inteq (version 1.0)

fredholm_solve: Solve a Fredholm equation of the first and second kind

Description

Solve a Fredholm equation of the first and second kind

Usage

fredholm_solve(
  k,
  f = function(x) x,
  a = -0,
  b = 1,
  num = 41L,
  smin = 0,
  smax = 1,
  snum = 41L,
  gamma = 0.001
)

Value

data-frame with evaluation points 'ygrid' and calculations 'ggrid'

Arguments

k

kernel function of two time scales

f

left hand side function with f(a)=0

a

lower bound of the grid for the integral approximation

b

upper bound of the grid for the integral approximation

num

number of points for the grid of the integral approximation

smin

lower bound of enforcement values for equation

smax

upper bound of enforcement values for equation

snum

number of points for the grid for the equation

gamma

regularization parameter

Examples

# Define the kernel function
k <- function(s, t) {
    ifelse(abs(s - t) <= 3, 1 + cos(pi * (t - s) / 3), 0)
}

# Define the right-hand side function
f <- function(s) {
    sp <- abs(s)
    sp3 <- sp * pi / 3
    ((6 - sp) * (2 + cos(sp3)) + (9 / pi) * sin(sp3)) / 2
}

# Define the true solution for comparison
trueg <- function(s) {
    k(0, s)
}

# Solve the Fredholm equation
res <- fredholm_solve(
    k, f, -3, 3, 1001L,
    smin = -6, smax = 6, snum = 2001L,
    gamma = 0.01
)

# Plot the results on the same graph using base graphics
plot(
    res$ygrid, res$ggrid,
    type = "l",
    col = "blue",
    xlim = c(-3, 3),
    #ylim = c(-1, 1),
    xlab = "s",
    ylab = "g(s)",
    main = "Fredholm Equation Solution"
)
# add the true solution
lines(res$ygrid, trueg(res$ygrid), col = "red", lty = 2)
legend( 
    "topright",
    legend = c("Estimated Value", "True Value"),
    col = c("blue", "red"),
    lty = c(1, 2)
)