solving integral eq. numerically

[liketen]

hi everyone

I have met a problem soving integral equation,

in [0, 1], ∫{(s²+t²)^½}×u(t)dt = {(s²+1)^1.5 -s³}/3

question requires to solve by using Simpson rule and Gaussian elimination with partial pivoting.

after some compositing Simpson method, how to set the system for Ax=y?

I should have dropt Numerical Analysis...

 

[jcy]

in [0, 1], ∫{(s²+t²)^½}×u(t)dt = {(s²+1)^1.5 -s³}/3

You can try the following method.

1. break up your line into N equally-spaced segments with endpoints
{t0,t1,t2,.....,t_n} where |t_{n+1}-t_n| = 1/N

2. expand u(t) = ∑{n=1}^N b_n Π_n(t).

Here, Π_n(t) = 1 for (t_{n-1},t_n)
= 0 otherwise

b_n are the unknown coefficients you want to solve for.

Obviously, if the solution u(t) is smooth, then you can approximate
u(t) by the summation if you choose b_n correctly and N large enough.

Now your equation is

∫{(s²+t²)^½}×∑{n=1}^N b_n Π_n(t).dt = {(s²+1)^1.5 -s³}/3

or interchanging summation and integration

∑{n=1}^N b_n ∫{(s²+t²)^½}× Π_n(t).dt = {(s²+1)^1.5 -s³}/3

3. Now, you have to solve for the b_n. You have N unknowns b_n, so you would like to make a matrix equation with N equations. Therefore, you can enforce the last equation at s_m for m = 1,2,...,N. The best s_m are at the midpoints of the
pulse functions, s_m = (t_{m}+t_{m-1})/2

So you get N equations, where the m-th equation is

∑{n=1}^N b_n ∫{(sm²+t²)^½}× Π_n(t).dt = {(sm²+1)^1.5 -sm³}/3

Or, in matrix terminology

Smn * b_n = v_m

4. Compute the entries

Smn = ∫_t{m-1}^t_m {(sm²+t²)^½} dt
Vm = {(sm²+1)^1.5 -sm³}/3

Use Simpsons Rule to compute the simple Smn integrations numerically.

5. Solve the matrix equation
Smn * bn = vm
using Gaussian elimination

Hope this helps. Sorry for the notation, I didn't have a lot of time.

John

 

[danidani]

thanks for John reply!

 

[shaista]

WHERE CAN I GET THE REQUIRED HELP BOOK ABOUT INTERGRAT SOLVING USING NUMERICAL METHODS I REQUIRE THEM BADLY AND URGENTLY PLEASE HELP ME I WILL BE VERY THANK FULL