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Re: Numerical equation solving

  • To: mathgroup at smc.vnet.net
  • Subject: [mg116203] Re: Numerical equation solving
  • From: Gary Wardall <gwardall at gmail.com>
  • Date: Sat, 5 Feb 2011 05:44:05 -0500 (EST)
  • References: <iigpkp$21b$1@smc.vnet.net>

On Feb 4, 5:58 am, florian.mau... at schott.com wrote:
> Hi everybody,
>
> I have a quite challenging question about numerical equation solving with
> MATHEMATICA:
>
> f1 - f2 + ((f2 - f1) t)/t1 = (r1^2 rho vt^2)/(2 \[Pi] r2^4) (3/2 + l/(2
> r2) k/((2 rho vt)/(\[Pi] r2 eta))^(1/4))
>
> The equation given above is implicit when solving to variable vt. The
> variable vt itself is a differential operator (D[v,t]), so vt must be
> replaced with D[v,t]. As I am interested in v the solution has to be
> integrated with respect to t (i.e. Integrate[D[v,t],t,{0,t1}] or Integrate
> [D[v,t],t,{0,t1}]). Finally, I want to visualize in a contour plot the
> values of f1 and t1 for which the integrated equation fulfills a certain
> number q. All variables are positive real numbers.
> To maybe better explain my problem I have summarized all steps in one
> MATHEMATICA command (I know that the syntax is not correct):
>
> ContourPlot[q = NIntegrate[D[v, t]/.Solve[f1-f2 + ((f2 - f1) t)/t1 ==
> (r1^2 rho D[v, t]^2)/(2 \[Pi] r2^4) (3/2 + l/(2 r2) k/((2 rho D[v,
> t])/(\[Pi] r2 eta))^(1/4)), D[v, t]],t, {0, t1}], {t1, 0.1, a}, {f1, 0, b}]
>
> How can I solve solve this problem with MATHEMATICA? Thanks in advance for
> your support!
>
> Mr.Mason


There might be a solution. If you can change your equation to and ODE
or a system of ODE's.

Here is a much simpler example:


Given: Sin[x+y]+y=x
Solve for y.

Solution:

The solution for y by the following method will be implicit.

Let yp be the first derivative of y with respect to x.

Then:

Cos[x+y]*[1+yp]+yp=1

Solving for yp:

Cos[x+y]+Cos[x+y]*yp+yp=1
Cos[x+y]+(Cos[x+y]+1)*yp=1
(Cos[x+y]+1)*yp=1-Cos[x+y]
yp=(1-Cos[x+y])/(Cos[x+y]+1)

Now find a point on the graph of y, numerically or otherwise:

Note that:
Sin[0+0]+0=0
So y(0)=0

Now solve the ODE:

yp=(1-Cos[x+y])/(Cos[x+y]+1) when y(0)=0

I hope this will work for you.

Good Luck


Gary Wardall


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