# bvp: Boundary Value Problems In pracma: Practical Numerical Math Functions

 bvp R Documentation

## Boundary Value Problems

### Description

Solves boundary value problems of linear second order differential equations.

### Usage

``````bvp(f, g, h, x, y, n = 50)
``````

### Arguments

 `f, g, h` functions on the right side of the differential equation. If `f, g` or `h` is a scalar instead of a function, it is assumed to be a constant coefficient in the differential equation. `x` `x[1], x[2]` are the interval borders where the solution shall be computed. `y` boundary conditions such that `y(x[1]) = y[1], y(x[2]) = y[2]`. `n` number of intermediate grid points; default 50.

### Details

Solves the two-point boundary value problem given as a linear differential equation of second order in the form:

`y'' = f(x) y' + g(x) y + h(x)`

with the finite element method. The solution `y(x)` shall exist on the interval `[a, b]` with boundary conditions `y(a) = y_a` and `y(b) = y_b`.

### Value

Returns a list `list(xs, ys)` with the grid points `xs` and the values `ys` of the solution at these points, including the boundary points.

### Note

Uses a tridiagonal equation solver that may be faster then `qr.solve` for large values of `n`.

### References

Kutz, J. N. (2005). Practical Scientific Computing. Lecture Notes 98195-2420, University of Washington, Seattle.

`shooting`

### Examples

``````##  Solve y'' = 2*x/(1+x^2)*y' - 2/(1+x^2) * y + 1
##  with y(0) = 1.25 and y(4) = -0.95 on the interval [0, 4]:
f1 <- function(x) 2*x / (1 + x^2)
f2 <- function(x)  -2 / (1 + x^2)
f3 <- function(x) rep(1, length(x))     # vectorized constant function 1
x <- c(0.0,   4.0)
y <- c(1.25, -0.95)
sol <- bvp(f1, f2, f3, x, y)
## Not run:
plot(sol\$xs, sol\$ys, ylim = c(-2, 2),
xlab = "", ylab = "", main = "Boundary Value Problem")
# The analytic solution is
sfun <- function(x) 1.25 + 0.4860896526*x - 2.25*x^2 +
2*x*atan(x) - 1/2 * log(1+x^2) + 1/2 * x^2 * log(1+x^2)
xx <- linspace(0, 4)
yy <- sfun(xx)
lines(xx, yy, col="red")
grid()
## End(Not run)
``````

pracma documentation built on Nov. 10, 2023, 1:14 a.m.