Re: help in solving a double integration....

*To*: mathgroup at smc.vnet.net*Subject*: [mg66087] Re: [mg66062] help in solving a double integration....*From*: Daniel Lichtblau <danl at wolfram.com>*Date*: Sat, 29 Apr 2006 03:40:27 -0400 (EDT)*References*: <200604281032.GAA03124@smc.vnet.net>*Sender*: owner-wri-mathgroup at wolfram.com

ashesh wrote: > Hi all, > > Would like to solve the following double integration. > > int_x_0^t int_y_0^t F(x) exp(-a|x-y|) G(y) dy dx > > F(x) = (3x/t); 0 < x <(t/3) > = (2t-3y)/t; (t/3) < x < (2t/3) > = 0; otherwise > > G(y) = F(y) > > Where a, t are real numbers. And there is absolute value of x-y, i.e., > it is |x-y| in the exponent. > > I am not sure as to how these fucntions, F(x) and G(y) can be fed to > mathematica, as they have different values in different ranges. > > Hope some one can help me solve this integration. Define your function using Piecewise. f[x_,t_] := Piecewise[{{3*x/t,0<x<t/3}, {2*t-3*y/t,t/3<x<2*t/3}}, 0] I pass some assumptions to integrate regarding the parameters {a,t}. I think assumptions on both are needed in the current version of Mathematica though the development version does not seem to worry about a. InputForm[ii = Integrate[f[x,t]*Exp[-a*Abs[x-y]]*f[y,t], {x,0,t}, {y,0,t}, Assumptions->{t>0,a>0}]] Out[8]//InputForm= (-81 + 324*E^((a*t)/3) - 243*E^((2*a*t)/3) + 54*a*t - 108*a*E^((a*t)/3)*t + 108*a*E^((2*a*t)/3)*t - 108*a*t^2 + 216*a*E^((a*t)/3)*t^2 + 72*a^2*E^((a*t)/3)*t^2 - 108*a*E^((2*a*t)/3)*t^2 - 72*a^2*E^((2*a*t)/3)*t^2 - 144*a^2*E^((a*t)/3)*t^3 + 144*a^2*E^((2*a*t)/3)*t^3 + 16*a^3*E^((2*a*t)/3)*t^3 + 72*a^2*E^((a*t)/3)*t^4 - 72*a^2*E^((2*a*t)/3)*t^4 - 36*a^3*E^((2*a*t)/3)*t^4 + 24*a^3*E^((2*a*t)/3)*t^5)/ (9*a^4*E^((2*a*t)/3)*t^2) Sanity check: evaluate numerically at some parameter values and compare to direct quadrature result for same. In[22]:= ii /. {t->4.,a->2.} Out[22]= 39.0948 In[23]:= With[{t=4,a=2},NIntegrate[f[x,t]*Exp[-2*Abs[x-y]]*f[y,t], {x,0,t}, {y,0,t}]] Out[23]= 39.0948 Daniel Lichtblau Wolfram Research

**References**:**help in solving a double integration....***From:*"ashesh" <ashesh.cb@gmail.com>