Solving a weakly singular integral equation

*To*: mathgroup at smc.vnet.net*Subject*: [mg53783] Solving a weakly singular integral equation*From*: Zaeem Burq <Z.Burq at ms.unimelb.edu.au>*Date*: Thu, 27 Jan 2005 05:41:21 -0500 (EST)*Sender*: owner-wri-mathgroup at wolfram.com

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