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Solving a weakly singular integral equation - Take 2.
*To*: mathgroup at smc.vnet.net
*Subject*: [mg54485] Solving a weakly singular integral equation - Take 2.
*From*: Zaeem Burq <Z.Burq at ms.unimelb.edu.au>
*Date*: Mon, 21 Feb 2005 03:44:55 -0500 (EST)
*Sender*: owner-wri-mathgroup at wolfram.com
Dear all,
I am still trying to solve a 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]/Sqrt[2 Pi] (* Gaussian probability density
function*)
c[t_]:= Sqrt[2.1 t Log[Log[1/t]] ]
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 kernel has a $(t-s)^1.5$ in the denominator, i.e.,
singularities along the diagonal (s=t). These singularities are
appropriately killed by the function p, but Mathematica seems to have
trouble with them:
Since c is differentiable (except at zero), the factor c[t] - c[s] in the
numerator of the kernel is O(t-s) when t-s is small and t > 0. This means
that the denominator (t-s)^1.5 is only as bad as Sqrt[t-s]. This combined
with the exponential function, means that the kernel is actually at-least
Holder continuous of order -0.5. Even when t=0, the exponential function
kills the pole at zero. It is known from other sources, that this integral
equation has a unique solution.
I wrote down the following routine based on a paper in the Mathematica
Journal: 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
\(c[t_] := Sqrt[2.1\ t\ Log[Log[1/t]]];\)\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]];\)\[IndentingNewLine]
Plot[approxsoln[x], {x, 0, .01}]\)
The PROBLEMS:
1. I have had to truncated the integral away from zero - i.e., s runs from
0 to t, but t runs from 0.0000000001 to 0.01. It should run from 0 to 0.1.
This is not entirely satisfactory, as I am mainly interested in behaviour
near zero.
2. As the code stands, Mathematica tells me that the numerical integrals
converge too slowly.
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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