Partially Invariant Solutions for Two-dimensional Ideal MHD Equations

1990 ◽  
Vol 45 (11-12) ◽  
pp. 1219-1229 ◽  
Author(s):  
D.-A. Becker ◽  
E. W. Richter
Keyword(s):  
Differential Equations ◽  
Linear Equations ◽  
Mhd Equations ◽  
Two Dimensional ◽  
Invariant Solutions ◽  
First Order ◽  
Partially Invariant Solutions ◽  
Quasilinear Systems ◽  
Ideal Mhd ◽  
Non Linear Equations

AbstractA generalization of the usual method of similarity analysis of differential equations, the method of partially invariant solutions, was introduced by Ovsiannikov. The degree of non-invariance of these solutions is characterized by the defect of invariance d. We develop an algorithm leading to partially invariant solutions of quasilinear systems of first-order partial differential equations. We apply the algorithm to the non-linear equations of the two-dimensional non-stationary ideal MHD with a magnetic field perpendicular to the plane of motion.

scholarly journals A linear algorithm for solving non-linear isothermal ice-shelf equations

2014 ◽  
Vol 7 (2) ◽  
pp. 1829-1864
Author(s):  
A. Sargent ◽  
J. L. Fastook
Keyword(s):  
Differential Equations ◽  
Iterative Algorithm ◽  
Linear Equations ◽  
Direct Methods ◽  
Two Dimensional ◽  
Ice Shelf ◽  
Linear Algorithm ◽  
Manufactured Solutions ◽  
Non Linear ◽  
Non Linear Equations

Abstract. A linear non-iterative algorithm is suggested for solving nonlinear isothermal steady-state Morland–MacAyeal ice shelf equations. The idea of the algorithm is in replacing the problem of solving the non-linear second order differential equations for velocities with a system of linear first order differential equations for stresses. The resulting system of linear equations can be solved numerically with direct methods which are faster than iterative methods for solving corresponding non-linear equations. The suggested algorithm is applicable if the boundary conditions for stresses can be specified. The efficiency of the linear algorithm is demonstrated for one-dimensional and two-dimensional ice shelf equations by comparing the linear algorithm and the traditional iterative algorithm on derived manufactured solutions. The linear algorithm is shown to be as accurate as the traditional iterative algorithm but significantly faster. The method may be valuable as the way to increase the efficiency of complex ice sheet models a part of which requires solving the ice shelf model as well as to solve efficiently two-dimensional ice-shelf equations.

On non-oscillation for certain system of non-linear ordinary differential equations

2017 ◽  
Vol 24 (2) ◽  
pp. 277-285 ◽  
Author(s):  
Zdeněk Opluštil
Keyword(s):  
Differential Equations ◽  
Linear Equations ◽  
Dimensional System ◽  
Two Dimensional ◽  
Linear Ordinary Differential Equations ◽  
Second Order Differential Equations ◽  
Integrable Functions ◽  
Non Linear ◽  
Two Dimensional System ◽  
Non Linear Equations

AbstractWe consider the following two-dimensional system of non-linear equations:u^{\prime}=g(t)|v|^{\frac{1}{\alpha}}\operatorname{sgn}v,\quad v^{\prime}=-p(t% )|u|^{\alpha}\operatorname{sgn}u,where {\alpha>0}, and {g\colon{[0,+\infty[}\rightarrow{[0,+\infty[}} and {p\colon{[0,+\infty[}\rightarrow\mathbb{R}} are locally integrable functions. Moreover, we assume that the coefficient g is non-integrable on {[0,+\infty]}. We establish new non-oscillation criteria for the considered system, which generalize known results for the corresponding linear system and for second order differential equations. In particular, the presented criteria are in compliance with the results of Hille and Nehari.

A bicharacteristic formulation of the ideal MHD equations

2010 ◽  
Vol 77 (2) ◽  
pp. 169-191
Author(s):  
HARI SHANKER GUPTA ◽  
PHOOLAN PRASAD
Keyword(s):  
Transport Equation ◽  
Linear Equations ◽  
Coupled System ◽  
Mhd Equations ◽  
Two Dimensional ◽  
First Order ◽  
Two Dimensional Flow ◽  
Spatial Coordinates ◽  
Aligned Magnetic Field ◽  
The Ideal

AbstractOn a characteristic surface Ω of a hyperbolic system of first-order equations in multi-dimensions (x, t), there exits a compatibility condition which is in the form of a transport equation along a bicharacteristic on Ω. This result can be interpreted also as a transport equation along rays of the wavefront Ωt in x-space associated with Ω. For a system of quasi-linear equations, the ray equations (which has two distinct parts) and the transport equation form a coupled system of underdetermined equations. As an example of this bicharacteristic formulation, we consider two-dimensional unsteady flow of an ideal magnetohydrodynamics gas with a plane aligned magnetic field. For any mode of propagation in this two-dimensional flow, there are three ray equations: two for the spatial coordinates x and y and one for the ray diffraction. In spite of little longer calculations, the final four equations (three ray equations and one transport equation) for the fast magneto-acoustic wave are simple and elegant and cannot be derived in these simple forms by use of a computer program like REDUCE.

A Simplified Algorithm for Finding Partially Invariant Solutions of Quasilinear Systems

1994 ◽  
Vol 49 (3) ◽  
pp. 458-464
Author(s):  
Dirk-A. Becker
Keyword(s):  
Exact Solutions ◽  
Euler Equations ◽  
Similarity Solutions ◽  
Similarity Analysis ◽  
Two Dimensional ◽  
Invariant Solutions ◽  
Partially Invariant Solutions ◽  
Quasilinear Systems ◽  
Pde Systems ◽  
Simplified Algorithm

Abstract The method of partially invariant solutions of PDE systems was introduced by Ovsiannikov as a generalization of the classical similarity analysis. It offers a possibility to calculate exact solutions possessing a higher degree of freedom than similarity solutions. Ovsiannikov's algorithm, however, is somewhat hard to apply because one has to deal with three equation systems derived from the original PDE system. By means of the two-dimensional Euler equations, we show how the algorithm can be essentially simplified if classical similarity solutions are already known. Further, we prove a necessary criterion for the simplified algorithm to be senseful.

A Difference Method for Plane Problems in Magnetoelastodynamics

1972 ◽  
Vol 39 (3) ◽  
pp. 689-695 ◽  
Author(s):  
W. W. Recker
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Numerical Method ◽  
Difference Method ◽  
Necessary Condition ◽  
Two Dimensional ◽  
Symmetric Hyperbolic System ◽  
Von Neumann ◽  
First Order ◽  
Independent Variables

The two-dimensional equations of magnetoelastodynamics are considered as a symmetric hyperbolic system of linear first-order partial-differential equations in three independent variables. The characteristic properties of the system are determined and a numerical method for obtaining the solution to mixed initial and boundary-value problems in plane magnetoelastodynamics is presented. Results on the von Neumann necessary condition are presented. Application of the method to a problem which has a known solution provides further numerical evidence of the convergence and stability of the method.

scholarly journals On non-homogeneous canonical third-order linear differential equations

1994 ◽  
Vol 57 (2) ◽  
pp. 138-148 ◽  
Author(s):  
N. Parhi
Keyword(s):  
Differential Equations ◽  
Linear Equations ◽  
Sufficient Conditions ◽  
Linear Differential Equations ◽  
Third Order ◽  
Order Linear ◽  
Non Linear ◽  
Linear Differential ◽  
Non Linear Equations

AbstractIn this paper sufficient conditions have been obtained for non-oscillation of non-homogeneous canonical linear differential equations of third order. Some of these results have been extended to non-linear equations.

scholarly journals On a class of non-linear differential equations with exact solution

2012 ◽  
Vol 34 (1) ◽  
pp. 7-17
Author(s):  
Dao Huy Bich ◽  
Nguyen Dang Bich
Keyword(s):  
Exact Solution ◽  
Differential Equations ◽  
Exact Solutions ◽  
Solid Mechanics ◽  
Linear Equations ◽  
Linear Differential Equations ◽  
Second Order Differential Equations ◽  
Non Linear ◽  
Linear Differential ◽  
Non Linear Equations

The present paper deals with a class of non-linear ordinary second-order differential equations with exact solutions. A procedure for finding the general exact solution based on a known particular one is derived. For illustration solutions of some non-linear equations occurred in many problems of solid mechanics are considered.

Some Invariant Solutions of Two-Dimensional Elastodynamics in Linear Homogeneous Isotropic Materials

2013 ◽  
Vol 5 (2) ◽  
pp. 212-221
Author(s):  
Houguo Li ◽  
Kefu Huang
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Theoretical Method ◽  
Two Dimensional ◽  
Invariant Solutions ◽  
Partial Differential ◽  
Isotropic Materials ◽  
Group Theoretical Method ◽  
Infinitesimal Operators

AbstractInvariant solutions of two-dimensional elastodynamics in linear homogeneous isotropic materials are considered via the group theoretical method. The second order partial differential equations of elastodynamics are reduced to ordinary differential equations under the infinitesimal operators. Three invariant solutions are constructed. Their graphical figures are presented and physical meanings are elucidated in some cases.

scholarly journals The differential equations of ballistics

1933 ◽  
Vol 231 (694-706) ◽  
pp. 263-288
Keyword(s):  
Differential Equations ◽  
Applied Mathematics ◽  
Linear Equations ◽  
Second Order ◽  
Linear Differential Equations ◽  
Pressure Index ◽  
Internal Ballistics ◽  
Non Linear ◽  
Linear Differential ◽  
Non Linear Equations

The differential equations arising in most branches of applied mathematics are linear equations of the second order. Internal ballistics, which is the dynamics of the motion of the shot in a gun, requires, except with the simplest assumptions, the discussion of non-linear differential equations of the first and second orders. The writer has shown in a previous paper* how such non-linear equations arise when the pressure-index a in the rate-of-burning equation differs from unity, although only the simplified case of non-resisted motion was there considered. It is proposed in the present investigation to examine some cases of resisted motion taking the pressure-index equal to unity, to give some extensions of the previous work, and to consider, so far as is possible, the nature and the solution of the types of differential equations which arise.

Sign in / Sign up

Export Citation Format

Share Document