1d:T797,# Shooting Method The shooting method solves the boundary value problem for second-order differential equations. ```r shooting(f, t0, tfinal, y0, h, a, b, itermax = 20, tol = 1e-6, hmax = 0) ``` ## Arguments - `f`: function in the differential equation $y'' = f(x, y, y')$. - `t0, tfinal`: start and end points of the interval. - `y0`: starting value of the solution. - `h`: function defining the boundary condition as a function at the end point of the interval. - `a, b`: two guesses of the derivative at the start point. - `itermax`: maximum number of iterations for the secant method. - `tol`: tolerance to be used for stopping and in the `ode45` solver. - `hmax`: maximal step size, to be passed to the solver. ## Details A second-order differential equation is solved with boundary conditions `y(t0) = y0` at the start point of the interval, and `h(y(tfinal), dy/dt(tfinal)) = 0` at the end. The zero of `h` is found by a simple secant approach. ## Returns Returns a list with two components, `t` for grid (or `time') points between `t0` and `tfinal`, and `y` the solution of the differential equation evaluated at these points. ## References L. V. Fausett (2008). Applied Numerical Analysis Using MATLAB. Second Edition, Pearson Education Inc. ## Note Replacing secant with Newton's method would be an easy exercise. The same for replacing `ode45` with some other solver. ## See Also `bvp` ## Examples ```r #-- Example 1 f <- function(t, y1, y2) -2*y1*y2 h <- function(u, v) u + v - 0.25 t0 <- 0; tfinal <- 1 y0 <- 1 sol <- shooting(f, t0, tfinal, y0, h, 0, 1) ## Not run: plot(sol$t, sol$y[, 1], type='l', ylim=c(-1, 1)) xs <- linspace(0, 1); ys <- 1/(xs+1) lines(xs, ys, col="red") lines(sol$t, sol$y[, 2], col="gray") grid() ## End(Not run) #-- Example 2 f <- function(t, y1, y2) -y2^2 / y1 h <- function(u, v) u - 2 t0 <- 0; tfinal <- 1 y0 <- 1 sol <- shooting(f, t0, tfinal, y0, h, 0, 1) ```

shooting function