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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