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

## Description

Solves boundary value problems of linear second order differential equations.

## Usage

 `1` ```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

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19``` ```## 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) ```

### Example output

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pracma documentation built on Dec. 11, 2021, 9:57 a.m.