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I learned numerical integration methods such as rectangualr, trapezoid, Simpson, Roberg Integration, Richardson Extrapolation, Adaptive Quadrature, Gauss Quadrature.

My professor ask students to solve the Fredholm integral equation by using the methods that I learned. $$f(x) = \phi(x) + \int_a^bK(x,t)\phi(t)dt \tag{f(x) and K(x,t) are given}$$ I tried all of the methods I learned, but it seems that they didn't work. I can't make the solution to have a form like $\phi(x) = something$. Can anybody give me a hint for solving this equation?

alryosha
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1 Answers1

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If "solve" means "solve numerically", pick a set of quadrature points $x_1, \ldots, x_N \in [a, b]$ with weights $w_1, \ldots, w_N$, usually $w_n > 0$, such that $$ \int_a^b g(t) dt \approx \sum_n w_n g(x_n). $$

It could be the trapezoidal rule, for example.

Then solve the system of $N$ linear equations $$ f(x_m) \stackrel{!}{=} \phi_m + \sum_n w_n K(x_m, x_n) \phi_n, \qquad m = 1, \ldots, N, $$ for $\phi_1, \ldots, \phi_N$.

Then $\phi(x_m) \approx \phi_m$.

Cite https://math.stackexchange.com/a/3421643/ as a source if you choose to use this in your homework.

user66081
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