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Solving a weakly singular integral equation

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  • Subject: [mg53783] Solving a weakly singular integral equation
  • From: Zaeem Burq <Z.Burq at>
  • Date: Thu, 27 Jan 2005 05:41:21 -0500 (EST)
  • Sender: owner-wri-mathgroup at

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
  \(p[x_] := \[ExponentialE]\^\(\(-x\^2\)/2\)\/\@\(2  \[Pi]\);\)\n
  \(c[t_] := Sqrt[t]\  + \ 2  t/9;\)\n
  \(approxsoln[x_] = 0;\)\n
        t_] := \(\(\ \)\(\((c[t] - c[s])\)\ p[\(c[t] - c[s]\)\/\@\(t - 
\)\/\((t - s)\)\^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]; 

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, 

Room 201, Richard Berry Building
University of Melbourne,
Parkville, VIC 3052.

ph: 8344 4248.

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