Shooting Methods for ODE's

Finding the solution of a linear boundary problem is assisted by the linear structure of the equation and the use of two special initial value problems. Suppose that u(t) is the unique solution to the I.V.P.

with

.

Furthermore, suppose that v(t) is the unique solution to the I.V.P.

with

.

Then the linear combination

.

is a solution to

with

Program (Linear Shooting Method):

To approximate the solution of the boundary value problem

with

over the interval [a,b] by using the Runge-Kutta method of order n=4.

The method involves solving a two systems of equations over . First solve

with ,
and .

Then solve

with ,
and .

Finally, the desired solution x(t) is the linear combination

.

The subroutine Runge2D will be used to construct the two solutions , and .

Comments

Runge-Kutta-Fehlberg Method

One way to guarantee accuracy in the solution of an I.V.P. is to solve the problem twice using step sizes h and and compare answers at the mesh points corresponding to the larger step size. But this requires a significant amount of computation for the smaller step size and must be repeated if it is determined that the agreement is not good enough. The Runge-Kutta-Fehlberg method (denoted RKF45) is one way to try to resolve this problem. It has a procedure to determine if the proper step size h is being used. At each step, two different approximations for the solution are made and compared. If the two answers are in close agreement, the approximation is accepted. If the two answers do not agree to a specified accuracy, the step size is reduced. If the answers agree to more significant digits than required, the step size is increased. Each Runge-Kutta-Fehlberg step requires the use of the following six values: Then an approximation to the solution of the I.V.P....

Van Der Pol System

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Newton-Raphson Method

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Powell's Method

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Numerical Analysis

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