Introduction to partial differential equations - Revision history

Admin at 10:30, 28 May 2011

2011-05-28T10:30:13Z

2011-05-28T10:14:01Z

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	= References =		= References =

	Wikipedia: http://en.wikipedia.org/wiki/Differential_equations		* Wikipedia: http://en.wikipedia.org/wiki/Differential_equations

	Solving quasi-linear PDEs - some examples: http://staff.um.edu.mt/jmus1/PDE.pdf		* Solving quasi-linear PDEs - some examples: http://staff.um.edu.mt/jmus1/PDE.pdf

	Lots of PDE solution methodology info: http://eqworld.ipmnet.ru/en/methods/meth-pde.htm		* Lots of PDE solution methodology info: http://eqworld.ipmnet.ru/en/methods/meth-pde.htm



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2010-11-06T01:21:56Z

Introduction

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	The differential equation is linear if <math>F</math> is linear in the variables <math>x,y,u,\dots,u_{x},u_{y},\dots,u_{xx},u_{yy},\dots</math> and the coefficients depend only on the independent variables <math>x,y,\dots</math>.		The differential equation is linear if <math>F</math> is linear in the variables <math>x,y,u,\dots,u_{x},u_{y},\dots,u_{xx},u_{yy},\dots</math> and the coefficients depend only on the independent variables <math>x,y,\dots</math>.

	The differential equation is quasi-linear if <math>F</math> is linear in the highest order derivatives (degree <math>n</math>), but the coefficients for all other derivatives (up to degree <math>n-1</math>) depend on <math>u,u_{x},u_{y},\dots,u_{xx},u_{yy},\dots</math>. ~~(NOT SURE IF I UNDERSTOOD/EXPRESSED THIS CORRECTLY)~~		The differential equation is quasi-linear if <math>F</math> is linear in the highest order derivatives (degree <math>n</math>), but the coefficients for all other derivatives (up to degree <math>n-1</math>) depend on <math>u,u_{x},u_{y},\dots,u_{xx},u_{yy},\dots</math>.

	= Variety of Solutions =		= Variety of Solutions =

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 See also [[Introduction to ordinary differential equations]]
 = Introduction =
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2010-10-23T07:54:29Z

Applications: fixing typo

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	Partial differential equations, needless to say, are extremely useful for describing physical phenomena.		Partial differential equations, needless to say, are extremely useful for describing physical phenomena.

	An excellent example of this ~~are~~ the [[Governing Equations Lectures\|governing equations]] for combustion.		An excellent example of this is the set of [[Governing Equations Lectures\|governing equations]] for combustion.

	Using this set of partial differential equations, it is possible to describe the dynamics of a combusting system.		Using this set of partial differential equations, it is possible to describe the dynamics of a combusting system.

2010-10-23T07:54:12Z

Applications

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	Partial differential equations, needless to say, are extremely useful for describing physical phenomena.		Partial differential equations, needless to say, are extremely useful for describing physical phenomena.

	An excellent example of this are the [[~~Combustion governing equations~~\|governing equations for combustion]].		An excellent example of this are the [[Governing Equations Lectures\|governing equations]] for combustion.

	Using this set of partial differential equations, it is possible to describe the dynamics of a combusting system.		Using this set of partial differential equations, it is possible to describe the dynamics of a combusting system.


	= References =		= References =

2010-10-21T15:33:47Z

2010-10-21T15:32:37Z

References

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	= References =		= References =

			Wikipedia: http://en.wikipedia.org/wiki/Differential_equations

	Solving quasi-linear PDEs - some examples: http://staff.um.edu.mt/jmus1/PDE.pdf		Solving quasi-linear PDEs - some examples: http://staff.um.edu.mt/jmus1/PDE.pdf

2010-10-21T15:32:13Z

Nonlinear

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	In practice, nonlinear partial differential equations usually contain powers of partial derivatives, e.g. <math>u_{xx}^3</math>, products of partial derivatives, etc. An excellent example of a nonlinear equation is the [[Governing Equations Lectures\|Navier-Stokes equation]].		In practice, nonlinear partial differential equations usually contain powers of partial derivatives, e.g. <math>u_{xx}^3</math>, products of partial derivatives, etc. An excellent example of a nonlinear equation is the [[Governing Equations Lectures\|Navier-Stokes equation]].

	~~{{Quote\|... as Sir Cyril Hinshelwood has observed ... ﬂuid dynamicists were divided~~
	~~into hydraulic engineers who observed things that could not be explained and~~
	~~mathematicians who explained things that could not be observed.~~

	~~- James Lighthill }}~~

	== Homogeneity and Non-Homogeneity ==		== Homogeneity and Non-Homogeneity ==

2010-10-21T08:14:41Z

References