1d:T772,# Classical Runge-Kutta Classical Runge-Kutta of order 4. ```r rk4(f, a, b, y0, n) rk4sys(f, a, b, y0, n) ``` ## Arguments - `f`: function in the differential equation $y' = f(x, y)$; defined as a function $R \times R^m \rightarrow R^m$, where $m$ is the number of equations. - `a, b`: endpoints of the interval. - `y0`: starting values; for $m$ equations `y0` needs to be a vector of length `m`. - `n`: the number of steps from `a` to `b`. ## Details Classical Runge-Kutta of order 4 for (systems of) ordinary differential equations with fixed step size. ## Returns List with components `x` for grid points between `a` and `b` and `y` an n-by-m matrix with solutions for variables in columns, i.e. each row contains one time stamp. ## References Fausett, L. V. (2007). Applied Numerical Analysis Using Matlab. Second edition, Prentice Hall. ## Note This function serves demonstration purposes only. ## See Also `ode23`, `deval` ## Examples ```r ## Example1: ODE # y' = y*(-2*x + 1/x) for x != 0, 1 if x = 0 # solution is x*exp(-x^2) f <- function(x, y) { if (x != 0) dy <- y * (- 2*x + 1/x) else dy <- rep(1, length(y)) return(dy) } sol <- rk4(f, 0, 2, 0, 50) ## Not run: x <- seq(0, 2, length.out = 51) plot(x, x*exp(-x^2), type = "l", col = "red") points(sol$x, sol$y, pch = "*") grid() ## End(Not run) ## Example2: Chemical process f <- function(t, u) { u1 <- u[3] - 0.1 * (t+1) * u[1] u2 <- 0.1 * (t+1) * u[1] - 2 * u[2] u3 <- 2 * u[2] - u[3] return(c(u1, u2, u3)) } u0 <- c(0.8696, 0.0435, 0.0870) a <- 0; b <- 40 n <- 40 sol <- rk4sys(f, a, b, u0, n) ## Not run: matplot(sol$x, sol$y, type = "l", lty = 1, lwd = c(2, 1, 1), col = c("darkred", "darkblue", "darkgreen"), xlab = "Time [min]", ylab= "Concentration [Prozent]", main = "Chemical composition") grid() ## End(Not run) ```

rk4 function