Services & Resources / Wolfram Forums
-----
 /
MathGroup Archive
2005
*January
*February
*March
*April
*May
*June
*July
*August
*September
*October
*November
*December
*Archive Index
*Ask about this page
*Print this page
*Give us feedback
*Sign up for the Wolfram Insider

MathGroup Archive 2005

[Date Index] [Thread Index] [Author Index]

Search the Archive

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
>
>
> 



  • Prev by Date: Re: Num. integration problem in Mathematica
  • Next by Date: Re: Algebra of Einstein velocity addition
  • Previous by thread: Solving a weakly singular integral equation
  • Next by thread: Re: Solving a weakly singular integral equation