---
title: "Inteq R documentation"
authors:
  - name: Aaron Jehle
    orcid: 0000-0002-3809-8248
    affiliation: 1
  - name: Mark Clements
    affiliation: 1
  - name: Matthew W. Thomas
    affiliation: 2
affiliations:
 - name: Department of Medical Epidemiology and Biostatistics, Karolinska Institutet, Stockholm, Sweden
   index: 1
 - name: Bureau of Economics, Federal Trade Commission, Washington, DC, USA
   index: 2
date: "`r Sys.Date()`"
output:
  rmarkdown::html_document:
    toc: true
    toc_depth: 2
    toc_float: true
    theme: lumen
    keep_md: true
vignette: >
  %\VignetteIndexEntry{Inteq R documentation}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}

---

This R package estimates Volterra and Fredholm integral equations.
It is a translation into R of the Python package "inteq" by Matthew W. Thomas (https://github.com/mwt/inteq).

Installation:
//to be completed

## Supported equations

# Fredholm integral equations


This package provides the function fredholm_solve which approximates the solution, $g(x)$, to the Fredholm Integral Equation of the first kind:

\begin{align}
f(s) = \int_a^b K(s, y)\, g(y)\, dy,
\end{align}

using the method described in Twomey (1963). It will return a grid that is an approximate solution, as demonstrated by the following example:

\[
f(s) = \int_{-3}^{3} K(s, y)\, g(y)\, dy,
\]

where the kernel \( K(s, y) \) and left-hand side \( f(s) \) are defined as:

\[
K(s, y) = 
\begin{cases}
1 + \cos\left(\dfrac{\pi (y - s)}{3}\right), & \text{if } |s - y| \leq 3 \\
0, & \text{otherwise}
\end{cases},
\]

\[
f(s) = \frac{1}{2} \left[ (6 - |s|)\left(2 + \cos\left(\frac{|s| \pi}{3}\right)\right) + \frac{9}{\pi} \sin\left(\frac{|s| \pi}{3}\right) \right].
\]

The true solution is \( g(y) = K(0, y) \).

```{r, echo=FALSE}

library(inteq)

# 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)
)

#end of code chunk
```

# Volterra integral equations

This package provides the function volterra_solve which approximates the solution, g(x), to the Volterra Integral Equation of the first kind:

\begin{align}
f(s) = \int_a^s K(s,y) g(y) dy,
\end{align}

using the method in Betto and Thomas (2021), as demonstrated by the following example:

\[
f(s) = \int_0^s \cos(t-s) g(t) dt,
\]

with left-hand side

\[
f(s) = \int_0^s \cos(t-s) \frac{2 + t^2}{2} dt
\]

and true solution

\[
g(s) = \frac{2 + s^2}{2}.
\]


```{r, echo=FALSE}
k <- function(s,t) {
        cos(t-s)
}
trueg <- function(s) {
    (2+s**2)/2
}

res <- volterra_solve(k,a=0,b=1,num=1000)

plot(
    res$sgrid, res$ggrid,
    type = "l",
    col = "blue",
    xlim = c(0, 1),
    #ylim = c(-1, 1),
    xlab = "s",
    ylab = "g(s)",
    main = "Volterra Equation Solution first kind"
)
# add the true solution
lines(res$sgrid, trueg(res$sgrid), col = "red", lty = 2)
legend( 
    "topright",
    legend = c("Estimated Value", "True Value"),
    col = c("blue", "red"),
    lty = c(1, 2)
)

```

It also provides volterra_solve2 which approximates the solution, g(x), to the Volterra Integral Equation of the second kind:

\begin{align}
g(s) = f(s) + \int_a^s K(s,y) g(y) dy,
\end{align}

using the method in Linz (1969), as demonstrated by the following example:

\[
g(s) = 1 + \int_0^s (s-t) g(t) dt,
\]

with true solution

\[
g(s) = \cosh(s).
\]

```{r, echo=FALSE}
k <- function(s,t) {
        0.5 * (t-s)** 2 * exp(t-s)
}
free <- function(t) {
    0.5 * t**2 * exp(-t)
}
true <- function(t) {
    1/3 * (1 - exp(-3*t/2) * (cos(sqrt(3)/2*t) + sqrt(3) * sin(sqrt(3)/2*t)))
}

res <- volterra_solve2(k,free,a=0,b=6,num=100)

plot(
    res$sgrid, res$ggrid,
    type = "l",
    col = "blue",
    xlim = c(0, 6),
    #ylim = c(-1, 1),
    xlab = "s",
    ylab = "g(s)",
    main = "Volterra Equation Solution second kind"
)
# add the true solution
lines(res$sgrid, true(res$sgrid), col = "red", lty = 2)
legend( 
    "topright",
    legend = c("Estimated Value", "True Value"),
    col = c("blue", "red"),
    lty = c(1, 2)
)

```