scholarly journals Cotton-Type and Joint Invariants for Linear Elliptic Systems

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
A. Aslam ◽  
F. M. Mahomed
Keyword(s):  
Elliptic Equations ◽  
Hyperbolic Equations ◽  
Elliptic Systems ◽  
General System ◽  
Linear Elliptic Equation ◽  
Linear Complex ◽  
Independent Variables ◽  
Joint Invariants ◽  
Laplace Type ◽  
Linear Hyperbolic Equations

Cotton-type invariants for a subclass of a system of two linear elliptic equations, obtainable from a complex base linear elliptic equation, are derived both by spliting of the corresponding complex Cotton invariants of the base complex equation and from the Laplace-type invariants of the system of linear hyperbolic equations equivalent to the system of linear elliptic equations via linear complex transformations of the independent variables. It is shown that Cotton-type invariants derived from these two approaches are identical. Furthermore, Cotton-type and joint invariants for a general system of two linear elliptic equations are also obtained from the Laplace-type and joint invariants for a system of two linear hyperbolic equations equivalent to the system of linear elliptic equations by complex changes of the independent variables. Examples are presented to illustrate the results.

scholarly journals Laplace-Type Semi-Invariants for a System of Two Linear Hyperbolic Equations by Complex Methods

2011 ◽  
Vol 2011 ◽  
pp. 1-15 ◽  
Author(s):  
F. M. Mahomed ◽  
A. Qadir ◽  
A. Ramnarain
Keyword(s):  
Hyperbolic Equations ◽  
Hyperbolic Partial Differential Equations ◽  
Integration Theory ◽  
Equivalence Transformations ◽  
Complex Scalar ◽  
Hyperbolic Pdes ◽  
Independent Variables ◽  
Dependent Variables ◽  
Two Independent Variables ◽  
Linear Hyperbolic Equations

In 1773 Laplace obtained two fundamental semi-invariants, called Laplace invariants, for scalar linear hyperbolic partial differential equations (PDEs) in two independent variables. He utilized this in his integration theory for such equations. Recently, Tsaousi and Sophocleous studied semi-invariants for systems of two linear hyperbolic PDEs in two independent variables. Separately, by splitting a complex scalar ordinary differential equation (ODE) into its real and imaginary parts PDEs for two functions of two variables were obtained and their symmetry structure studied. In this work we revisit semi-invariants under equivalence transformations of the dependent variables for systems of two linear hyperbolic PDEs in two independent variables when such systems correspond to scalar complex linear hyperbolic equations in two independent variables, using the above-mentioned splitting procedure. The semi-invariants under linear changes of the dependent variables deduced for this class of hyperbolic linear systems correspond to the complex semi-invariants of the complex scalar linear hyperbolic equation. We show thatthe adjoint factorization corresponds precisely to the complex splitting. We also study the reductions and the inverse problem when such systems of two linear hyperbolic PDEs arise from a linear complex hyperbolic PDE. Examples are given to show the application of this approach.

Fast high order ADER schemes for linear hyperbolic equations

2004 ◽  
Vol 197 (2) ◽  
pp. 532-539 ◽  
Author(s):  
Thomas Schwartzkopff ◽  
Michael Dumbser ◽  
Claus-Dieter Munz
Keyword(s):  
Hyperbolic Equations ◽  
High Order ◽  
Linear Hyperbolic Equations
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