Re: Solving a weakly singular integral equation

*To*: mathgroup at smc.vnet.net*Subject*: [mg53801] Re: Solving a weakly singular integral equation*From*: "Jens-Peer Kuska" <kuska at informatik.uni-leipzig.de>*Date*: Fri, 28 Jan 2005 02:43:45 -0500 (EST)*Organization*: Uni Leipzig*References*: <ctahsj$b14$1@smc.vnet.net>*Sender*: owner-wri-mathgroup at wolfram.com

Hi, and why do you not use the analytic power of Mathematica ? p[x_] := Exp[- x^2/2]/(2 Pi) c[t] := Sqrt[t] + 2t/9 G[s_, t_] := (c[t] - c[s]) p[(c[t] - c[s])/Sqrt[t - s]] / (t - s)^(3/2) eqn = f[t] == c[t]p[c[t]/Sqrt[t]] / t^(3/2) - Integrate[G[s, t]*f[s], {s, 0, t}] and finaly compute a power series with logeqn = Series[#, {t, 0, 3}] & /@ eqn The result show there is a series term with 1/t that cause the singularity. Regards Jens "Zaeem Burq" <Z.Burq at ms.unimelb.edu.au> schrieb im Newsbeitrag news:ctahsj$b14$1 at smc.vnet.net... > > Dear all, > > I am trying to solve a non-linear Volterra type-2 integral equation by > using successive approximations method. I am mainly interested in the > behaviour of the solution near zero. > > The unknown function is f[t]. Define > > p[x_]:= Exp[- 0.5 * x^2]/(2 Pi) (* Gaussian probability density function*) > > c[t]:= Sqrt[t] + 2t/9 > > G[s_,t_]:= (c[t] - c[s]) p[(c[t]-c[s])/Sqrt[t-s]] / (t-s)^1.5 (* The > kernel *) > > The integral equation is: > > f[t] = c[t]p[c[t]/Sqrt[t]] / t^1.5 - \int_{0}^{t} G[s,t] f[s] ds > > As you can see, the perturbation function has an apparant singularity at > 0, and the kernel has singularities along the diagonal (s=t). All the > singularities are appropriately killed by the function p, but Mathematica > seems to have trouble with them. > > I wrote down the following routine: n is the number of iterations of the > approximation process: > > \!\(Clear[c, approxsoln, K]\n > n = 3\n > 3\n > \(p[x_] := \[ExponentialE]\^\(\(-x\^2\)/2\)\/\@\(2 \[Pi]\);\)\n > \(c[t_] := Sqrt[t]\ + \ 2 t/9;\)\n > \(approxsoln[x_] = 0;\)\n > \(G[s_, > t_] := \(\(\ \)\(\((c[t] - c[s])\)\ p[\(c[t] - c[s]\)\/\@\(t - > s\)]\)\ > \)\/\((t - s)\)\^1.5`;\)\n > \(For[j = 1, > j <= \ n, \(j++\), \[IndentingNewLine]values = > Table[{t, \(\(\ \)\(c[t] p[c[t]\/\@t]\)\)\/\((t)\)\^1.5`\ - \ > NIntegrate[G[s, t]*approxsoln[s], {s, 0, t}]}, {t, > 0.0000000001, .01, .01\/10}]; \n > approxsoln[t_] = > InterpolatingPolynomial[values, t]; > \[IndentingNewLine]Print[j]];\)\) > > Plot[approxsoln[x], {x, 0, .01}] > > I have truncated the integral away from zero, but this is not entirely > satisfactory, as I am mainly interested in behaviour near zero. > > I'd appreciate any help. > > Best, Zaeem. > > ________________________________ > > Zaeem Burq > PhD Stochastic Processes, > Dept. of Mathematics and Statistics, > Unimelb. > > Room 201, Richard Berry Building > University of Melbourne, > Parkville, VIC 3052. > > ph: 8344 4248. > > http://www.ms.unimelb.edu.au/~zab > > >