       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