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.

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 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.

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.

scholarly journals On the calculus of symbols.—Fourth memoir. With applications to the theory of non-linear differential equations

1864 ◽  
Vol 13 ◽  
pp. 423-432
Keyword(s):  
Differential Equations ◽  
Linear Function ◽  
Linear Equations ◽  
Linear Differential Equations ◽  
Linear Functions ◽  
Non Linear ◽  
Linear Differential ◽  
Non Linear Equations

In the preceding memoirs on the Calculus of Symbols, systems have been constructed for the multiplication and division of non-commutative symbols subject to certain laws of combination; and these systems suffice or linear differential equations. But when we enter upon the consideration of non-linear equations, we see at once that these methods do not apply. It becomes necessary to invent some fresh mode of calculation, and a new notation, in order to bring non-linear functions into a condition which admits of treatment by symbolical algebra. This is the object of the following memoir. Professor Boole has given, in his 'Treatise on Differential Equations,’ a method due to M. Sarrus, by which we ascertain whether a given non-linear function is a complete differential. This method, as will be seen by anyone who will refer to Professor Boole s treatise, is equivalent to finding the conditions that a non-linear function may be externally divisible by the symbol of differentiation. In the following paper I have given a notation by which I obtain the actual expressions for these conditions, and for the symbolical remainders arising in the course of the division, and have extended my investigations to ascertaining the results of the symbolical division of non-linear functions by linear functions of the symbol of differentiation. Let F ( x, y, y 1 , y 2 , y 3 . . . . y n ) be any non-linear function, in which y 1 , y 2 , y 3 . . . . y n denote respectively the first, second, third, . . . . n th differential of y with respect to ( x ).

scholarly journals Some solvable non-linear equations related to aerodynamic instability phenomena

1994 ◽  
Vol 16 (4) ◽  
pp. 11-14
Author(s):  
Nguyen Dang Bich ◽  
Nguyen Vo Thong
Keyword(s):  
Differential Equations ◽  
Linear Equations ◽  
Linear Differential Equations ◽  
Aerodynamic Forces ◽  
Non Linear ◽  
Aerodynamic Instability ◽  
Linear Differential ◽  
Non Linear Equations

This article proposes some forms of aerodynamic forces, looks for accurate solutions of core - pendent non-linear differential equations and analyses the characteristics of aerodynamic forces and of equation's solution. Found solutions proves that aerodynamic forces are formed from two elements: element of non-linear and dispersion, that single solutions are usually found and aerodynamic instability easy occurs.

Mathematical model of flat-vertical profile moisture transfer under trickle irrigation in conditions of incomplete saturation

2016 ◽  
Vol 3 (3) ◽  
pp. 35-40 ◽  
Author(s):  
M. Romashchenko ◽  
A. Shatkovsky ◽  
V. Onotsky
Keyword(s):  
Differential Equations ◽  
Moisture Transfer ◽  
Linear Equations ◽  
Point Sources ◽  
Richards Equation ◽  
Trickle Irrigation ◽  
Partial Derivatives ◽  
Linear Algebraic Equations ◽  
Non Linear ◽  
Non Linear Equations

Aim. To develop an effi cient method of building a numerical model for the process of moisture transfer under trickle irrigation, with the mathematical modeling of the method involving the system of differential equations in partial derivatives of Klute-Richards, and to perform computing experiments regarding fl at-vertical profi le moisture transfer with point sources. Methods. The mathematical apparatus of the theory of differential schemes of solving differential equations in partial derivatives, and Newton’s method of iterative approximate solving of non-linear equations. Results. A stable differential two-step symmetrized algorithm (TS-algorithm) along with the corresponding scheme of the method of numerical solution for initially-boundary task for Richards’ equation was created. The method was realized in the form of a computer program in C++ language, the computing experiments were performed with three deeper points, the humidity zones for volume moisture and potential were obtained. Conclusions. The numerical method was suggested, ensuring the effi cient solution to Richards’ non-linear equation in conditions of several deep point sources. The algorithm structure allows reducing the system of non-linear algebraic equations with many unknowns to solving independent non-linear equations with one unknown. The presented method may easily be expanded for three-dimensional cases. The results of computing experiments are in agreement with natural observations.

II.—The Stability of Solutions of Non-linear Difference-differential Equations

1950 ◽  
Vol 63 (1) ◽  
pp. 18-26 ◽  
Author(s):  
E. M. Wright
Keyword(s):  
Differential Equations ◽  
Linear Equations ◽  
Linear Perturbation ◽  
Stability Of Solutions ◽  
Essential Step ◽  
Non Linear ◽  
Independent Variable ◽  
The Stability ◽  
Similar Theory ◽  
Non Linear Equations

SynopsisPoincaré, Liapounoff, Perron and others have proved theorems about the order of smallness, as the independent variable tends to + ∞, of solutions of differential equations with non-linear perturbation terms. A similar theory exists for difference equations. By a simple use of transforms, we here extend the theorems, with suitable modifications, to difference-differential equations. The results are an essential step in the development of a general theory of non-linear equations of this type.

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