bvp4c_test, a MATLAB code which calls bvp4c(), which solves boundary value problems (BVP) in one spatial dimension.
A simple two point boundary value problem involves a second degree differential equation
y"(x) = f(x,y)which is to hold over some interval
a < x < bwith boundary conditions
y(a) = ya y(b) = yb
The solution to your boundary value problem begins by specifying an initial guess for the solution. If the problem is linear, or only mildly nonlinear, then a simple guess may be sufficient. For difficult problems, it may be necessary to exert some effort to preparing a good initial guess. The guess function is a MATLAB structure, which is typically defined by a call to bvpinit() as follows:
solinit = bvpinit ( xinit, yinit, parameters )where
Once the solution guess has been defined, the simplest call to bvp4c() has the form
sol = bvp4c ( odefun, bcfun, solinit )where
The simplest call to the user-written function odefun() has the form
dydx = odefun ( x, y )where
The simplest call to the user-written function bcfun() has the form
res = bcfun ( ya, yb )where
The bvp4c() function returns as output the quantity sol, which contains information that can be used to evaluate the solution components at any point in the domain. To do this, however, you must invoke the deval() function. For instance, if the solution has been computed over the interval [0,4], and we wish to evaluate the solution y(x) at 101 evenly spaced points within that interval, then the sequence of commands might be:
solinit = bvpinit ( xinit, yinit ); sol = bvp4c ( odefun, bcfun, solinit ); x = linspace ( 0.0, 4.0, 101 ); y = deval ( sol, x ); plot ( x, y(1,:) );Note that deval evaluates all M components of the solution. In the common case of a second order BVP, y(1,:) would contain the solution, and y(2,:) the derivative of the solution, at each point x.
The computer code and data files described and made available on this web page are distributed under the MIT license
bvp4c_test is available in a MATLAB version.
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SAMPLE 1 sets up a solution to the problem y'' + abs(y) = 0, y(0) = 0, y(4) = -2.
EXAMPLE 1 sets up a solution to a system of five first order ODE's. This is a sample problem for the MUSN program.
EXAMPLE 2 sets up a solution to a y''+3py/(p+x^2)^2=0, for which an analytic solution is known.
EXAMPLE 3 sets up a solution to Mathieu's equation, an eigenvalue problem y'' + (lambda-2*q*cos(2x)y)=0, y'(0) = 0, y'(pi) = 0, y(0) = 1, with q = 5, and lambda an unknown eigenvalue which we estimate to be 15. The special functional form is used to specify the initial guess for the solution.
EXAMPLE 4 sets up problem modeling the propagation of nerve impulses, in which the solution is expected to be periodic, with the period unknown.
BRATU sets up the Bratu equation, which includes a parameter lambda. Depending on the value of lambda, the equation may have 2, 1 or 0 solutions.