A note on the variational formulation for quasi-linear initial-value problems

Geffen, Nima; Yaniv, Sara

doi:10.1007/bf01595133

Cited by 3 publications

(6 citation statements)

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“…(7)) in the present analysis turns out as the vector potential of the solenoidal field A, or the stream function for the two-dimensional case [6], and hence is of interest in itself.…”

Section: Theoremmentioning

confidence: 99%

“…(4)) can account for appropriate initial conditions for time-dependent problems (e.g. the nonlinear wave equation [6]). Since the vector-field u need not be irrotational (Eq.…”

Section: Theoremmentioning

confidence: 99%

“…The two-dimensional nonlinear case was worked out in [6] with the aid of a Lagrange multiplier. It is extended here to three independent variables.…”

Section: Introductionmentioning

confidence: 99%

“…diffusion and wave equations) the Dirichlet problem is improperly posed, and different conditions on parts of the boundary only have to be prescribed. Variational formulations for initial (rather than boundary) value problems have been proposed for linear parabolic and hyperbolic equations [1][2][3][4][5].The two-dimensional nonlinear case was worked out in [6] with the aid of a Lagrange multiplier. It is extended here to three independent variables.…”

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confidence: 99%

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A variational formulation for constrained quasilinear vector systems

Geffen¹

1977

Quart. Appl. Math.

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Abstract.A variational formulation for multi-dimensional initial-and/or boundaryvalue problems for a system of quasilinear conservation equations with a rotationality condition in a vector form with the aid of a vector Lagrange multiplier is given. The duality between the physical and 'phase' (or hodograph) spaces emerges, and the Lagrange multiplier turns out to be the vector potential for the conserved field, and hence of some interest in itself. Application is given to a family of transonic flows in the physical and hodograph planes, and to a problem in nonlinear sound propagation. Introduction.Variational formulations equivalent to boundary-value problems for ordinary and elliptic partial differential equations have been of common use in particle and continuum mechanics and other branches of physics. For non-elliptic equations (e.g. diffusion and wave equations) the Dirichlet problem is improperly posed, and different conditions on parts of the boundary only have to be prescribed. Variational formulations for initial (rather than boundary) value problems have been proposed for linear parabolic and hyperbolic equations [1][2][3][4][5].The two-dimensional nonlinear case was worked out in [6] with the aid of a Lagrange multiplier. It is extended here to three independent variables. The three-dimensional case is presented in vector notation which brings out the geometric and physical meaning of the various terms and the vector Lagrange multiplier, and is invariant under coordinate transformations. For more than three independent space variables, the constraints and

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“…(7)) in the present analysis turns out as the vector potential of the solenoidal field A, or the stream function for the two-dimensional case [6], and hence is of interest in itself.…”

Section: Theoremmentioning

confidence: 99%

“…(4)) can account for appropriate initial conditions for time-dependent problems (e.g. the nonlinear wave equation [6]). Since the vector-field u need not be irrotational (Eq.…”

Section: Theoremmentioning

confidence: 99%

“…The two-dimensional nonlinear case was worked out in [6] with the aid of a Lagrange multiplier. It is extended here to three independent variables.…”

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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A variational formulation for constrained quasilinear vector systems

Geffen¹

1977

Quart. Appl. Math.

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“…Recently, Geffen[17] has derived a similar variational principal in terms of the velocities u = ~x and v = ~y. From(17), the Euler-Lagrange equation(16) can easily be obtained.…”

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confidence: 99%

Applications of a variational method for mixed differential equations

Khosla

Rubin

1981

J Eng Math

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Variational principles for elliptic boundary-value problems as well as linear initial-value problems have been derived by various investigators. For initial-value problems Tonti and Reddy have used a convolution type of bilinear form of the functional for the time-like coordinate. This introduces a certain amount of directionality thereby reflecting the initial-value nature of tire problem. In the present investigation the methods of Tonti and Reddy are used to derive the appropriate variational formulation for the transonic flow problem. A number of linear and non-linear examples have been investigated. As a test for the existence of directionality, finite-differences are used to discretize the variational integral. For initial-value problems of wave equation and diffusion equation type, fully implicit finite-difference approximations are recovered. The small-disturbance transonic equation leads to the Murman and Cole differencing theory ; when applied to the full potentialflow equations, the rotated difference scheme due to Jameson is obtained.

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Variational formulations for nonlinear wave propagation and unsteady transonic flow

Geffen

1977

Journal of Applied Mathematics and Physics (ZAMP)

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A note on the variational formulation for quasi-linear initial-value problems

Cited by 3 publications

References 3 publications

A variational formulation for constrained quasilinear vector systems

A variational formulation for constrained quasilinear vector systems

Applications of a variational method for mixed differential equations

Variational formulations for nonlinear wave propagation and unsteady transonic flow

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