Guaranteeing isoperimetry constraint for non-extremal functional in PDE.
$begingroup$
First of all, hello and thank you for your time.
Context
I am making a program that solves the differential equation for the time evolution of a system from the equations: $$F[mathbf{y}]=intlimits_{Omegasubsetmathbb{R}^n}f(mathbf{x},mathbf{y}, mathbf{nabla y})dmathbf{x}$$
$$frac{partial y_i}{partial t}=kDeltafrac{delta F}{delta y_i}$$
Where $Delta$ is the Laplacian and $$frac{delta F}{delta y_i}=sumlimits_kfrac{partial}{partial x_k}frac{partial f}{partial (partial_ky_i)}-frac{partial f}{partial y_i}$$ is the variational derivative ($partial _k$ is the partial derivative with respect to $x_k$). Also: $mathbf{x}inmathbb{R}^n, mathbf{y}inmathbb{R}^m$
The system should satisfy the global conservation constraints:
$$J_i[mathbf{y}]=intlimits_{Omegasubsetmathbb{R}^n}y_idmathbf{x}=k_i$$
Where the $k_i$ are constants.
I ran the program (without adding the Lagrange multipliers to the integral). And noticed that the $J_i$ increased with time, which was obviously not intended.
Question
I want to add the constraints to the solution. At first I naïvely thought that I could just modify the functional by adding lagrange mutipliers: $$K[mathbf{y}]=F[mathbf{y}]-sumlimits_ilambda_i J_i[mathbf{y}]$$
But when checking my reference book I noticed the Theorem said (I modified and omitted parts to take what's most relevant to the current question):
Suppose that $F$ has an extremum at $yin C^2[x_0, x_1]$ subject to the boundary conditions [...]. Then there exist two numbers $lambda_0, lambda_1$ not both zero such that $$frac{delta K}{delta y}=0$$
Where $K=lambda_0 F-lambda_1 J$
As the theorem says, this works when one wishes to find the extremum so my naïve assumption is probably wrong since the system I described only reaches the extremum of the functional when $mathbf{y}$ gets to the steady state (i.e. it approaches it asymptotically).
Is there a way to satisfy the constraint continuously throughout the time evolution of the system despite $F$ not being stationary?
I understand this may be more involved than what an answer in the site may allow so if you know of any good textbook where I could find it I would also be very grateful.
pde calculus-of-variations lagrange-multiplier euler-lagrange-equation variational-analysis
asked Jan 18 at 18:26
Salvador VillarrealSalvador Villarreal
1107
$endgroup$
add a comment |
$begingroup$
First of all, hello and thank you for your time.
Context
I am making a program that solves the differential equation for the time evolution of a system from the equations: $$F[mathbf{y}]=intlimits_{Omegasubsetmathbb{R}^n}f(mathbf{x},mathbf{y}, mathbf{nabla y})dmathbf{x}$$
$$frac{partial y_i}{partial t}=kDeltafrac{delta F}{delta y_i}$$
Where $Delta$ is the Laplacian and $$frac{delta F}{delta y_i}=sumlimits_kfrac{partial}{partial x_k}frac{partial f}{partial (partial_ky_i)}-frac{partial f}{partial y_i}$$ is the variational derivative ($partial _k$ is the partial derivative with respect to $x_k$). Also: $mathbf{x}inmathbb{R}^n, mathbf{y}inmathbb{R}^m$
The system should satisfy the global conservation constraints:
$$J_i[mathbf{y}]=intlimits_{Omegasubsetmathbb{R}^n}y_idmathbf{x}=k_i$$
Where the $k_i$ are constants.
I ran the program (without adding the Lagrange multipliers to the integral). And noticed that the $J_i$ increased with time, which was obviously not intended.
Question
I want to add the constraints to the solution. At first I naïvely thought that I could just modify the functional by adding lagrange mutipliers: $$K[mathbf{y}]=F[mathbf{y}]-sumlimits_ilambda_i J_i[mathbf{y}]$$
But when checking my reference book I noticed the Theorem said (I modified and omitted parts to take what's most relevant to the current question):
Suppose that $F$ has an extremum at $yin C^2[x_0, x_1]$ subject to the boundary conditions [...]. Then there exist two numbers $lambda_0, lambda_1$ not both zero such that $$frac{delta K}{delta y}=0$$
Where $K=lambda_0 F-lambda_1 J$
As the theorem says, this works when one wishes to find the extremum so my naïve assumption is probably wrong since the system I described only reaches the extremum of the functional when $mathbf{y}$ gets to the steady state (i.e. it approaches it asymptotically).
Is there a way to satisfy the constraint continuously throughout the time evolution of the system despite $F$ not being stationary?
I understand this may be more involved than what an answer in the site may allow so if you know of any good textbook where I could find it I would also be very grateful.
pde calculus-of-variations lagrange-multiplier euler-lagrange-equation variational-analysis
asked Jan 18 at 18:26
Salvador VillarrealSalvador Villarreal
1107
$endgroup$
add a comment |
$begingroup$
First of all, hello and thank you for your time.
Context
I am making a program that solves the differential equation for the time evolution of a system from the equations: $$F[mathbf{y}]=intlimits_{Omegasubsetmathbb{R}^n}f(mathbf{x},mathbf{y}, mathbf{nabla y})dmathbf{x}$$
$$frac{partial y_i}{partial t}=kDeltafrac{delta F}{delta y_i}$$
Where $Delta$ is the Laplacian and $$frac{delta F}{delta y_i}=sumlimits_kfrac{partial}{partial x_k}frac{partial f}{partial (partial_ky_i)}-frac{partial f}{partial y_i}$$ is the variational derivative ($partial _k$ is the partial derivative with respect to $x_k$). Also: $mathbf{x}inmathbb{R}^n, mathbf{y}inmathbb{R}^m$
The system should satisfy the global conservation constraints:
$$J_i[mathbf{y}]=intlimits_{Omegasubsetmathbb{R}^n}y_idmathbf{x}=k_i$$
Where the $k_i$ are constants.
I ran the program (without adding the Lagrange multipliers to the integral). And noticed that the $J_i$ increased with time, which was obviously not intended.
Question
I want to add the constraints to the solution. At first I naïvely thought that I could just modify the functional by adding lagrange mutipliers: $$K[mathbf{y}]=F[mathbf{y}]-sumlimits_ilambda_i J_i[mathbf{y}]$$
But when checking my reference book I noticed the Theorem said (I modified and omitted parts to take what's most relevant to the current question):
Suppose that $F$ has an extremum at $yin C^2[x_0, x_1]$ subject to the boundary conditions [...]. Then there exist two numbers $lambda_0, lambda_1$ not both zero such that $$frac{delta K}{delta y}=0$$
Where $K=lambda_0 F-lambda_1 J$
As the theorem says, this works when one wishes to find the extremum so my naïve assumption is probably wrong since the system I described only reaches the extremum of the functional when $mathbf{y}$ gets to the steady state (i.e. it approaches it asymptotically).
Is there a way to satisfy the constraint continuously throughout the time evolution of the system despite $F$ not being stationary?
I understand this may be more involved than what an answer in the site may allow so if you know of any good textbook where I could find it I would also be very grateful.
pde calculus-of-variations lagrange-multiplier euler-lagrange-equation variational-analysis
asked Jan 18 at 18:26
Salvador VillarrealSalvador Villarreal
1107
$endgroup$
First of all, hello and thank you for your time.
Context
I am making a program that solves the differential equation for the time evolution of a system from the equations: $$F[mathbf{y}]=intlimits_{Omegasubsetmathbb{R}^n}f(mathbf{x},mathbf{y}, mathbf{nabla y})dmathbf{x}$$
$$frac{partial y_i}{partial t}=kDeltafrac{delta F}{delta y_i}$$
Where $Delta$ is the Laplacian and $$frac{delta F}{delta y_i}=sumlimits_kfrac{partial}{partial x_k}frac{partial f}{partial (partial_ky_i)}-frac{partial f}{partial y_i}$$ is the variational derivative ($partial _k$ is the partial derivative with respect to $x_k$). Also: $mathbf{x}inmathbb{R}^n, mathbf{y}inmathbb{R}^m$
The system should satisfy the global conservation constraints:
$$J_i[mathbf{y}]=intlimits_{Omegasubsetmathbb{R}^n}y_idmathbf{x}=k_i$$
Where the $k_i$ are constants.
I ran the program (without adding the Lagrange multipliers to the integral). And noticed that the $J_i$ increased with time, which was obviously not intended.
Question
I want to add the constraints to the solution. At first I naïvely thought that I could just modify the functional by adding lagrange mutipliers: $$K[mathbf{y}]=F[mathbf{y}]-sumlimits_ilambda_i J_i[mathbf{y}]$$
But when checking my reference book I noticed the Theorem said (I modified and omitted parts to take what's most relevant to the current question):
Suppose that $F$ has an extremum at $yin C^2[x_0, x_1]$ subject to the boundary conditions [...]. Then there exist two numbers $lambda_0, lambda_1$ not both zero such that $$frac{delta K}{delta y}=0$$
Where $K=lambda_0 F-lambda_1 J$
As the theorem says, this works when one wishes to find the extremum so my naïve assumption is probably wrong since the system I described only reaches the extremum of the functional when $mathbf{y}$ gets to the steady state (i.e. it approaches it asymptotically).
Is there a way to satisfy the constraint continuously throughout the time evolution of the system despite $F$ not being stationary?
I understand this may be more involved than what an answer in the site may allow so if you know of any good textbook where I could find it I would also be very grateful.
pde calculus-of-variations lagrange-multiplier euler-lagrange-equation variational-analysis
pde calculus-of-variations lagrange-multiplier euler-lagrange-equation variational-analysis
asked Jan 18 at 18:26
Salvador VillarrealSalvador Villarreal
1107
asked Jan 18 at 18:26
Salvador VillarrealSalvador Villarreal
1107
edited Jan 21 at 16:53
Salvador Villarreal
asked Jan 18 at 18:26
Salvador VillarrealSalvador Villarreal
1107
asked Jan 18 at 18:26
Salvador VillarrealSalvador Villarreal
1107
asked Jan 18 at 18:26
Salvador VillarrealSalvador Villarreal
1107
1107
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