The Solution of Non-Linear Equations and of Differential Equations with Two-Point Boundary Conditions

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.

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

Vietnam Journal of Mechanics ◽

10.15625/0866-7136/34/1/424 ◽

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.

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The differential equations of ballistics

Philosophical Transactions of the Royal Society of London. Series A, Containing Papers of a Mathematical or Physical Character ◽

10.1098/rsta.1933.0008 ◽

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.

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

Proceedings of the Royal Society of London ◽

10.1098/rspl.1863.0087 ◽

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 ).

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Some solvable non-linear equations related to aerodynamic instability phenomena

Vietnam Journal of Mechanics ◽

10.15625/0866-7136/10176 ◽

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.

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

Geoscientific Model Development Discussions ◽

10.5194/gmdd-7-1829-2014 ◽

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.

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

Georgian Mathematical Journal ◽

10.1515/gmj-2016-0068 ◽

2017 ◽

Vol 24 (2) ◽

pp. 277-285 ◽

Cited By ~ 1

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.

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THE NUMERICAL MODELLING OF TIDES IN A SHALLOW SEMI-ENCLOSED BASIN BY A MODIFIED ELLIPTIC METHOD

Marine Research in Indonesia ◽

10.14203/mri.v21i0.389 ◽

2018 ◽

Vol 21 ◽

pp. 1-47

Author(s):

Mulia M. Sidjabat

Keyword(s):

Numerical Model ◽

Numerical Modelling ◽

Boundary Conditions ◽

Coefficient Of Friction ◽

Iterative Scheme ◽

Linear Equations ◽

Tidal Constituent ◽

Non Linear ◽

Non Linear Equations ◽

Good Agreement

A numerical model which renders possible the numerical solution of the nonlinear tidal equations by obtaining a solution for each individual tidal component is developed. For this purpose, a set of time independent non-linear equations for each tidal constituent is constructed. Each of these sets of equations is interrelated through the non-linear frictional terms, the approximation of which is accomplished by an iterative scheme. The method is tested for several models before it is applied to the real basin (Bight of Abaco). In order to evaluate the model and to construct the boundary conditions along the opening, a series of tidal observations were undertaken. The viability of the method is indicated by the fact that the results of computations using a coefficient of friction r = 0.0034 give good agreement with observations for all components and over all stations.

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Existence and uniqueness of solutions for the first-order non-linear differential equations with three-point boundary conditions

Filomat ◽

10.2298/fil1905387m ◽

2019 ◽

Vol 33 (5) ◽

pp. 1387-1395

Author(s):

Misir Mardanov ◽

Yagub Sharifov ◽

Kamala Ismayilova

Keyword(s):

Differential Equations ◽

Boundary Conditions ◽

Existence And Uniqueness ◽

Contraction Mapping Principle ◽

Point Boundary ◽

Uniqueness Of Solutions ◽

First Order ◽

Non Linear ◽

Linear Differential ◽

Non Linear System

In this paper the existence and uniqueness of the solutions to boundary value problems for the first order non-linear system of the ordinary differential equations with three-point boundary conditions are investigated. For the first time the Green function is constructed and the considered problem is reduced to the equivalent integral equations that allow us to prove the existence and uniqueness theorems in differ from existing works, applying the Banach contraction mapping principle and Schaefer?s fixed point theorem. An example is given to illustrate the obtained results.

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

Agricultural science and practice ◽

10.15407/agrisp3.03.035 ◽

2016 ◽

Vol 3 (3) ◽

pp. 35-40 ◽

Cited By ~ 1

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.

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