Entry
Math: Calculus: Definition: What is the definition of a partial differential equation?
Apr 16th, 2005 03:30
Knud van Eeden,
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--- Knud van Eeden --- 13 April 2005 - 06:25 pm ----------------------
Math: Calculus: Definition: What is the definition of a partial
differential equation?
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If in an equation containing a dependent variable u(x,y,z, ....) there
are
more than one independent variables x, y, z, ... and there also occur
derivatives of u to x, u to y, u to z, ... then you call this equation
a
partial differential equation.
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e.g.
u = ux + uy
where u is an dependent variable of x and y ux is the partial
derivative of
this u to x uy is the partial derivative of this u to y
This basically asks, via this equation, 'can you find a function u of
two
independent variables such that if you take the partial derivative of
this
function u to x, and also the partial derivative of thus function u to
y
and then add it, you will get that function u as a result?'
---
e.g.
ut = 3 . ux
where u is an dependent variable of t and x ut is the partial
derivative of
this u to t ux is the partial derivative of this u to x
This basically asks, via this equation, 'can you find a function u of
two
independent variables t and x, such that if you take the partial
derivative
of this function u to t, that will give the same result as taking the
partial derivative of thus function u to x and then multiply it with
three.
Can you please tell me that function u?'
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e.g.
utt = 3 . uxx
where u is an dependent variable of t and x utt is the second partial
derivative of this u to t uxx is the second partial derivative of this
u to
x
This basically asks, via this equation, 'can you find a function u of
two
independent variables t and x, such that if you take the second partial
derivative of this function u to t, that will give the same result as
taking the second partial derivative of thus function u to x and then
multiply it with three. Can you please tell me that function u?'
---
e.g.
utt = 3 . uxx + 4 . uyy + 6 . uzz
where u is an dependent variable of t and x utt is the second partial
derivative of this u to t uxx is the second partial derivative of this
u to
x uyy is the second partial derivative of this u to y uzz is the second
partial derivative of this u to z
This basically asks, via this equation, 'can you find a function u of
four
independent variables t, x, y and z such that if you take the second
partial derivative of this function u to t, then take the second
partial
derivative of this function u to x and then multiply it with three.
then
take the second partial derivative of this function u to y and then
multiply it with four. then take the second partial derivative of this
function u to z and then multiply it with six. and then add the latter
three expressions, it will give equality. Can you please tell me that
function u?'
---
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Internet: see also:
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Math: Calculus: Differential equation: Partial: Link: Overview: Can
you give an overview of links?
http://www.faqts.com/knowledge_base/view.phtml/aid/35621/fid/813
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