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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
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