User:Egm6322.s09.Three.ge/Report1

Test =General nonlinear PDEs=

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

then $$\left \{ {x}_{i} \right \}$$has n independent variables.

Example 1: 1-D Case With Time Variable (i.e. 1-D time dependent problem)

If $$(x, t)=({x}_{1},{x}_{2})$$,then


 * $${x}_{1}=x$$
 * $$ {x}_{2}=t$$

$${x}_{1}$$ is the spatial variable; $${x}_{2}$$ is the temporal variable.

Example 2: 3-D Time Dependent Problems

$$ (x,y,z,t)=({x}_{1},{x}_{2},{x}_{3},{x}_{4})$$


 * $$ x=x_{1}$$
 * $$ y=x_{2}$$
 * $$ z=x_{3}$$
 * $$ t=x_{4}$$

Components of PDE
The unknown functions are expressed as $${u}_{1}$$,$${u}_{2}$$,...

e.g.

Navier-Stokes Equation in 3D is a typical PDE where (u1,u2,u3) represents the velocity field along (x,y,z) coordinates and (x1,x2,x3,x4) represent the independent variables (x,y,z,t).

$${u}_{i}(\left \{ {x}_{j} \right \});$$ $$i=1,2,3$$ $$j=1,2,3,4$$

In this case, the PDE has one unknown function u ;

n independent variable $$\left \{ {x}_{i} \right \}$$, i=1,2,...,n;

m th partial derivative: $$\frac{\partial^m u}{\partial x_{i_1}\,...,\partial x_{i_m}}$$

i1,...,im=1,...,n

=Order= The order of a PDE is given by the order of the highest derivative. For example,

$$ 5u_{x}-7u_{y}=0$$

is a first order PDE because highest derivatives are the first derivatives of u with respect to x and y.

$$u_{x}=\frac{\partial u} {\partial x}; u_{y}=\frac{\partial u} {\partial y}$$

By that definition, the following PDE's

$$ 6\left( u_{x}\right)^{3}+2\left(u_{y}\right)^2+u^\frac{1}{2}+x^2+\sin xy=0 $$

$$ 6\left( u_{x}\right)^{3}+2\left(u_{y}\right)^2+u^\frac{1}{2}=0 $$

are also first order.

The operator D(u) is given by D(u)=... and D(.)=

A PDE can also be of higher order. For instance, the equation $$div\left(grad \cdot u \right)$$ given in cartesian form as $$u_{xx}+u_{yy}=0$$ is clearly of second order.

More generally, the form of a second order PDE is $$ div\left(\kappa\cdot grad\right)+ f\left(x,y\right)=0 $$ where $$\kappa$$ is a second order tensor

The values of $$\kappa$$ may be put into a matrix.

$$\kappa_{ij}=\begin{bmatrix} \kappa_{11} &\kappa_{22} \\ \kappa_{21} & \kappa_{22} \end{bmatrix} $$

It is important to note that while the values of $$\kappa$$ may be a matrix, this does not mean that $$\kappa$$ is the matrix. In a different coordinate system the values in the matrix would be different, but they would still represent the same second order tensor $$\kappa$$.

If $$\kappa$$ is constant, the equation is a linear 2nd order PDE.

If $$\kappa=\kappa(x,y)$$, the equation is a linear 2nd order PDE.

If $$\kappa=\kappa(x,y,u)$$, the equation is a quasi-linear 2nd order PDE.

=Linearity= A simple definition of linearity of a PDE is if it's variables and partial derivatives are of first degree. (Selvadurai 2000,p.74) A more thorough definition is that a linear PDE is both additive and homogeneous.(Kolmogorov & Fomin, p.123).

Definition
 u  is an unknown function; and  L  is an operator.

 L(u)  is linear with respect to  u  if

$$L(\alpha\,\!u+\beta\,\!u)=\alpha\,\!L(u)+\beta\,\!L(v)$$

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

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

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

If  L  is a operator, and $$L(u)=2{u}_{xx}+3{u}_{yy}-7x^2-x$$

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

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

$$ L(\alpha u+ \beta v) \ne \alpha L(u) +\beta L(v) $$

and then, the operator  L  is nonlinear.

Nonlinear PDEs
$$F( \left \{ {x}_{i} \right \}, \left \{ \frac{\partial u}{\partial x_i} \right \},\frac{\partial^2 u}{\partial x_i\,\partial x_j},...)=0$$, i=1,2,...,n

This general form of PDE has n independent variables: $$\left \{ {x}_{i} \right \}$$ and 1 unknown function u.

$$\left \{ {x}_{i} \right \}$$ has n arguments

$$\left \{ \frac{\partial u}{\partial x_i} \right \}$$ is computation of gradient of u, and has n arguments

$$\frac{\partial^2 u}{\partial x_i\,\partial x_j}$$ is the Hessian derivative, and has $$\frac{n(n+1)}{2}$$ arguments

$${H}_{ij}$$ is the Hessian Matrix, and $${H}_{ij}:=\frac{\partial^2 u}{\partial x_i\,\partial x_j}$$, (":=" means "equal by definition")

$$H^T=H$$



The PDE F[(x,y),u,(ux,uy),(uxx,uxy,uyy)]=0 has 2 independent variables (x,y), and 1 unknown function u.

e.g. $$2{u}_{xx}+3{u}_{yy}=ax^2+bx$$

Laplace Equation: $${u}_{xx}+{u}_{yy}=0$$

$$5({u}_{xx})^2+7({u}_{yy})^2=0$$ is a nonlinear PDE since the second derivatives uxx and uyy are squared.