# 0.6 Solution of the partial differential equations  (Page 2/13)

 Page 2 / 13

Second order PDE. The classification of second order PDEs as elliptic, parabolic, and hyperbolic arise from a transformation of the independent variables. The classification apply to quasilinear (i.e., linear in the highest order derivatives) but we will only discuss linear equations with constant coefficients here. Numerical solutions are needed for quasilinear systems. Again let $u$ denote the dependent variables and $t$ , $x$ , $y$ , $z$ as the independent variables. Examples of the different classes of equations are

$\begin{array}{c}0=\frac{{\partial }^{2}u}{\partial {x}^{2}}+\frac{{\partial }^{2}u}{\partial {y}^{2}}+\frac{{\partial }^{2}u}{\partial {z}^{2}}+\rho ={\nabla }^{2}u+\rho ,\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\mathrm{elliptic}\phantom{\rule{0.277778em}{0ex}}\mathrm{equation}\hfill \\ \frac{\partial u}{\partial t}=\frac{{\partial }^{2}u}{\partial {x}^{2}}+\frac{{\partial }^{2}u}{\partial {y}^{2}}+\frac{{\partial }^{2}u}{\partial {z}^{2}}+\rho ={\nabla }^{2}u+\rho ,\phantom{\rule{1.em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\mathrm{parabolic}\phantom{\rule{0.277778em}{0ex}}\mathrm{equation}\hfill \\ \frac{{\partial }^{2}u}{\partial {t}^{2}}=\frac{{\partial }^{2}u}{\partial {x}^{2}}+\frac{{\partial }^{2}u}{\partial {y}^{2}}+\frac{{\partial }^{2}u}{\partial {z}^{2}}+\rho ={\nabla }^{2}u+\rho ,\phantom{\rule{1.em}{0ex}}\mathrm{hyperbolic}\phantom{\rule{0.277778em}{0ex}}\mathrm{equatio}n\hfill \end{array}$

The $\rho$ term represents sources. When the cgs system of units is used in electrostatics and $\rho$ is the charge density, the source is expressed as $4\pi \phantom{\rule{0.166667em}{0ex}}\rho$ . The factor $4\pi$ is absent with the mks or SI system of units. The parabolic PDEs are sometimes called the diffusion equation or heat equation. In the limit of steady-state conditions, the parabolic equations reduce to elliptic equations. The hyperbolic PDEs are sometimes called the wave equation. A pair of first order conservation equations can be transformed into a second order hyperbolic equation.

Convective-diffusion equation. The above equations represented convection without diffusion or diffusion without convection. When both the first and second spatial derivatives are present, the equation is called the convection-diffusion equation.

$\frac{\partial u}{\partial t}+\frac{\partial u}{\partial x}=\frac{1}{{N}_{Pe}}\frac{{\partial }^{2}u}{\partial {x}^{2}}$

Usually a dimensionless group such as the Reynolds number, or Peclet number appears as a factor to quantify the relative contribution of convection and diffusion.

## Systems described by the poisson and laplace equation

We saw earlier that an irrotational vector field can be expressed as the gradient of a scalar and if in addition the vector field is solenoidal, then the scalar potential is the solution of the Laplace equation.

$\begin{array}{cc}\mathbf{v}=-\nabla \varphi ,\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\hfill & \mathrm{irrotational}\phantom{\rule{0.277778em}{0ex}}\mathrm{flow}\hfill \\ \nabla •\mathbf{v}=\Theta =-{\nabla }^{2}\varphi \hfill \\ \nabla •\mathbf{v}=0=-{\nabla }^{2}\varphi ,\phantom{\rule{1.em}{0ex}}\hfill & \mathrm{incompressible},\mathrm{irrotational}\phantom{\rule{0.277778em}{0ex}}\mathrm{flow}\hfill \end{array}$

Also, if the velocity field is solenoidal then the velocity can be expressed as the curl of the vector potential and the Laplacian of the vector potential is equal to the negative of the vorticity. If the flow is irrotational, then the vorticity is zero and the vector potential is a solution of the Laplace equation.

$\begin{array}{cc}\mathbf{v}=\nabla ×\mathbf{A},\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\hfill & \mathrm{incompressible}\phantom{\rule{0.277778em}{0ex}}\mathrm{flow}\hfill \\ \nabla ×\mathbf{v}=\mathbf{w}=-{\nabla }^{2}\mathbf{A},\phantom{\rule{1.em}{0ex}}\hfill & \mathrm{for}\phantom{\rule{0.277778em}{0ex}}\nabla •\mathbf{A}=0\hfill \\ {\nabla }^{2}\psi =-w,\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\hfill & \mathrm{in}\phantom{\rule{0.277778em}{0ex}}\mathrm{two}\phantom{\rule{0.277778em}{0ex}}\mathrm{dimensions}\hfill \\ \nabla ×\mathbf{v}=0=-{\nabla }^{2}\mathbf{A},\phantom{\rule{1.em}{0ex}}\hfill & \mathrm{irrotational}\phantom{\rule{0.277778em}{0ex}}\mathrm{flow}\phantom{\rule{0.277778em}{0ex}}\mathrm{and}\phantom{\rule{1.em}{0ex}}\nabla •\mathbf{A}=0\hfill \\ {\nabla }^{2}\psi =0,\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\hfill & \mathrm{for}\phantom{\rule{0.277778em}{0ex}}\mathrm{two}\phantom{\rule{0.277778em}{0ex}}\mathrm{dimensional},\mathrm{irrotational},\mathrm{incompressible}\mathrm{flow}\hfill \end{array}$

Other systems, which are solution of the Laplace equation, are steady state heat conduction in a homogenous medium without sources and in electrostatics and static magnetic fields. Also, the flow of a single-phase, incompressible fluid in a homogenous porous media has a pressure field that is a solution of the Laplace equation.

## Systems described by the diffusion equation

Diffusion phenomena occur with viscous flow, thermal conduction, and molecular diffusion. Heat conduction and diffusion without convection are described by the diffusion equation. Convection is always present in fluid flow but its contribution to the momentum balance is neglected for creeping (low Reynolds number) flow or cases where the velocity is perpendicular to the velocity gradient. In this case

How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
for teaching engĺish at school how nano technology help us
Anassong
How can I make nanorobot?
Lily
Do somebody tell me a best nano engineering book for beginners?
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
how can I make nanorobot?
Lily
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!