Form initial guess for BVP solvers
solinit = bvpinit(x,yinit)
solinit = bvpinit(x,yinit,parameters)
solinit = bvpinit(sol,[anew bnew])
solinit = bvpinit(sol,[anew bnew],parameters)
x is a vector that specifies an initial mesh. If you want to solve the BVP on [a,b], then specify x(1) as a and x(end) as b. The solver adapts this mesh to the solution, so a guess like xb=nlinspace(a,b,10) often suffices. However, in difficult cases, you should place mesh points where the solution changes rapidly. The entries of x must be in
Increasing order if a<b
Decreasing order if a>b
For two-point boundary value problems, the entries of x must be distinct. That is, if a<b, the entries must satisfy x(1) < x(2) < ... < x(end). If a>b, the entries must satisfy x(1) > x(2) > ... > x(end)
For multipoint boundary value problem, you can specify the points in [a,b] at which the boundary conditions apply, other than the endpoints a and b, by repeating their entries in x. For example, if you set
x = [0, 0.5, 1, 1, 1.5, 2];
the boundary conditions apply at three points: the endpoints 0 and 2, and the repeated entry 1. In general, repeated entries represent boundary points between regions in [a,b]. In the preceding example, the repeated entry 1 divides the interval [0,2] into two regions: [0,1] and [1,2].
yinit is a guess for the solution. It can be either a vector, or a function:
Vector – For each component of the solution, bvpinit replicates the corresponding element of the vector as a constant guess across all mesh points. That is, yinit(i) is a constant guess for the ith component yinit(i,:) of the solution at all the mesh points in x.
Function – For a given mesh point, the guess function must return a vector whose elements are guesses for the corresponding components of the solution. The function must be of the form
y = guess(x)
where x is a mesh point and y is a vector whose length is the same as the number of components in the solution. For example, if the guess function is a function, bvpinit calls
y(:,j) = guess(x(j))
at each mesh point.
For multipoint boundary value problems, the guess function must be of the form
y = guess(x, k)
where y an initial guess for the solution at x in region k. The function must accept the input argument k, which is provided for flexibility in writing the guess function. However, the function is not required to use k.
solinit is a structure with the following fields. The structure can have any name, but the fields must be named x, y, and parameters.
Ordered nodes of the initial mesh.
Initial guess for the solution with solinit.y(:,i) a guess for the solution at the node solinit.x(i).
Optional. A vector that provides an initial guess for unknown parameters.
solinit = bvpinit(sol,[anew bnew]) forms an initial guess on the interval [anew bnew] from a solution sol on an interval [a,b]. The new interval must be larger than the previous one, so either anew <= a < b <= bnew or anew >= a > b >= bnew. The solution sol is extrapolated to the new interval. If sol contains parameters, they are copied to solinit.