scholarly journals A variational approach to the shock structure problem

2005 ◽  
Vol 32 (1) ◽  
pp. 39-63
Author(s):  
Srboljub Simic
Keyword(s):  
Variational Approach ◽  
Least Squares Method ◽  
Direct Method ◽  
General Procedure ◽  
Ritz Method ◽  
Shock Structure ◽  
Approximate Analytical Solutions ◽  
Variational Form ◽  
Structure Problem ◽  
Shock Thickness

A variational approach to the shock structure problem is proposed. The set of governing equations, consisted of n first-order ordinary differential equations accompanied with 2n boundary conditions at ??, is put into variational form by means of least-squares method. The corresponding variational principle is adjusted for application of Ritz method. This direct method is used for construction of approximate analytical solutions to the shock structure problem and derivation of the estimates for the shock thickness. General procedure is applied to the study of Burgers' equation and equations of gas dynamics.

A modified tangent-gas approximation for two-dimensional steady flow

1958 ◽  
Vol 4 (6) ◽  
pp. 600-606 ◽  
Author(s):  
G. Power ◽  
P. Smith
Keyword(s):  
Least Squares ◽  
Steady Flow ◽  
Variational Approach ◽  
Least Squares Method ◽  
Two Dimensional ◽  
Subsonic Flows ◽  
Von Karman ◽  
Straight Line ◽  
Volume Relationship ◽  
Pressure Volume Relationship

A set of two-dimensional subsonic flows past certain cylinders is obtained using hodograph methods, in which the true pressure-volume relationship is replaced by various straight-line approximations. It is found that the approximation obtained by a least-squares method possibly gives best results. Comparison is made with values obtained by using the von Kármán-Tsien approximation and also with results obtained by the variational approach of Lush & Cherry (1956).

scholarly journals Theoretical and experimental aspects of the shock structure problem

1966 ◽  
pp. 973-979 ◽  
Author(s):  
H. W. Liepmann ◽  
R. Narasimha ◽  
M. Chahine
Keyword(s):  
Shock Structure ◽  
Structure Problem

APPLICATION DIRECT METHOD CALCULUS OF VARIATION FOR KLEIN-GORDON FIELD

2015 ◽  
Vol 77 (23) ◽  
Author(s):  
Saiman Saiman ◽  
Rinto Agustino ◽  
Hamdani Hamdani
Keyword(s):  
Hydrogen Atom ◽  
Functional Form ◽  
Gordon Equation ◽  
Direct Method ◽  
Ritz Method ◽  
Field Equations ◽  
De Broglie Waves ◽  
Klein Gordon ◽  
Quantum Wave ◽  
Quantum Wave Equation

Klein-Gordon field is often used to study the dynamics of elementary particles. The Klein–Gordon equation was first considered as a quantum wave equation by Schrödinger in his search for an equation describing de Broglie waves. The equation was found in his notebooks from late 1925, and he appears to have prepared a manuscript applying it to the hydrogen atom. Yet, because it fails to take into account the electron's spin, the equation failed to predict the fine structure of the hydrogen atom, and overestimated the overall magnitude of the splitting pattern energy. This paper will describe in detail using the Direct Method of Calculus Variation as an alternative to solve the Klien-Gordon field equations. The Direct Method simplified the calculation because the variables are calculated and expressed in functional form of energy. The result of the calculation of Klien-Gordon Feld provided the existence of the minimizer, i.e.  with  and . Explicit form of the minimizer was calculated by the Ritz method through rows of convergent density

Kinetic models and the shock structure problem

1975 ◽  
Vol 8 (4) ◽  
pp. 620-625 ◽  
Author(s):  
V. I. Zhuk ◽  
V. A. Rykov ◽  
E. M. Shakhov
Keyword(s):  
Kinetic Models ◽  
Shock Structure ◽  
Structure Problem

scholarly journals Free Vibrations of a Trapezoidal Plate with an Internal Line Hinge

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
María Virginia Quintana ◽  
Ricardo Oscar Grossi
Keyword(s):  
Plate Theory ◽  
Ritz Method ◽  
Free Vibrations ◽  
Rectangular Domain ◽  
Internal Line ◽  
Thin Plates ◽  
Mode Shapes ◽  
Approximate Analytical Solutions ◽  
Elastically Restrained Edges ◽  
Elastically Restrained

This paper deals with a general variational formulation for the determination of natural frequencies and mode shapes of free vibrations of laminated thin plates of trapezoidal shape with an internal line hinge restrained against rotation. The analysis was carried out by using the kinematics corresponding to the classical laminated plate theory (CLPT). The eigenvalue problem is obtained by employing a combination of the Ritz method and the Lagrange multipliers method. The domain of the plate is transformed into a rectangular domain in the computational space by using nonorthogonal triangular coordinates and the transverse displacements are approximated with a set of simple polynomials automatically generated and expressed in the triangular coordinates. The developed algorithm allows obtaining approximate analytical solutions for mentioned plate with different geometries, aspect ratio, position of the line hinge, and boundary conditions including translational and rotational elastically restrained edges. It allows studying the influence of the mentioned line on the vibration frequencies and respective mode shapes. The algorithm can easily be programmed and it is numerically stable. Additionally, as a particular case, the results of triangular plates can be easily generated.

ANALYTICAL SOLUTION FOR NONLINEAR INTERACTION OF EULER BEAM RESTING ON A TENSIONLESS SOIL

2021 ◽  
Vol 8 (1) ◽  
Author(s):  
Tamer HMA Kasem ◽  
Mohamed El-Shabrawy
Keyword(s):  
Exact Solutions ◽  
Analytical Solutions ◽  
Nonlinear Interaction ◽  
Ritz Method ◽  
Bending Moment ◽  
Approximate Solutions ◽  
Compatibility Conditions ◽  
Euler Beam ◽  
Approximate Analytical Solutions ◽  
Lift Off

The nonlinear interaction between an elastic Euler beam and a tensionless soil foundation is studied. Exact analytical solutions of the challenging problem are rather complicated. The basic obstacle is imposing compatibility conditions at lift-off points. These points are determined as a part of the solution although being needed to get the solution itself. In the current work, solutions are derived using the approximate Rayleigh-Ritz method. The principal of vanishing variation of potential energy is adopted. The solution is approximated using a set of suitable trial functions. Lift-off points are identified through an iterative procedure and compatibility conditions are satisfied implicitly. Results are presented for various cases, including clamped support and free end condition. Various distributed loading conditions are analyzed. Exact solutions are revised briefly. Accurate high order approximate analytical solutions are obtained using MAXIMA symbolic manipulator. The convergence of approximate solutions towards the exact solutions is verified. For each case detailed results of deflection, bending moment and shear are presented.

Propagation of Cylindrically Symmetric Gaussian Beams in a Higher-Order Nonlinear Medium

1997 ◽  
Vol 06 (02) ◽  
pp. 209-234 ◽  
Author(s):  
Zlatko Jovanoski ◽  
Rowland A. Sammut
Keyword(s):  
Gaussian Beam ◽  
Analytical Solutions ◽  
Nonlinear Medium ◽  
Variational Approach ◽  
Beam Propagation ◽  
Propagation Distance ◽  
Gaussian Beams ◽  
Cylindrically Symmetric ◽  
Approximate Analytical Solutions ◽  
Symmetric Perturbation

The propagation of a cylindrically symmetric Gaussian beam in a cubic-quintic nonlinear medium is analysed via a variational approach. Explicit conditions for stationary beam propagation are determined and their stability to symmetric perturbation of the spot width is established. Approximate analytical solutions are secured for the spot width modulation with propagation distance. A comparison is made with beams propagating in a medium exhibiting a two-level saturation.

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