frac{\partial u}{\partial \eta}\frac{\partial }{\partial \xi}\left(\frac{\partial \psi}{\partial x}\right)\frac{\partial \phi}{\partial x}+\frac{\partial }{\partial \eta}\left(\frac{\partial \phi}{\partial x}\right )\frac{\partial u}{\partial \xi}\frac{\partial \psi}{\partial x}+\frac{\partial u}{\partial \eta}\frac{\partial }{\partial \eta}\left(\frac{\partial \psi}{\partial x}\right)\frac{\partial \psi}{\partial x}$$

EXPRESSION FOR $$\frac{\partial^2 u}{\partial y^2}$$
Similarly,

$$\frac{\partial^2 u}{\partial y^2}=\frac{\partial }{\partial y}\left (\frac{\partial u}{\partial y} \right )$$ $$ \therefore$$ from chain rule we have

$$ \frac{\partial^2 u}{\partial y^2}=\frac{\partial }{\partial \xi}\left (\frac{\partial u}{\partial y} \right )\frac{\partial \phi}{\partial y} + \frac{\partial }{\partial \eta}\left (\frac{\partial u}{\partial y} \right )\frac{\partial \psi}{\partial y}$$

$$\frac{\partial^2 u}{\partial y^2}=\frac{\partial }{\partial \xi}\left (\frac{\partial u}{\partial \xi}\frac{\partial \phi}{\partial y} + \frac{\partial u}{\partial \eta}\frac{\partial \psi}{\partial y} \right )\frac{\partial \phi}{\partial y} + \frac{\partial }{\partial \eta}\left (\frac{\partial u}{\partial \xi}\frac{\partial \phi}{\partial y} + \frac{\partial u}{\partial \eta}\frac{\partial \psi}{\partial y} \right )\frac{\partial \psi}{\partial y} $$

Let,

$$\frac{\partial }{\partial \xi}\left (\frac{\partial u}{\partial \xi}\frac{\partial \phi}{\partial y} + \frac{\partial u}{\partial \eta}\frac{\partial \psi}{\partial y} \right )\frac{\partial \phi}{\partial y} =I$$

and

$$\frac{\partial }{\partial \eta}\left (\frac{\partial u}{\partial \xi}\frac{\partial \phi}{\partial y} + \frac{\partial u}{\partial \eta}\frac{\partial \psi}{\partial y} \right )\frac{\partial \psi}{\partial y}=II

$$

Differentiating $$I$$ We have

$$I=\frac{\partial }{\partial \xi}\left(\frac{\partial u}{\partial \xi}\right)\left(\frac{\partial \phi}{\partial y}\right)^2 + \frac{\partial }{\partial \xi}\left(\frac{\partial \phi}{\partial y}\right )\frac{\partial u}{\partial \xi}\frac{\partial \phi}{\partial y}$$

$$ + \frac{\partial }{\partial \xi}\left(\frac{\partial u}{\partial \eta}\right)\frac{\partial \psi}{\partial y}\frac{\partial \phi}{\partial y} + \frac{\partial u}{\partial \eta}\frac{\partial }{\partial \xi}\left(\frac{\partial \psi}{\partial y}\right)\frac{\partial \phi}{\partial y} $$

and

$$II=\frac{\partial }{\partial \eta}\left(\frac{\partial u}{\partial \xi}\right)\frac{\partial \phi}{\partial y}\frac{\partial \psi}{\partial y} + \frac{\partial }{\partial \eta}\left(\frac{\partial \phi}{\partial y}\right )\frac{\partial u}{\partial \xi}\frac{\partial \psi}{\partial y} $$

$$+\frac{\partial }{\partial \eta}\left(\frac{\partial u}{\partial \eta}\right)\left(\frac{\partial \psi}{\partial y}\right)^2 + \frac{\partial u}{\partial \eta}\frac{\partial }{\partial \eta}\left(\frac{\partial \psi}{\partial y}\right)\frac{\partial \psi}{\partial y}

$$

$$\Rightarrow I=\frac{\partial^2 u}{\partial \xi^2}\left(\frac{\partial \phi}{\partial y}\right)^2 + \frac{\partial }{\partial \xi}\left(\frac{\partial \phi}{\partial y}\right )\frac{\partial u}{\partial \xi}\frac{\partial \phi}{\partial y} + \frac{\partial^2 u}{\partial \xi{\partial \eta}}\frac{\partial \psi}{\partial y}\frac{\partial \phi}{\partial y} + \frac{\partial u}{\partial \eta}\frac{\partial }{\partial \xi}\left(\frac{\partial \psi}{\partial y}\right)\frac{\partial \phi}{\partial y}

$$

$$\Rightarrow II=\frac{\partial^2 u}{\partial \xi{\partial \eta}}\frac{\partial \psi}{\partial y}\frac{\partial \phi}{\partial y} + \frac{\partial }{\partial \eta}\left(\frac{\partial \phi}{\partial y}\right )\frac{\partial u}{\partial \xi}\frac{\partial \psi}{\partial y} + \frac{\partial^2 u}{\partial \eta^2}\left(\frac{\partial \psi}{\partial y}\right)^2 + \frac{\partial u}{\partial \eta}\frac{\partial }{\partial \eta}\left(\frac{\partial \psi}{\partial y}\right)\frac{\partial \psi}{\partial y} $$

The complete eypansion of $$u_{yy}=$$

$$ I+II=\frac{\partial^2 u}{\partial y^2}=\frac{\partial^2  u}{\partial \xi^2}\left(\frac{\partial \phi}{\partial y}\right)^2+2\frac{\partial^2 u}{\partial \xi{\partial \eta}}\frac{\partial \psi}{\partial y}\frac{\partial \phi}{\partial y}+\frac{\partial^2 u}{\partial \eta^2}\left(\frac{\partial \psi}{\partial y}\right)^2$$

$$+\frac{\partial }{\partial \xi}\left(\frac{\partial \phi}{\partial y}\right )\frac{\partial u}{\partial \xi}\frac{\partial \phi}{\partial y} +\frac{\partial u}{\partial \eta}\frac{\partial }{\partial \xi}\left(\frac{\partial \psi}{\partial y}\right)\frac{\partial \phi}{\partial y}+\frac{\partial }{\partial \eta}\left(\frac{\partial \phi}{\partial y}\right )\frac{\partial u}{\partial \xi}\frac{\partial \psi}{\partial y}+\frac{\partial u}{\partial \eta}\frac{\partial }{\partial \eta}\left(\frac{\partial \psi}{\partial y}\right)\frac{\partial \psi}{\partial y}$$

EXPRESSION FOR $$\frac{\partial^2 u}{\partial x\partial y}$$
$$\frac{\partial^2 u}{\partial x\partial y}=\frac{\partial }{\partial x}\left (\frac{\partial u}{\partial y} \right )$$

$$\Rightarrow \frac{\partial^2 u}{\partial x\partial y}=\frac{\partial }{\partial \xi}\left (\frac{\partial u}{\partial \xi}\frac{\partial \phi}{\partial y} + \frac{\partial u}{\partial \eta}\frac{\partial \psi}{\partial y} \right )\frac{\partial \phi}{\partial x} + \frac{\partial }{\partial \eta}\left (\frac{\partial u}{\partial \xi}\frac{\partial \phi}{\partial y} + \frac{\partial u}{\partial \eta}\frac{\partial \psi}{\partial y} \right )\frac{\partial \psi}{\partial x}$$

$$I=\frac{\partial }{\partial \xi}\left (\frac{\partial u}{\partial \xi}\frac{\partial \phi}{\partial y} + \frac{\partial u}{\partial \eta}\frac{\partial \psi}{\partial y} \right )\frac{\partial \phi}{\partial x}$$

$$II=\frac{\partial }{\partial \eta}\left (\frac{\partial u}{\partial \xi}\frac{\partial \phi}{\partial y} + \frac{\partial u}{\partial \eta}\frac{\partial \psi}{\partial y} \right )\frac{\partial \psi}{\partial x}$$

$$\Rightarrow I=\frac{\partial^2 u }{\partial \xi^2}\frac{\partial \phi}{\partial x}\frac{\partial \phi}{\partial y} + \frac{\partial u}{\partial \xi}\frac{\partial }{\partial \xi}\left (\frac{\partial \phi}{\partial y} \right )\frac{\partial \phi}{\partial x} + \frac{\partial^2 u}{\partial \xi\partial \eta}\frac{\partial \psi}{\partial x}\frac{\partial \phi}{\partial x} + \frac{\partial u}{\partial \eta}\frac{\partial }{\partial \xi}\left (\frac{\partial \psi}{\partial y} \right )\frac{\partial \phi}{\partial x} $$

$$II=\frac{\partial^2 u}{\partial \xi\partial \eta}\frac{\partial \psi}{\partial x}\frac{\partial \phi}{\partial x} + \frac{\partial u}{\partial \xi}\frac{\partial }{\partial \eta}\left (\frac{\partial \phi}{\partial y} \right )\frac{\partial \psi}{\partial x} + \frac{\partial^2 u }{\partial \eta^2}\frac{\partial \psi}{\partial y}\frac{\partial \psi}{\partial x} + \frac{\partial u}{\partial \eta}\frac{\partial }{\partial \eta}\left (\frac{\partial \psi}{\partial y} \right )\frac{\partial \psi}{\partial x} $$

$$\Rightarrow \frac{\partial^2 u}{\partial x\partial y}=I+II=\frac{\partial^2 u }{\partial \xi^2}\frac{\partial \phi}{\partial x}\frac{\partial \phi}{\partial y}+ \frac{\partial^2 u}{\partial \xi\partial \eta}\left(\frac{\partial \psi}{\partial x}\frac{\partial \phi}{\partial x}+\frac{\partial \psi}{\partial x}\frac{\partial \phi}{\partial x}\right)+\frac{\partial^2 u }{\partial \eta^2}\frac{\partial \psi}{\partial y}\frac{\partial \psi}{\partial x}$$

$$+\frac{\partial u}{\partial \xi}\frac{\partial }{\partial \xi}\left (\frac{\partial \phi}{\partial y} \right )\frac{\partial \phi}{\partial x}+ \frac{\partial u}{\partial \eta}\frac{\partial }{\partial \xi}\left (\frac{\partial \psi}{\partial y} \right )\frac{\partial \phi}{\partial x}+ \frac{\partial u}{\partial \xi}\frac{\partial }{\partial \eta}\left (\frac{\partial \phi}{\partial y} \right )\frac{\partial \psi}{\partial x}+ \frac{\partial u}{\partial \eta}\frac{\partial }{\partial \eta}\left (\frac{\partial \psi}{\partial y} \right )\frac{\partial \psi}{\partial x}$$

The matrix form of the above expressions
We Have

$$\begin{bmatrix} \frac{\partial }{\partial x} \\ \frac{\partial }{\partial y} \end{bmatrix} =J^T\begin{bmatrix} \frac{\partial }{\partial \overline x} \\ \frac{\partial }{\partial \overline y} \end{bmatrix}$$

Let this be 1

and

$$\begin{bmatrix} \frac{\partial }{\partial x} \\ \frac{\partial }{\partial y} \end{bmatrix}

\begin{bmatrix} \frac{\partial }{\partial x} & \frac{\partial }{\partial y} \end{bmatrix}= \begin{bmatrix} \partial_{xx} &\partial_{xy} \\ \partial_{yx} &\partial_{yy} \end{bmatrix}$$

Let this be 2

Also

$$\because \left (AB \right )^T=B^TA^T$$

$$\begin{bmatrix} \partial_x & \partial_y \end{bmatrix}= \begin{bmatrix} \partial_ \bar x & \partial_\bar y \end{bmatrix}J$$

Let this be 3

Applying equation 2 and 3 in equation 1 ,we have

$$\begin{bmatrix} \partial_x \\ \partial_y \end{bmatrix} \begin{bmatrix} \partial_x & \partial_y \end{bmatrix} = J^T \begin{bmatrix} \partial_\bar x \\ \partial_\bar y \end{bmatrix} \begin{bmatrix} \partial_\bar x & \partial_\bar y \end{bmatrix} J$$

$$\Rightarrow \begin{bmatrix} \partial_{xx} &\partial_{xy} \\ \partial_{yx} &\partial_{yy} \end{bmatrix} =\begin{bmatrix} J_{11} & J_{21}\\ J_{12} & J_{22} \end{bmatrix} \begin{bmatrix} \partial_{\bar x\bar x} &\partial_{\bar x\bar y} \\ \partial_{\bar y\bar x} &\partial_{\bar y\bar y} \end{bmatrix} \begin{bmatrix} J_{11} &J_{12} \\ J_{11}& J_{22} \end{bmatrix}$$

$$\Rightarrow \begin{bmatrix} \partial_{xx} &\partial_{xy} \\ \partial_{yx} &\partial_{yy} \end{bmatrix}$$

$$= \begin{bmatrix}

\partial_{\bar x \bar x} (J_{11})^2 +2 \partial_{\bar x \bar y} J_{21} J_{11} + \partial_{\bar y \bar y } {J_{21}}^2 & \partial_{\bar x \bar x} J_{11} J_{21} + \partial_{\bar x \bar y} J_{22} J_{11} + \partial_{\bar x \bar y} J_{12} J_{21} + \partial_{\bar y \bar y} J_{22} J_{21} \\

\partial_{\bar x \bar x} J_{11} J_{21} + \partial_{\bar x \bar y} J_{21} J_{12} + \partial_{\bar x \bar y} J_{11} J_{22} + \partial_{\bar y \bar y} J_{21} J_{22} & \partial_{\bar x \bar x} (J_{12})^2 +2 \partial_{\bar x \bar y} J_{12} J_{22} + \partial_{\bar y \bar y } {J_{22}}^2

\end{bmatrix}$$

$$\therefore u_{xx}=\partial_{\bar x \bar x}(J_{11})^2+\partial_{\bar x \bar y}J_{11}J_{21}+\partial_{\bar y \bar y}(J_{21})^2=\partial_{\xi \xi}(\phi_{x})^2+2\partial_{\xi \eta}\phi_{x}\psi_{x}+\partial_{\eta \eta}(\psi_{x})^2$$

$$u_{xy}=\partial_{\bar x \bar x} J_{11} J_{21} + \partial_{\bar x \bar y} J_{22} J_{11} + \partial_{\bar x \bar y} J_{12} J_{21} + \partial_{\bar y \bar y} J_{22} J_{21} =u_{\xi\xi}\phi_x\phi_y + u_{\xi\eta}\left( \phi_x\psi_y + \phi_y\psi_x \right) + u_{\eta\eta}\psi_x\psi_y $$

$$u_{yy}=\partial_{\bar x \bar x} (J_{12})^2 +2 \partial_{\bar x \bar y} J_{12} J_{22} + \partial_{\bar y \bar y } {J_{22}}^2 =u_{\xi\xi}\phi_y^2 + 2u_{\xi\eta}\phi_y\psi_y + u_{\eta\eta}\psi_y^2 + $$