Systems of BVP with unbalanced BCs
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I have a system of ODEs forming a BVP. I was thinking of using Finite Difference to solve it but for some functions I do not have the BCs on both sides (though in total they are enough to make the problem well stated). I can easily solve the problem with bvp4c in Matlab but I was wondering how to set it up for the Finite Difference method considering indeed that some BCs on the left side are missing.
I thought of shooting but apaprently it can be used only if there is a side where only one BC is missing.
Thank you
The system is:
begin{align}
frac{dx(z)}{dz} &= L(z)cos(theta(z)) \
frac{dy(z)}{dz} &= L(z)sin(theta(z)) \
frac{dtheta(z)}{dz} &= L(z) K(z) \
frac{dK(z)}{dz} &= frac{4 L(z) M(z) U(z)}{3pi} - frac{L(z) K(z) N_3(z)}{3pi} \
frac{dU(z)}{dz} &= frac{L(z) N_3(z) U(z)}{3pi} \
frac{dN_1(z)}{dz} &= - L(z) K(z) N_3(z) + frac{grho a_0^2pi L(z)cos(theta(z))}{eta U_0 U(z)} \
frac{dN_3(z)}{dz} &= L(z) K(z) N_1(z) + frac{grho a_0^2 pi L(z)sin(theta(z))}{eta U_0 U(z)} \
frac{dM(z)}{dz} &= - L(z) N_1(z) + frac{grho a_0^2 pi L(z) K(z)sin(theta(z))} {4eta U_0 U(z)^2} \
frac{dL(z)}{dz} &= 0
end{align}
and the BC are:
$$
begin{aligned}
x(0) &= 0 \
y(0) &= 0 \
theta(0) &= -pi/2 \
U(0) &= 1
end{aligned} quad
begin{aligned}
x(1) &= 10 \
y(1) &= 0 \
theta(1) &= 0 \
U(1) &= 1 \
K(1) &= 0
end{aligned}
$$
boundary-value-problem
edited Dec 6 '18 at 9:00
Dylan
12.9k31027
asked Nov 30 '18 at 12:31
acalore88acalore88
11
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add a comment |
$begingroup$
I have a system of ODEs forming a BVP. I was thinking of using Finite Difference to solve it but for some functions I do not have the BCs on both sides (though in total they are enough to make the problem well stated). I can easily solve the problem with bvp4c in Matlab but I was wondering how to set it up for the Finite Difference method considering indeed that some BCs on the left side are missing.
I thought of shooting but apaprently it can be used only if there is a side where only one BC is missing.
Thank you
The system is:
begin{align}
frac{dx(z)}{dz} &= L(z)cos(theta(z)) \
frac{dy(z)}{dz} &= L(z)sin(theta(z)) \
frac{dtheta(z)}{dz} &= L(z) K(z) \
frac{dK(z)}{dz} &= frac{4 L(z) M(z) U(z)}{3pi} - frac{L(z) K(z) N_3(z)}{3pi} \
frac{dU(z)}{dz} &= frac{L(z) N_3(z) U(z)}{3pi} \
frac{dN_1(z)}{dz} &= - L(z) K(z) N_3(z) + frac{grho a_0^2pi L(z)cos(theta(z))}{eta U_0 U(z)} \
frac{dN_3(z)}{dz} &= L(z) K(z) N_1(z) + frac{grho a_0^2 pi L(z)sin(theta(z))}{eta U_0 U(z)} \
frac{dM(z)}{dz} &= - L(z) N_1(z) + frac{grho a_0^2 pi L(z) K(z)sin(theta(z))} {4eta U_0 U(z)^2} \
frac{dL(z)}{dz} &= 0
end{align}
and the BC are:
$$
begin{aligned}
x(0) &= 0 \
y(0) &= 0 \
theta(0) &= -pi/2 \
U(0) &= 1
end{aligned} quad
begin{aligned}
x(1) &= 10 \
y(1) &= 0 \
theta(1) &= 0 \
U(1) &= 1 \
K(1) &= 0
end{aligned}
$$
boundary-value-problem
edited Dec 6 '18 at 9:00
Dylan
12.9k31027
asked Nov 30 '18 at 12:31
acalore88acalore88
11
$endgroup$
add a comment |
$begingroup$
I have a system of ODEs forming a BVP. I was thinking of using Finite Difference to solve it but for some functions I do not have the BCs on both sides (though in total they are enough to make the problem well stated). I can easily solve the problem with bvp4c in Matlab but I was wondering how to set it up for the Finite Difference method considering indeed that some BCs on the left side are missing.
I thought of shooting but apaprently it can be used only if there is a side where only one BC is missing.
Thank you
The system is:
begin{align}
frac{dx(z)}{dz} &= L(z)cos(theta(z)) \
frac{dy(z)}{dz} &= L(z)sin(theta(z)) \
frac{dtheta(z)}{dz} &= L(z) K(z) \
frac{dK(z)}{dz} &= frac{4 L(z) M(z) U(z)}{3pi} - frac{L(z) K(z) N_3(z)}{3pi} \
frac{dU(z)}{dz} &= frac{L(z) N_3(z) U(z)}{3pi} \
frac{dN_1(z)}{dz} &= - L(z) K(z) N_3(z) + frac{grho a_0^2pi L(z)cos(theta(z))}{eta U_0 U(z)} \
frac{dN_3(z)}{dz} &= L(z) K(z) N_1(z) + frac{grho a_0^2 pi L(z)sin(theta(z))}{eta U_0 U(z)} \
frac{dM(z)}{dz} &= - L(z) N_1(z) + frac{grho a_0^2 pi L(z) K(z)sin(theta(z))} {4eta U_0 U(z)^2} \
frac{dL(z)}{dz} &= 0
end{align}
and the BC are:
$$
begin{aligned}
x(0) &= 0 \
y(0) &= 0 \
theta(0) &= -pi/2 \
U(0) &= 1
end{aligned} quad
begin{aligned}
x(1) &= 10 \
y(1) &= 0 \
theta(1) &= 0 \
U(1) &= 1 \
K(1) &= 0
end{aligned}
$$
boundary-value-problem
edited Dec 6 '18 at 9:00
Dylan
12.9k31027
asked Nov 30 '18 at 12:31
acalore88acalore88
11
$endgroup$
I have a system of ODEs forming a BVP. I was thinking of using Finite Difference to solve it but for some functions I do not have the BCs on both sides (though in total they are enough to make the problem well stated). I can easily solve the problem with bvp4c in Matlab but I was wondering how to set it up for the Finite Difference method considering indeed that some BCs on the left side are missing.
I thought of shooting but apaprently it can be used only if there is a side where only one BC is missing.
Thank you
The system is:
begin{align}
frac{dx(z)}{dz} &= L(z)cos(theta(z)) \
frac{dy(z)}{dz} &= L(z)sin(theta(z)) \
frac{dtheta(z)}{dz} &= L(z) K(z) \
frac{dK(z)}{dz} &= frac{4 L(z) M(z) U(z)}{3pi} - frac{L(z) K(z) N_3(z)}{3pi} \
frac{dU(z)}{dz} &= frac{L(z) N_3(z) U(z)}{3pi} \
frac{dN_1(z)}{dz} &= - L(z) K(z) N_3(z) + frac{grho a_0^2pi L(z)cos(theta(z))}{eta U_0 U(z)} \
frac{dN_3(z)}{dz} &= L(z) K(z) N_1(z) + frac{grho a_0^2 pi L(z)sin(theta(z))}{eta U_0 U(z)} \
frac{dM(z)}{dz} &= - L(z) N_1(z) + frac{grho a_0^2 pi L(z) K(z)sin(theta(z))} {4eta U_0 U(z)^2} \
frac{dL(z)}{dz} &= 0
end{align}
and the BC are:
$$
begin{aligned}
x(0) &= 0 \
y(0) &= 0 \
theta(0) &= -pi/2 \
U(0) &= 1
end{aligned} quad
begin{aligned}
x(1) &= 10 \
y(1) &= 0 \
theta(1) &= 0 \
U(1) &= 1 \
K(1) &= 0
end{aligned}
$$
boundary-value-problem
boundary-value-problem
edited Dec 6 '18 at 9:00
Dylan
12.9k31027
asked Nov 30 '18 at 12:31
acalore88acalore88
11
edited Dec 6 '18 at 9:00
Dylan
12.9k31027
asked Nov 30 '18 at 12:31
acalore88acalore88
11
edited Dec 6 '18 at 9:00
Dylan
12.9k31027
edited Dec 6 '18 at 9:00
Dylan
12.9k31027
edited Dec 6 '18 at 9:00
Dylan
12.9k31027
12.9k31027
asked Nov 30 '18 at 12:31
acalore88acalore88
11
asked Nov 30 '18 at 12:31
acalore88acalore88
11
asked Nov 30 '18 at 12:31
acalore88acalore88
11
11
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