Re: system of differential equations mathematica help

*To*: mathgroup at smc.vnet.net*Subject*: [mg129381] Re: system of differential equations mathematica help*From*: Alexei Boulbitch <Alexei.Boulbitch at iee.lu>*Date*: Thu, 10 Jan 2013 02:20:33 -0500 (EST)*Delivered-to*: l-mathgroup@mail-archive0.wolfram.com*Delivered-to*: l-mathgroup@wolfram.com*Delivered-to*: mathgroup-newout@smc.vnet.net*Delivered-to*: mathgroup-newsend@smc.vnet.net

Hi, Your equations admit 2 first integrals: a[x]*u[x]=C1 u[x]^2/2+C2=-p[x] With the help of the boundary conditions one gets C1=0.1 and C2=-0.995. You may eliminate u and p and end up with a single equation: 50a''-0.005a^(-2)+2a^(-3/2)=2.005 (Please check its correctness yourself. I did it fast and may have introduced minor errors). This one is more easy to solve. Even with this equation, however, NDSolve reports problems: s = NDSolve[{a''[x] - 0.005*a[x]^-2 + 2*a[x]^(-3/2) == 2.005, a[0] == 1, a[10] == 1}, a[x], {x, 0, 10}] Power::infy: Infinite expression 1/0.^2 encountered. >> Power::infy: Infinite expression 1/0.^(3/2) encountered. >> Infinity::indet: Indeterminate expression 2.005 +ComplexInfinity+ComplexInfinity encountered. >> Here one way is to try to regularize it so that one has no infinity at a->0: s = NDSolve[{a''[x] - 0.005*(a[x]^2 + 0.001)^-1 + 2*(a[x]^2 + 0.001)^(-3/4) == 2.005, a[0] == 1, a[10] == 1}, a[x], {x, 0, 10}] This is already better, as you may see by evaluating this: Plot[a[x] /. s, {x, 0, 10}] But there are still warnings: FindRoot::cvmit: Failed to converge to the requested accuracy or precision within 100 iterations. >> NDSolve::berr: There are significant errors {-0.95868,-0.852012} in the boundary value residuals. Returning the best solution found. >> But in this point you may already look for some appropriate method to get a better solution. Try to go through the tutorials: Menu/Help/ tutorial/IntroductionToNumericalDifferentialEquations Menu/Help/ tutorial/NumericalSolutionOfDifferentialEquations And Menu/Help/ tutorial/NDSolveOverview There you will find several approaches to apply in difficult cases as yours. Have fun, Alexei Alexei BOULBITCH, Dr., habil. IEE S.A. ZAE Weiergewan, 11, rue Edmond Reuter, L-5326 Contern, LUXEMBOURG Office phone : +352-2454-2566 Office fax: +352-2454-3566 mobile phone: +49 151 52 40 66 44 e-mail: alexei.boulbitch at iee.lu