mqr.utils.

fredholm2

mqr.utils.fredholm2(t0, fn_K, fn_g, lmda, x, w)

Solve a Fredholm equation of the second kind.

Evaluates \(f\) at t0 in:

\[f(t) = g(t) + λ \int_Q K(t, s) f(s) ds\]

where the definite integral is approximated over the region \(Q\) defined by the quadrature points x and weights w.

Uses the quadrature points and weights provided to form linear equations, and then interpolates the result using Nystrom’s method. See [1] for a summary.

Parameters:

t0float

Point to evaluate the function f(t).

fn_KCallable[]

Function that returns values of K(t, s); see equation above.

fn_gCallable[]

Function that returns values of g(t); see equation above.

lmbdafloat

Multiple of the integral term; see equation above.

xarray_like

Quadrature abcissa points.

warray_like

Quadrature weights.

References

[1]

Press, W. H. (2007). Numerical recipes 3rd edition: The art of scientific computing. Cambridge university press.