- #1
radiogaga35
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Given the boundary value problem (primes denote differentiation w.r.t x):
[tex]\begin{array}{l}
y'' = f(x,y,y') \\
y(a) = \alpha \\
y(b) = \beta \\
\end{array}[/tex]
the nonlinear shooting method may be implemented to solve the problem. A bisection algorithm may be used or, with a little more effort, Newton's method may be implemented (in which case one solves a fourth order, and not a second order, IVP - http://www.math.utah.edu/~pa/6620/shoot.pdf" ).
But what happens if the form of the boundary conditions changes to:
[tex]\begin{array}{l}
y'(a) = \alpha \\
y(b) = \beta \\
\end{array}[/tex]
Is one still justified in reducing the BVP to a 2nd order IVP, but this time with initial SLOPE fixed at [tex]y'(a) = \alpha [/tex] and then trying different values of [tex]y(a)[/tex] in order to achieve the condition [tex]y(b) = \beta [/tex]? (As opposed to varying initial slope to achieve second condition).
Furthermore, if this will indeed work, then a bisection method should be easy to implement, but what about adapting Newton's method for this case? Can anyone point me to an appropriate reference that discusses this matter?
Thank you! :-)
[tex]\begin{array}{l}
y'' = f(x,y,y') \\
y(a) = \alpha \\
y(b) = \beta \\
\end{array}[/tex]
the nonlinear shooting method may be implemented to solve the problem. A bisection algorithm may be used or, with a little more effort, Newton's method may be implemented (in which case one solves a fourth order, and not a second order, IVP - http://www.math.utah.edu/~pa/6620/shoot.pdf" ).
But what happens if the form of the boundary conditions changes to:
[tex]\begin{array}{l}
y'(a) = \alpha \\
y(b) = \beta \\
\end{array}[/tex]
Is one still justified in reducing the BVP to a 2nd order IVP, but this time with initial SLOPE fixed at [tex]y'(a) = \alpha [/tex] and then trying different values of [tex]y(a)[/tex] in order to achieve the condition [tex]y(b) = \beta [/tex]? (As opposed to varying initial slope to achieve second condition).
Furthermore, if this will indeed work, then a bisection method should be easy to implement, but what about adapting Newton's method for this case? Can anyone point me to an appropriate reference that discusses this matter?
Thank you! :-)
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