Classroom Tips and Techniques: Numeric Solution of a Two-Point BVP
Robert J. Lopez Emeritus Professor of Mathematics and Maple Fellow Maplesoft
Introduction
A recent query from a colleague in engineering posed a nonlinear two-point boundary value problem that could only be solved numerically. In this month's article, we'll explore the solution of a similar problem, highlighting the way Maple can be used to obtain the required solution.
Initializations
Statement of the BVP
Consider the nonlinear ODE
and the related nonlinear boundary conditions
that pose a two-point boundary value problem on the interval
Using dsolve to Obtain a Solution
A naive use of Maple's dsolve command did not provide a solution to the problem originally posed, running at great length and then crashing. However, inclusion of the approxsoln parameter very quickly led to the desired solution. This parameter provides an initial guess for an iterative Rayleigh-Ritz solution process that converges very quickly to a solution. In the original engineering problem there were two possible solutions, only one of which was physically meaningful.
If we apply that same strategy to the problem posed in the previous section, assuming that perhaps a solution might be close to , we obtain the graph shown in Figure 1.
Figure 1 Solution of nonlinear two-point BVP starting with initial guess
In principle, it should be possible to solve this BVP via a shooting method, using the two boundary conditions as the governing equations that determine and . In the sequel, we apply this idea to the solution process, and also come to grips with another disturbing facet of the problem we selected, namely, finding all possible solutions.
A Solution from First Principles
We begin by defining a procedure that, given values for , returns the list This procedure implements a shooting method, starting from the initial values of and , numerically integrating a solution for the resulting initial value problem, and returning the endpoint values.
In terms of and , the boundary conditions can be expressed via the expressions
It is now possible to graph the functions and , resulting in Figure 2.
Figure 2 Graph of the equations for and , as determined by the boundary conditions of the BVP
Figure 2 suggests that there are actually four solutions of the given BVP.
Unfortunately, the fsolve command in Maple 11 does not permit systems of procedure calls, as would be required by our definitions of and through the procedure Fortunately, this shortcoming is corrected in the next version of Maple. Meanwhile, we need to solve a pair of nonlinear algebraic equations whose terms are calculated via the solution of an initial value problem.
We propose three methods for this. First, we implement Broyden's method as a generalization of the Secant algorithm for solving a single equation. Second, we apply a minimization technique to , a sum of squares. Finally, because of the special nature of the first boundary condition, we can eliminate from the second equation, and apply the Secant method to the single resulting equation.
In anticipation of all three iterative techniques, we define the following initial points, each taken from Figure 2.
Broyden's Method
Newton's method for solving the system of equations is captured by the fixed-point scheme
where is the Jacobian matrix for F. Since this requires computation of derivatives, it would be tedious to implement a method for repeatedly computing . As an alternative, we consider Broyden's method wherein is only computed once. Thereafter, an approximation to is computed, an approximation that converges to where is a solution to the system
After obtaining either analytically or approximately by numeric differentiation, Broyden's method thereafter replaces with
as an approximation to
Our implementation of Broyden's method takes the vector as impute, along with an initial point, a stepsize used for numerically computing , and the integer used to define the computational tolerance . Our code is simplistic, in that it does not type-checking, and does not make provision for generalization to other systems. It is provided as a sketch to show how such a computation might be implemented.
Using our Broyden procedure, we compute solutions for and .
With pairs determined, we can solve the corresponding four initial value problems, and graph their solutions. The following commands generate the solutions and graphs,
and the graphs themselves appear in Figure 3.
Figure 3 Graphs of four possible solutions generated as solutions of initial value problems from pairs computed by Broyden's method
The four solutions found in Figure 3 can be obtained directly with Maple's dsolve command, provided an appropriate approximate solution is provided. If we list the four approximate solutions
then the following loop computes the corresponding dsolve solutions.
Figure 4 displays these solutions.
Figure 4 Solutions obtained by dsolve using the approxsoln parameter
The graphs in Figures 3 and 4 are essentially the same.
Minimizing a Sum of Squares
A pair that minimizes the sum of squares
is a solution to the equations . We can compute such minimizing pairs with the NLPSolve command in Maple's Optimization package. This command provides the nonlinear simplex (Nelder-Mead) method, which does not require computation of derivatives. The computed pairs and the values of at the minima are given by the following loop.
The first number returned in each solution is the value of at the computed extremum.
To facilitate a comparison with the values computed by Broyden's method, we express the nonlinear simplex solutions as the vectors
The Euclidean norms of the differences between the two solutions are given by
The agreement is substantial.
Secant Method
If the first boundary condition is solve for , we obtain
with which can be eliminated from the second boundary condition. Thus, we have the single equation
to solve. Figure 5 provides a graph of the left-hand side of this equation.
Figure 5 Graph of
There are four intercepts for the graph in Figure 5, consistent with our earlier findings on the number of solutions of the nonlinear BVP. The equation can be solved with the Secant method, which we have coded as the following procedure. Our procedure includes a calculation of , and returns both and .
An application of the Secant method leads to the following four solutions.
As we did earlier, we compare these solutions with those computed by Broyden's method.
The solutions appear to be identical.
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