User:Egm6322.s09.mafia

=General Non-Linear PDEs=

Partial differential equations (PDEs) are a class of mathematical expressions that include one or more independent variables and their respective derivatives. These equations describe a host of physical occurrences in the real world. Examples of PDEs in everyday life can be as simple as tracking the change in temperature for a can of soda that is placed on a kitchen counter after being removed from the refrigerator (see Heat Conduction Equation, ). More complex examples of PDEs can be found in Schrodinger's Equation which describes the quantum state of a system, , or, the Monge Ampere equation that occurs most often in studies of differential geometry. A brief introduction to the basic tenets in the study of partial differential equations is presented here. Information given on this Wiki page are from lecture notes of EGM 6322 taught by Dr. Vu-Quoc.

Defining Variables
For these notes, we will assume we have n independent variables:

$$  \left \{ {x}_{i} \right \}=   \left \{ {x}_{1},...,{x}_{n}   \right \} $$.

An example of this would be single dimensional case with a time variable.

' $$ (  x,t )=( {x}_{1}, { x}_{2 } )  $$'

where $$ {x}_{1} = {x}  $$, a spatial variable

and $$ {x}_{2} = {t}  $$, a temporal variable.

A more complex situation would be a three dimensional, time dependent problem:

$$(x,y,z,t)=({x}_{1},...,{x}_{4})$$.

In this case, $$ {x}_{1} = {x}  $$, $$  {x}_{2} = {y}  $$, $$  {x}_{3} = {z}  $$, and $$  {x}_{4} = {t}  $$.

Definition of Functions
The unknown function $$ u $$ maps numbers from the domain $$ \Omega $$ to the real numbers $$ \mathbb{R} $$ In two dimensions, this means:

$$ \left( x,y \right) \in \Omega $$

$$ u \left( x,y \right) \in \mathbb{R} $$

and

$$ \Omega \rightarrow \mathbb{R}$$

$$ \left( x,y \right) \mapsto u \left( x,y \right) $$

which means that $$\Omega $$ is the domain of u, $$ \mathbb{R}$$ is the range of u, and $$u \left ( \Omega \right )$$ is the image of $$\Omega $$ under the mapping of $$ u $$.

Example Functions
For these notes, we will assume we have unknown functions: $$ {u}_{1},  {u}_{2} ,...  $$.

The Navier Stokes Equation
An example of this is the Navier-Stokes Equations in 3-D:

$$ {u}_{i} ( \left \{  {x}_{j} \right \} ) $$

where $$ {u}_{i} $$ is the velocity field in x, y, z; $$ {u}_{1}, {u}_{2}, {u}_{3} $$, dependent upon variables $$ {x}_{1}, ... {x}_{4}$$, where $$ {x}_{1} = {x}  $$, $$  {x}_{2} = {y}  $$, $$  {x}_{3} = {z}  $$, and $$  {x}_{4} = {t}  $$.

One Unknown Function
For our present discussion, this can be restricted to one unknown function $$ u $$, containing $$ n $$ independent variables $$ {{x}_{i}} i=1,...,n $$

The mth partial derivative of this function can be expressed as:

$$ \frac{\partial^m u}{\partial x_i\ ,...,\partial x_m} $$ where

$$ {{i}_{1},...,{i}_{m}} $$ is the subset of m indices among n possible indices

$$ {i}_{i}, ..., {i}_{m}=1,...,n $$.

Another Example
Another example of a non-linear PDE is:

$$ F \left ( \left \{ {x}_{i} \right \}, \left \{ \frac{\partial u}{\partial x_i} \right \}, \left \{ \frac{\partial^2 u}{\partial x_i \partial x_j} \right \}, ... \right ) =0$$

where $$ \left \{ {x}_{i} \right \}$$ contains $$ n $$ arguments,

$$ \left \{ \frac{\partial u}{\partial x_i} \right \}$$ are the components of the $$ \nabla u$$, and

$$ \left \{ \frac{\partial^2 u}{\partial x_i \partial x_j} \right \}$$ are the components of the Hessian of $$ u $$.

The Hessian
The Hessian is a symmetric $$ n \times n $$ matrix, thus H = HT

and $${H}_{n \times n}$$ := $$ {\left [ {H}_{ij}\right ]}_{n \times n}$$.

The Hessian is defined as:

$$ {H}_{ij} := \left \{ \frac{\partial^2 u}{\partial x_i \partial x_j} \right \}$$.

This has many interesting properties.



The Laplace Equation
The Laplace Equation is a single function of two (or more) variables:

$$ F \left ( \left (x,y \right) ,u,\left ({u}_{x},{u}_{y} \right), \left ({u}_{xx},{u}_{xy},{u}_{yy} \right ), ...   \right )=0 $$

Example of the Laplace Equation include:

$$2{u}_{xx}+3{u}_{yy}=a{x}^{2}+bx$$

$${u}_{xx} +{u}_{yy}=0$$

The Definition of Linearity
If $$ u $$ is an unknown function, the operator $$ L $$ is considered a linear operator with respect to $$ u $$ if:

$$ L \left ( \alpha u + \beta v \right ) = \alpha L \left ( u \right )+ \beta L \left ( v \right ) $$

Example of Checking for Linearity
Let's take the function:

$$2{u}_{xx}+3{u}_{yy}-7{x}^{2}+x=0$$

The operator is :

$$ L= 2 \frac{\partial^2 }{\partial x^2} + 3 \frac{\partial^2 }{\partial y^2} -7 x^2-x$$

where $$-7 x^2-x:= f(x)$$

To check the linearity, we assume $$\alpha$$ and $$\beta$$ are real number constants, and evaluate: $$ L(\alpha u+ \beta v)= 2 \frac{\partial^2 (\alpha u+ \beta v)}{\partial x^2} + 3 \frac{\partial^2 (\alpha u+ \beta v)}{\partial y^2} + f(x)$$

which is equivalent to:

$$ L(\alpha u+ \beta v)=2 {\left (\alpha u + \beta v \right )}_{xx}+3 {\left (\alpha u +\beta v\right )}_{yy}+f(x) $$

If we distribute the integers, as well as the second derivative operators, the equation becomes:

$$ L(\alpha u+ \beta v)=2 \alpha {u}_{xx}+2 \beta {v}_{xx}+3 \alpha {u}_{yy} + 3 \beta {v}_{yy}+f(x) $$.

Grouping the terms by the constants $$\alpha$$ and $$\beta$$ concludes:

$$ L(\alpha u+ \beta v)= \alpha \left [2 {u}_{xx}+3{u}_{yy} \right ] +\beta \left [ 2 {v}_{xx} +3 {v}_{yy} \right ] +f(x) $$.

The operator $$ L $$ is nonlinear because

$$ \alpha L(u)+ \beta L(v)= \alpha \left [2 {u}_{xx} +3 {u}_{yy} +f(x) \right ] +\beta \left [2 {v}_{xx}+3 {v}_{yy} + f(x) \right ] $$

and

$$ f(x) \ne \alpha f(x) +\beta f(x) $$


 * $$J=\begin{bmatrix} \dfrac{\partial y_1}{\partial x_1} & \cdots & \dfrac{\partial y_1}{\partial x_n} \\ \vdots & \ddots & \vdots \\ \dfrac{\partial y_m}{\partial x_1} & \cdots & \dfrac{\partial y_m}{\partial x_n} \end{bmatrix}. $$

Definition of the order of PDE
Order of highest derivative in the PDE.

Example of first order PDE
linear: $$5u_x-7u_y=0$$

non-linear: $$6(u_x)^3+2(u_y)^2+(u)^{1/2}+x^2+sin xy=0$$

we can assume that $$D(u)=6(u_x)^3+2(u_y)^2+(u)^{1/2}$$

More Examples of 1st Order PDEs

Examples of 1st order linear and nonlinear PDEs are given below. For more information, see the Wiki page for Partial Differential Equations.

(LINEAR) $$ \frac{du}{dx}\ + \frac{du}{dy}\ + \frac{du}{dz}\ = 0 $$ (LINEAR) $$ \dot x \ + 42\dot y \ = 0 $$ (LINEAR) $$ t \frac{dy}{dt}\ + 4 \frac{dy}{dx}\ = 0 $$ (NON-LINEAR) $$ \left(\frac{du}{dx}\ \right)^2 + \left(\frac{du}{dy}\ \right)^3  = 0 $$ (NON-LINEAR) $$ u + \sqrt[3]{\left(\frac{du}{dy}\ \right)}  = 0 $$

As an addition proof, it should be noted that the linear PDEs given above are also separable (the non-linear PDEs are not separable expressions). More information about the linearity of PDEs is given in the following references:,.

Homework
Show D(u) is not linear.

(Solution)

Again, $$D \left( u \right) = 6 \left( u_x \right)^3 + 2 \left( u_y \right)^2 + \left( u \right)^{1/2}$$.

D(u) is linear if the following formula holds:

$$D \left( \alpha\cdot u + \beta\cdot v \right) = \alpha\cdot D \left( u \right) + \beta\cdot D \left( v \right) $$

OR,

$$6 \left( \alpha\cdot u + \beta\cdot v \right)^3 + 2 \left( \alpha\cdot u + \beta\cdot v  \right)^2 +  \left( \alpha\cdot u + \beta\cdot v  \right)^{1/2} = \alpha\cdot \left [ 6 \left( u_x  \right)^3 + 2 \left( u_y  \right)^2 +  \left( u  \right)^{1/2} \right ] + \beta\cdot \left [ 6 \left( v_x  \right)^3 + 2 \left( v_y  \right)^2 +  \left( v  \right)^{1/2} \right ]$$

The expressions on either side of the equality sign in the above equation are NOT identical. Therefore, D(u) is NOT linear.

Example of second order PDE
$$div(gradu)+f(x,y)=0$$

Notice: tensorial notation = coord free notation.

Cartesian Coordinate (x,y)
Let's take the function:

$$u_xx+u_yy+f(x,y)=0$$

We can also write it as div( $$\kappa$$ $$\cdot$$$$grad u$$$$+f(x,y)=0$$

Here, $$\kappa$$ is a continuity tensor (2nd order tensor).

$$\kappa$$ = $$\kappa$$ $$e_i$$ $$\otimes$$ $$e_j$$ 2nd-order tensor

$$v$$ = $$v_i$$ $$e_i$$ vector,1st-order tensor

Operators
$$grad$$ $$u$$ $$=$$ $$\frac{\partial u}{\partial x_i}$$ $$e_i$$

$$div$$ $$v$$ $$=$$ $$\frac{\partial v_i}{\partial x_i}$$ $$e_i$$

$$\kappa$$ $$\cdot$$ $$grad$$ $$u$$ =( $$\kappa_{ij}$$ $$e_i$$ $$\otimes$$ $$e_j$$ ) $$\cdot$$ ($$\frac{\partial u_k}{\partial x_k}$$ $$e_k$$ )

=$$\kappa_{ij}$$ $$\frac{\partial u}{\partial x_k}$$ ( $$e_i$$ $$\otimes$$ $$e_j$$ ) $$\cdot$$ $$e_k$$

Here, $$e_i$$ $$\otimes$$ $$e_j$$ $$\cdot$$ $$e_k$$ = $$e_i$$ ( $$e_j$$ $$\cdot$$ $$e_k$$ )

Kronecker delta
Definition: If $$e_j$$ $$\otimes$$ $$e_k$$ = $$\delta_{jk}$$, $$\delta_{jk}$$ is called Kronecker delta.

$$ \delta_{jk} = \begin{cases} 1 & for \ j=k \\ 0 & for \ j \neq k \end{cases} $$ = $$\kappa_{jk} \frac{\partial u}{\partial x_i} $$ $$e_i$$

$$u(x,y)$$ is a scaler function $$\Rightarrow$$ 0th order tensor $$gradu$$ is a vector field $$\Rightarrow$$ 1st order tensor

$$grad$$  ($$\cdot$$) increases tensor order by 1

$$div$$  ($$\cdot$$) decreases tensor order by 1

We can let $$v$$ : = $$\kappa$$ $$\cdot$$ $$gradu$$

$$div$$ $$v$$ = $$\frac{\partial }{\partial x_i}$$ $$\kappa_{jk}$$ $$\frac{\partial u}{\partial x_j}$$

Leopold Kronecker (1823 - 1891)

Kronecker Delta was named after Leopold Kronecker, a 19th century mathematician from Prussia. Born in 1823, Kronecker studied at the University of Berlin and later went on to teach at what is now known as Humboldt University in 1883. He specialized in the areas of number theory, finitism, and continuity before dying in Berlin in 1891. source:

Homework
Expand the function above.

What's more
$$\kappa_ij$$ = $$\begin{bmatrix} \kappa_{11} & \kappa_{12} \\ \kappa_{21} & \kappa_{22} \end{bmatrix} $$

Notice: in PDE, $$\kappa_{ij}=\kappa_{ji}$$ or $$\kappa^T=$$ $$\kappa$$

$$v$$ = $$v_i$$ $$e_i$$ $$\rightarrow$$ $$\begin{Bmatrix} v_i\end{Bmatrix} _{3 \times 1 or 2 \times 1} $$

And if we change $$e_i$$ coordinate into different system $$\underline{\overline{e_i}}$$

$$\overline{v_i}$$ $$\underline{\overline{e_i}}$$ $$\to$$ $$\overline{\begin{Bmatrix} v_i\end{Bmatrix}}$$

Linearity
if $$\kappa$$ $$= const$$ $$\Rightarrow$$ is a linear 2nd order PDE

$$\kappa$$ $$= \underline{\kappa} (x,y)$$ $$\Rightarrow$$ is also a linear 2nd order PDE

$$\kappa$$ $$= \underline{\kappa} (x,y,u)$$ $$\Rightarrow$$ is a quasilinear 2nd order PDE

Homework
Show that $$\kappa$$ $$= \underline{\kappa} (x,y)$$ $$\Rightarrow$$ is also a linear 2nd order PDE