Re: NDSolve -- indexing of dependent variable that is arbitrary list

*To*: mathgroup at smc.vnet.net*Subject*: [mg112238] Re: NDSolve -- indexing of dependent variable that is arbitrary list*From*: Greylander <greylander at gmail.com>*Date*: Mon, 6 Sep 2010 04:14:48 -0400 (EDT)*References*: <i5vnsi$2es$1@smc.vnet.net>

I have found a workaround for this problem, but I am interested if there is a more straightforward way. Instead of using Part[], as in y[t][[i]] and y''[t][[i]] (see previous examples quoted below), I can use "y[i]" as a dependent variable. This allows my dependent variable to effectively have arbitrary structure based on the indexing: "y[i,j,k...]". In my example, the differential equations would be constructed with Table[ y[i]''[t] == Sum[ f[ y[i][t], y[j][t] ], {j ,1, N} ], {i , 1, N} ] If f above is an inverse square force law, then the above could be a 1- dimensional N-body problem with interaction force f. For a 3- dimensional N-body problem, with interaction force f, I could use: Table[ y[i,k]''[t] == Sum[ f[k][ { y[i,1][t], y[i,2][t], y[i, 3] }, { y[j,1][t], y[j,2][t], y[j,3] } ], {j, 1, N} ], {i, 1, N} ], {k, 1, 3} ] where the 3D coordinates of each body are given by { y[i,1], y[i,2], y[i,3] }, and the vector force function f[a,b]:={f[1][a,b],f[2] [a,b],f[3][a,b]}. The above technique allows for arbitrary structure to the independent variable, but without ever using "y" as a variable that has list- structure. This is not obvious and not elegant. I wonder if someone can show me a way to do this for arbitrary list structure that would not require the y[i,k] notation, where I can set up the equation starting with y''[t] == ...? However, this is now an academic question, because I now have a way to achieve the desired results. On Sep 5, 5:28 am, Greylander <greylan... at gmail.com> wrote: > Hello, this is related to my previous question about NDSolve -- n- > body problem. The question is the generalized version of that > question, and hopefully I am expressing the question better ere. > > If your dependent variable, y[t] (for example), has arbitrary list > structure (list of lists of lists..), how can you set up an equation > such as > > y''[t] = f(y[t]) > > where f(y[t]) is any arbitrary function of the elements of y[t]? > > The simple examples provided in the documentation only show cases of > f(y[t]) that use vector or matrix operations that operate on y[t] as a > whole without explicitly referencing any elements of y[t]. Any form > of using Part[] that i have tried always acts on the literal "y[t]" > instead of 'waiting' until to act on y[t] that have been given a value > (i.e. y[0] is properly initialized with the full list structure. I > would, for example, like to create an equation or set of equations, > for NDSolve like this: > > Table[ y''[t][[i]] == Sum[ f[ y[t][[i]], y[t][[j]] {j ,1, = N} ], > {i ,1, N} ], > y'[0] = Table[ g[i], {i, 1, N} ], > y[0] = Table[ h[i], {i, 1, N} ] > > or > > y''[t] = Table[ Sum[ f[ y[t][[i]], y[t][[j]] {j 1 N} ], {i 1 > N} ], > y'[0] = Table[ g[i], {i, 1, N} ], > y[0] = Table[ h[i], {i, 1, N} ] > > But neither of the above will work because the [[i]] or [[j]] indexing > is applied to the literal y''[t] and y[t], and not to the list > structure they should have later. > > Thanks for any help.