On a class of differential equations which model tide-well systems

1970 ◽  
Vol 3 (3) ◽  
pp. 391-411 ◽  
Author(s):  
B. J. Noye
Keyword(s):  
Differential Equation ◽  
Exact Solution ◽  
Steady State ◽  
Differential Equations ◽  
Sea Level ◽  
Approximate Solutions ◽  
Near Resonance ◽  
Non Linear ◽  
Image Position ◽  
Steady State Solutions

For three possible types of tide-well systems the non-dimensional head response, Y(τ), to a sinusoidal fluctuation of the sea-level is given by the differential equation Estimates of the non-dimensional well response Z(τ) = sinτ - Y(τ) are found by considering the steady state solutions of the above equation. With n = 2 the equation is linear and an exact solution can be found; for n ≠ 2 the equation is non-linear and several methods which give approximate solutions are described. The methods used can be extended to cover other values of n; for example, with n = 4 the equation corresponds to one governing oscillations near resonance in open pipes.

scholarly journals The Laplace-Adomian-Pade Technique for the ENSO Model

2013 ◽  
Vol 2013 ◽  
pp. 1-4 ◽  
Author(s):  
Yi Zeng
Keyword(s):  
Differential Equation ◽  
Exact Solution ◽  
Approximate Solution ◽  
Differential Equations ◽  
Nonlinear Differential Equation ◽  
Initial Conditions ◽  
Approximate Solutions ◽  
Decomposition Methods ◽  
High Accuracy ◽  
Padé Method

The Laplace-Adomian-Pade method is used to find approximate solutions of differential equations with initial conditions. The oscillation model of the ENSO is an important nonlinear differential equation which is solved analytically in this study. Compared with the exact solution from other decomposition methods, the approximate solution shows the method’s high accuracy with symbolic computation.

scholarly journals On the non-existence of L2 solutions of a class of non-linear differential equations

1965 ◽  
Vol 14 (4) ◽  
pp. 257-268 ◽  
Author(s):  
J. Burlak
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Linear Differential Equations ◽  
Half Line ◽  
Non Linear ◽  
Linear Differential ◽  
Image Position

In 1950, Wintner (11) showed that if the function f(x) is continuous on the half-line [0, ∞) and, in a certain sense, is “ small when x is large ” then the differential equationdoes not have L2 solutions, where the function y(x) satisfying (1) is called an L2 solution if

Analytical theory of non-linear oscillations

1974 ◽  
Vol 76 (1) ◽  
pp. 285-296 ◽  
Author(s):  
Chike Obi
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Existence Theorem ◽  
Wide Class ◽  
Periodic Oscillations ◽  
Real Domain ◽  
Non Linear ◽  
General Existence ◽  
Linear Differential ◽  
Image Position

In this paper, we improve on the results of two previous papers (8, 9) by establishing a general existence theorem (section 1·3, below) for a class of periodic oscillations of a wide class of non-linear differential equations of the second order in the real domain which are perturbations of the autonomous differential equationwhere g(x) is strictly non-linear. We then, by way of illustrating the power of the theorem, apply it to the problems which Morris (section 2·2 below), Shimuzu (section 2·3 below) and Loud (section 2·5 below) set themselves on the existence of periodic oscillations of certain differential equations which are perturbations of equations of the form (1·1·1).

On the relation between distinct particular solutions of equation

1993 ◽  
Vol 123 (5) ◽  
pp. 917-926
Author(s):  
Guan Ke-ying ◽  
W. N. Everitt
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Linear Differential Equations ◽  
Particular Solutions ◽  
Non Linear ◽  
Linear Differential ◽  
Image Position

SynopsisThere exists a relation (1.5) between any n + 2 distinct particular solutions of the differential equationIn this paper, we show that when and only when n = 0, 1 and 2, this relation can be represented by the following form:provided the form of this relation function Φn depends only on n and is independent of the coefficients of the equation. This result reveals interesting properties of these non-linear differential equations.

A further instability theorem for a certain fifth-order differential equation

1979 ◽  
Vol 86 (3) ◽  
pp. 491-493 ◽  
Author(s):  
J. O. C. Ezeilo
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Trivial Solution ◽  
Constant Coefficient ◽  
Order Differential Equation ◽  
Sufficient Conditions ◽  
Non Linear ◽  
Image Position ◽  
Fifth Order

In our previous consideration in (1) of the constant-coefficient fifth-order differential equation:an attempt was made to identify (though not exhaustively) different sufficient conditions on a1,…,a5 for the instability of the trivial solution x = 0 of (1·1). It was our expectation that the conditions so identified could be generalized in some form or other to equations (1·1) in which a1,…,a5 were not necessarily constants, thereby giving rise to instability theorems for some non-linear fifth-order differential equations; and this turned out in fact to be so except only for the case:with R0 = R0(a1, a2, a3, a4) > 0 sufficiently large, about which we were unable at the time to derive any worthwhile generalization to any equation (1·1) in which a1, …,a5 are not all constants.

scholarly journals Approximate Method for Solving the Linear Fuzzy Delay Differential Equations

2015 ◽  
Vol 2015 ◽  
pp. 1-9 ◽  
Author(s):  
S. Narayanamoorthy ◽  
T. L. Yookesh
Keyword(s):  
Differential Equation ◽  
Exact Solution ◽  
Approximate Solution ◽  
Differential Equations ◽  
Approximate Method ◽  
Delay Differential Equations ◽  
Decomposition Method ◽  
Adomian Decomposition Method ◽  
Adomian Decomposition ◽  
Delay Differential

We propose an algorithm of the approximate method to solve linear fuzzy delay differential equations using Adomian decomposition method. The detailed algorithm of the approach is provided. The approximate solution is compared with the exact solution to confirm the validity and efficiency of the method to handle linear fuzzy delay differential equation. To show this proper features of this proposed method, numerical example is illustrated.

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 A computational method for solving differential equations with quadratic nonlinearity by using Bernoulli polynomials

2019 ◽  
Vol 23 (Suppl. 1) ◽  
pp. 275-283
Author(s):  
Kubra Bicer ◽  
Mehmet Sezer
Keyword(s):  
Differential Equations ◽  
Linear Equations ◽  
Bernoulli Polynomials ◽  
Approximate Solutions ◽  
Mean Value ◽  
Computational Method ◽  
Mean Value Theorem ◽  
Residual Functions ◽  
Non Linear ◽  
Linear Differential

In this paper, a matrix method is developed to solve quadratic non-linear differential equations. It is assumed that the approximate solutions of main problem which we handle primarily, is in terms of Bernoulli polynomials. Both the approximate solution and the main problem are written in matrix form to obtain the solution. The absolute errors are applied to numeric examples to demonstrate efficiency and accuracy of this technique. The obtained tables and figures in the numeric examples show that this method is very sufficient and reliable for solution of non-linear equations. Also, a formula is utilized based on residual functions and mean value theorem to seek error bounds.

scholarly journals Oscillation Criteria for Second Order Neutral Differential Equations

1996 ◽  
Vol 48 (4) ◽  
pp. 871-886 ◽  
Author(s):  
Horng-Jaan Li ◽  
Wei-Ling Liu
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Delay Differential Equations ◽  
Second Order ◽  
Neutral Delay ◽  
Oscillation Criteria ◽  
Neutral Differential Equations ◽  
Neutral Delay Differential Equation ◽  
Image Position ◽  
Delay Differential

AbstractSome oscillation criteria are given for the second order neutral delay differential equationwhere τ and σ are nonnegative constants, . These results generalize and improve some known results about both neutral and delay differential equations.

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