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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
>
>
>
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