Systems of Conservation Laws: Two-Dimensional Riemann Problems. Progress in Nonlinear Differential Equations and Their Applications, Vol. 38

2002 ◽  
Vol 55 (5) ◽  
pp. B97-B97 ◽  
Author(s):  
Yuxi Zheng ◽  
TH Moulden
Keyword(s):  
Differential Equations ◽  
Conservation Laws ◽  
Nonlinear Differential Equations ◽  
Two Dimensional ◽  
Riemann Problems ◽  
Systems Of Conservation Laws

Water Waves in a Nonhomogeneous Incompressible Fluid

1977 ◽  
Vol 44 (4) ◽  
pp. 523-528 ◽  
Author(s):  
A. E. Green ◽  
P. M. Naghdi
Keyword(s):  
Surface Tension ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Incompressible Fluid ◽  
Water Waves ◽  
Nonlinear Differential Equations ◽  
Long Waves ◽  
Direct Approach ◽  
Hydraulic Jumps ◽  
Two Dimensional

After a brief discussion of some undesirable features of a number of different partial differential equations often employed in the existing literature on water waves, a relatively simple restricted theory is constructed by a direct approach which is particularly suited for applications to problems of fluid sheets. The rest of the paper is concerned with a derivation of a system of nonlinear differential equations (which may include the effects of gravity and surface tension) governing the two-dimensional motion of incompressible in-viscid fluids for propagation of fairly long waves in a nonhomogeneous stream of water of variable initial depth, as well as some new results pertaining to hydraulic jumps. The latter includes an additional class of possible solutions not noted previously.

scholarly journals On the Question of Nonlinear Fluctuations of Heavy Ropes

2020 ◽  
Vol 15 ◽  
Keyword(s):  
Differential Equations ◽  
Dynamic Systems ◽  
Mathematical Description ◽  
Displacement Vector ◽  
Nonlinear Differential Equations ◽  
The Other ◽  
Two Dimensional ◽  
Small Oscillations ◽  
Curvilinear Motion ◽  
Nonlinear Fluctuations

Due to the methods of the dynamics of curvilinear motion the system of nonlinear differential equations describing two-dimensional arbitrary displacements of a heavy rope, one end of which is free, and the other is fastened is obtained. The problem is solved in the language of displacement vector ()()(),,,xyxtxtxtξξξ=+ij, where (),xxtξ and (),yxtξ accordingly, arbitrary offsets of the rope points in the axes of coordinates. It is shown that in the limit of small oscillations the obtained equations are reduced to the description of the weak offsets of the horizontal string fixed on both ends, and small displacements of the heavy vertical filament, fluctuating due to the force of gravity. The main goal of the paper is the mathematical description of the strong non-equilibrium dynamic systems.

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