scholarly journals Generation of Three Dimensional Body Fitted Coordinates Using Hyperbolic Partial Differential Equations.

1984 ◽  
Author(s):  
J. L. Steger
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Three Dimensional ◽  
Hyperbolic Partial Differential Equations ◽  
Partial Differential

Eigenstructure of First-Order Velocity-Stress Equations for Waves in Elastic Solids of Trigonal 32 Symmetry

2010 ◽  
Vol 77 (6) ◽  
Author(s):  
Lixiang Yang ◽  
Yung-Yu Chen ◽  
S.-T. John Yu
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Three Dimensional ◽  
Coefficient Matrix ◽  
Hyperbolic Partial Differential Equations ◽  
Wave Polarization ◽  
Partial Differential ◽  
Plane Wave Solution ◽  
Wave Speeds ◽  
First Order

This paper reports the eigenstructure of a set of first-order hyperbolic partial differential equations for modeling waves in solids with a trigonal 32 symmetry. The governing equations include the equation of motion and partial differentiation of the elastic constitutive relation with respect to time. The result is a set of nine, first-order, fully coupled, hyperbolic partial differential equations with velocity and stress components as the unknowns. Shown in the vector form, the model equations have three 9×9 coefficient matrices. The wave physics are fully described by the eigenvalues and eigenvectors of these matrices; i.e., the nontrivial eigenvalues are the wave speeds, and a part of the corresponding left eigenvectors represents wave polarization. For a wave moving in a certain direction, three wave speeds can be identified by calculating the eigenvalues of the coefficient matrix in a rotated coordinate system. In this process, without using the plane-wave solution, we recover the Christoffel matrix and thus validate the formulation. To demonstrate this approach, two- and three-dimensional slowness profiles of quartz are calculated. Wave polarization vectors for wave propagation in several compression directions as well as noncompression directions are discussed.

scholarly journals GROUP PROPERTIES OF A ONE-DIMENSIONAL NONLINEAR POROELASTICITY EQUATION

2019 ◽  
Vol 4 (1) ◽  
pp. 149-155
Author(s):  
Kholmatzhon Imomnazarov ◽  
Ravshanbek Yusupov ◽  
Ilham Iskandarov
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Lie Algebra ◽  
Three Dimensional ◽  
Group Classification ◽  
Second Order ◽  
Partial Differential ◽  
One Dimensional ◽  
Preliminary Group

This paper studies a class of partial differential equations of second order , with arbitrary functions and , with the help of the group classification. The main Lie algebra of infinitely infinitesimal symmetries is three-dimensional. We use the method of preliminary group classification for obtaining the classifications of these equations for a one-dimensional extension of the main Lie algebra.

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