Shooting Method for Boundary Value Problems
This module enables the user to compare different approximate
solutions computed using the shooting method for boundary value
problems for ordinary differential equations. A general boundary value
problem (BVP) consists of an ordinary differential equation
(ODE) with side conditions specified at more than one point.
This module illustrates the solution of second-order scalar ODEs of the
form u″ = f(t, u,
u′) on an interval [a,
b] with boundary conditions u(a) =
α and u(b) =
β. The shooting method replaces the given
BVP by a sequence of initial value problems (IVPs) for the same ODE
with initial conditions u(a) =
α and u′(a) =
x, where x is a guessed initial slope that is
successively refined until the desired boundary condition at b
is satisfied. Let u(b; x) be the
value at b of the solution produced by a given IVP solver for
initial conditions u(a) = α and
u′(a) = x. The shooting method
employs an iterative method for solving nonlinear equations to find an
initial slope x* such that u(b; x*) =
β. The solution to this final IVP then coincides
with the solution to the original BVP.
The user begins by selecting from the menu provided an ODE and a
specific solution to be sought (if there is more than one). Boundary
values u(a) = α and
u(b) = β are indicated by black
dots on the graph. Next the user specifies the order of the
Runge-Kutta method to be used as the IVP solver and the number of steps
to take in traversing the interval [a, b].
When the user clicks Solve the resulting approximate solution to
the BVP is drawn on the graph. To compare this solution with other
approximate solutions, the user can make additional choices of method
order and number of steps, then click Solve to add each new
approximate solution to the graph. The solutions for different
parameters are color coded so that the color changes from blue to red
as the number of steps increases and from light to dark as the method
order increases. To clear all of the solutions from the graph, click
Reset.
To illustrate the details of the individual steps of the shooting
method for a particular choice of parameters, see the alternative
Shooting Method module.
Reference: Michael T. Heath, Scientific Computing,
An Introductory Survey, 2nd edition, McGraw-Hill, New York,
2002. See Section 10.3.
Developers: Evan VanderZee and Michael Heath
Department of Computer Science