Nonlinear Differential Equations Equivalent to Solvable Nonlinear Equations

AbstractThe aim of our paper is to study oscillatory and asymptotic properties of solutions of nonlinear differential equations of the third order with deviating argument. In particular, we prove a comparison theorem for properties A and B as well as a comparison result on property A between nonlinear equations with and without deviating arguments. Our assumptions on nonlinearity f are related to its behavior only in a neighbourhood of zero and/or of infinity.

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SOME LINEAR AND NONLINEAR EQUATIONS OF HIGHER ORDER

Science Evolution ◽

10.21603/2500-1418-2016-1-1-46-52 ◽

2016 ◽

pp. 46-52

Author(s):

Nadezda Chuesheva ◽

Aleksandr Chueshev ◽

Aleksandr Chueshev

Keyword(s):

Differential Equations ◽

Exact Solutions ◽

Nonlinear Equations ◽

Existence And Uniqueness ◽

Nonlinear Differential Equations ◽

Uniqueness Of Solutions ◽

Partial Derivatives ◽

Petviashvili Equation ◽

De Vries ◽

Linear And Nonlinear

In this article we consider boundary value problems for some linear and nonlinear differential equations with partial derivatives of the sixth, fifth, fourth and third orders. We write out conditions on equation coefficients for which existence and uniqueness of solutions from Sobolev's space occur. If these conditions on equation coefficients are not valid, then there are given examples when solution is not unique, or is not unstable, or does not belong to Sobolev's space from existence and uniqueness theorem even for analytical coefficients and analytical right side of differential equation. After S.P. Novikov’s fundamental study in 1974 the interest to the nonlinear Korteweg-de Vries equation, Kadomtsev-Petviashvili equation and other nonlinear equations significantly grew. In this study of such equations we used methods of algebraic geometry integration and expansion method. In these studies exact solutions of special nonlinear equations series in partial derivatives play a big role. Solvability of similar equations was also studied in articles of A.I. Kozhanov, N.A. Larkin and other authors. The aim of this article is to find some exact solutions for special series partial differential equations. Solution graphs of such problems for linear equations and for the Korteweg-de Vries, Burgers-Korteweg-deVries, and Kadomtsev-Petviashvili equations are constructed.

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Differential equations of motion of a material point in the perpendicular plane to the plane of the gravitating disk

Indonesian Journal of Electrical Engineering and Computer Science ◽

10.11591/ijeecs.v24.i3.pp1307-1314 ◽

2021 ◽

Vol 24 (3) ◽

pp. 1307

Author(s):

Zhenisgul Rakhmetullina ◽

Indira Uvaliyeva ◽

Farida Amenova

Keyword(s):

Analytical Solution ◽

Differential Equations ◽

Nonlinear Equations ◽

Nonlinear Differential Equations ◽

Equations Of Motion ◽

Material Point ◽

Point System ◽

Disk Material ◽

Analytical Review ◽

Celestial Bodies

This paper presents an analytical solution of the differential equations of motion of a material point in the plane perpendicular to the plane of the gravitating disk. The differential equations of the problem under study and the applied Gilden's method are described in the works of A. Poincaré. Differential equations refer to nonlinear equations. The analysis of methods for solving nonlinear differential equations was carried out. The methodology of applying the Gilden method to the solution of the differential equations under consideration can be applied in studies of the problem of the motion of celestial bodies in the “disk-material point” system in perpendicular planes. To identify the various properties of the gravitating disk, an analytical review of the state of the problem of the motion of a material point in the field of a gravitating disk is carried out. Summing up the presented review on the problem under study, a conclusion is made. The substantive formulation of the problem is described, which is formulated as follows: the study of the influence of disk-shaped bodies on the motion of a material point and methods for their solution.

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Exact solutions of two nonlinear equations and hypercircle estimates

The Journal of the Australian Mathematical Society Series B Applied Mathematics ◽

10.1017/s0334270000002575 ◽

1980 ◽

Vol 22 (1) ◽

pp. 98-103 ◽

Cited By ~ 2

Author(s):

V. G. Hart

Keyword(s):

Differential Equations ◽

Exact Solutions ◽

Nonlinear Equations ◽

Nonlinear Differential Equations

AbstractIn a recent paper estimates of the solutions of two nonlinear differential equations were made by use of the hypercircie method. Here exact solutions are given which are compared with those estimates.

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A NEW ALGORITHM FOR SOLVING NONLINEAR BOUNDARY VALUE PROBLEMS ARISING IN HEAT TRANSFER

International Journal of Modeling Simulation and Scientific Computing ◽

10.1142/s1793962311000499 ◽

2011 ◽

Vol 02 (03) ◽

pp. 355-373 ◽

Cited By ~ 13

Author(s):

S. S. MOTSA

Keyword(s):

Heat Transfer ◽

Differential Equations ◽

Closed Form ◽

Boundary Value Problems ◽

Nonlinear Equations ◽

Nonlinear Boundary ◽

Numerical Solutions ◽

Nonlinear Differential Equations ◽

Analytic Solutions ◽

Nonlinear Boundary Value Problems

In this paper, a very efficient and easy-to-use successive linearization approach for solving nonlinear differential equations is proposed. The implementation of the method is demonstrated by solving three nonlinear differential equations of different complexities arising in heat transfer. New closed form explicit analytic solutions of some of the governing nonlinear equations are obtained and compared with the results of the proposed method and with numerical solutions from the MATLAB in-built routine bvp4c.

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Application of the exp(-φ)-expansion method to the Pochhammer-Chree equation

Filomat ◽

10.2298/fil1809347k ◽

2018 ◽

Vol 32 (9) ◽

pp. 3347-3354 ◽

Cited By ~ 1

Author(s):

Nematollah Kadkhoda ◽

Michal Feckan ◽

Yasser Khalili

Keyword(s):

Partial Differential Equations ◽

Differential Equations ◽

Exact Solutions ◽

Present Article ◽

Analytical Solutions ◽

Nonlinear Differential Equations ◽

Nonlinear Partial Differential Equations ◽

Rational Functions ◽

Expansion Method ◽

Exponential Functions

In the present article, a direct approach, namely exp(-?)-expansion method, is used for obtaining analytical solutions of the Pochhammer-Chree equations which have a many of models. These solutions are expressed in exponential functions expressed by hyperbolic, trigonometric and rational functions with some parameters. Recently, many methods were attempted to find exact solutions of nonlinear partial differential equations, but it seems that the exp(-?)-expansion method appears to be efficient for finding exact solutions of many nonlinear differential equations.

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Solving nonlinear differential equations with differentiable quantum circuits

Physical Review A ◽

10.1103/physreva.103.052416 ◽

2021 ◽

Vol 103 (5) ◽

Author(s):

Oleksandr Kyriienko ◽

Annie E. Paine ◽

Vincent E. Elfving

Keyword(s):

Differential Equations ◽

Nonlinear Differential Equations ◽

Quantum Circuits

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Nonlinear Differential Equations Modeling the Antarctic Circumpolar Current

Journal of Mathematical Fluid Mechanics ◽

10.1007/s00021-021-00618-7 ◽

2021 ◽

Vol 23 (4) ◽

Author(s):

Jifeng Chu ◽

Kateryna Marynets

Keyword(s):

Fixed Point ◽

Differential Equations ◽

Boundary Conditions ◽

Topological Degree ◽

Antarctic Circumpolar Current ◽

Fixed Point Theorems ◽

Nonlinear Differential Equations ◽

Existence Results ◽

Circumpolar Current ◽

The Antarctic

AbstractThe aim of this paper is to study one class of nonlinear differential equations, which model the Antarctic circumpolar current. We prove the existence results for such equations related to the geophysical relevant boundary conditions. First, based on the weighted eigenvalues and the theory of topological degree, we study the semilinear case. Secondly, the existence results for the sublinear and superlinear cases are proved by fixed point theorems.

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