Defining a Jacobian Matrix
reaching out my post. I have a trouble of defining a Jacobian matrix for my problem. Basically, I have 4 differential equations to be solved.
$dot x_1(t)=x_2(t),$
$dot x_2(t)=p_2(t)−sqrt 2 x_1(t)e^{-αt},$
$dot p_1(t)=sqrt 2p_2(t)e^{-αt}+x_1(t)$
$dot p_2(t)=−p_1(t)$
with initial and boundary values of: $x_1(0)=1, p_2(0)=0,p_1(1)=0,p_2(1)=0$
Following classic shooting method strategy, I suggest some values for $x_2(0), p_1(0)$, solve the system of diff equations using Dormand Prince method, and obtain some values at the boundary. Here I check, if the solution on my boundaries meets the criteria of $p_1(1)=0,p_2(1)=0$. If not, I have to suggest new values for initial $x_2(0), p_1(0)$, to be iterated again. Since this is two-parametric shooting, I can't use simple linear interpolation as in one-parametric shooting, so I have to use Newton's method for that. Here is my question: How to implement Newton's method to iterate towards the right solution? I tried myself, and doesn't work. Can you at least help me with defining a Jacobian, for the given problem of iterative solution to $x_2(0), p_1(0)$
numerical-methods boundary-value-problem numerical-optimization initial-value-problems newton-raphson
asked Nov 20 at 11:44
Farid Hasanov
13
add a comment |
reaching out my post. I have a trouble of defining a Jacobian matrix for my problem. Basically, I have 4 differential equations to be solved.
$dot x_1(t)=x_2(t),$
$dot x_2(t)=p_2(t)−sqrt 2 x_1(t)e^{-αt},$
$dot p_1(t)=sqrt 2p_2(t)e^{-αt}+x_1(t)$
$dot p_2(t)=−p_1(t)$
with initial and boundary values of: $x_1(0)=1, p_2(0)=0,p_1(1)=0,p_2(1)=0$
Following classic shooting method strategy, I suggest some values for $x_2(0), p_1(0)$, solve the system of diff equations using Dormand Prince method, and obtain some values at the boundary. Here I check, if the solution on my boundaries meets the criteria of $p_1(1)=0,p_2(1)=0$. If not, I have to suggest new values for initial $x_2(0), p_1(0)$, to be iterated again. Since this is two-parametric shooting, I can't use simple linear interpolation as in one-parametric shooting, so I have to use Newton's method for that. Here is my question: How to implement Newton's method to iterate towards the right solution? I tried myself, and doesn't work. Can you at least help me with defining a Jacobian, for the given problem of iterative solution to $x_2(0), p_1(0)$
numerical-methods boundary-value-problem numerical-optimization initial-value-problems newton-raphson
asked Nov 20 at 11:44
Farid Hasanov
13
add a comment |
reaching out my post. I have a trouble of defining a Jacobian matrix for my problem. Basically, I have 4 differential equations to be solved.
$dot x_1(t)=x_2(t),$
$dot x_2(t)=p_2(t)−sqrt 2 x_1(t)e^{-αt},$
$dot p_1(t)=sqrt 2p_2(t)e^{-αt}+x_1(t)$
$dot p_2(t)=−p_1(t)$
with initial and boundary values of: $x_1(0)=1, p_2(0)=0,p_1(1)=0,p_2(1)=0$
Following classic shooting method strategy, I suggest some values for $x_2(0), p_1(0)$, solve the system of diff equations using Dormand Prince method, and obtain some values at the boundary. Here I check, if the solution on my boundaries meets the criteria of $p_1(1)=0,p_2(1)=0$. If not, I have to suggest new values for initial $x_2(0), p_1(0)$, to be iterated again. Since this is two-parametric shooting, I can't use simple linear interpolation as in one-parametric shooting, so I have to use Newton's method for that. Here is my question: How to implement Newton's method to iterate towards the right solution? I tried myself, and doesn't work. Can you at least help me with defining a Jacobian, for the given problem of iterative solution to $x_2(0), p_1(0)$
numerical-methods boundary-value-problem numerical-optimization initial-value-problems newton-raphson
asked Nov 20 at 11:44
Farid Hasanov
13
reaching out my post. I have a trouble of defining a Jacobian matrix for my problem. Basically, I have 4 differential equations to be solved.
$dot x_1(t)=x_2(t),$
$dot x_2(t)=p_2(t)−sqrt 2 x_1(t)e^{-αt},$
$dot p_1(t)=sqrt 2p_2(t)e^{-αt}+x_1(t)$
$dot p_2(t)=−p_1(t)$
with initial and boundary values of: $x_1(0)=1, p_2(0)=0,p_1(1)=0,p_2(1)=0$
Following classic shooting method strategy, I suggest some values for $x_2(0), p_1(0)$, solve the system of diff equations using Dormand Prince method, and obtain some values at the boundary. Here I check, if the solution on my boundaries meets the criteria of $p_1(1)=0,p_2(1)=0$. If not, I have to suggest new values for initial $x_2(0), p_1(0)$, to be iterated again. Since this is two-parametric shooting, I can't use simple linear interpolation as in one-parametric shooting, so I have to use Newton's method for that. Here is my question: How to implement Newton's method to iterate towards the right solution? I tried myself, and doesn't work. Can you at least help me with defining a Jacobian, for the given problem of iterative solution to $x_2(0), p_1(0)$
numerical-methods boundary-value-problem numerical-optimization initial-value-problems newton-raphson
numerical-methods boundary-value-problem numerical-optimization initial-value-problems newton-raphson
asked Nov 20 at 11:44
Farid Hasanov
13
asked Nov 20 at 11:44
Farid Hasanov
13
edited Nov 20 at 12:40
asked Nov 20 at 11:44
Farid Hasanov
13
asked Nov 20 at 11:44
Farid Hasanov
13
asked Nov 20 at 11:44
Farid Hasanov
13
13
add a comment |
add a comment |
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