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