I'm trying to solve the attached integral equation (Eq. 5), numerically. The value of particle acceleration 
and fluid acceleration 
are known up to the time step n where t = n 
and the integral is from time 0 to t. I'm currently using the basic definition of integral to solve it (namely, Eq. 4). The integral can be evaluated numerically except at a region where τ is approaching t. There, the integrand becomes infinite and needs to be treated separately. Therefore, I split the integral into two terms: Z1 and Z2 where Z1 is the series term expressed in Eq. 5 and Z2 is the last term in Eq. 5. Here is the code that I wrote:
if n > 2
Z1= 0;
for m = 2:n-1
Z1 = Z1+ ((d2z_p(n-m) - d2z_f(n-m))/sqrt(m)) * dt^(1/2)
end
Z2 = (d2z_p(n-1) - d2z_f(n-1)) * dt^(1/2);
Z_1 = Z1 + Z2;
end
Z_1 is supposed to be the solution of the integral, but I know that it doesn't give the right answer either because Eq. 4 is not the right one to use, or my code is wrong. Anyone can help me on how to solve the given integral?
