On the Cauchy problem for some abstract nonlinear differential equations

1996 ◽  
Vol 3 (2) ◽  
pp. 279-290 ◽  
Author(s):  
Hamza A. S. Abujabal ◽  
Mahmoud M. El-Borai
Keyword(s):  
Cauchy Problem ◽  
Differential Equations ◽  
Nonlinear Differential Equations ◽  
The Cauchy Problem

scholarly journals The Parametrization of the Cauchy Problem for Nonlinear Differential Equations with Contrast Structures

2018 ◽  
pp. 486-510
Author(s):  
Evgenii B. Kuznetsov ◽  
Sergey S. Leonov ◽  
Ekaterina D. Tsapko
Keyword(s):  
Cauchy Problem ◽  
Differential Equations ◽  
Nonlinear Differential Equations ◽  
Computational Time ◽  
Method Of Solution ◽  
The Cauchy Problem ◽  
Contrast Structures ◽  
Interior Layers ◽  
Boundary And Interior Layers ◽  
Best Argument

Introduction. The paper provides an analysis of numerical methods for solving the Cauchy problem for nonlinear ordinary differential equations with contrast structures (interior layers). Similar equations simulate various applied problems of hydro- and aeromechanics, chemical kinetics, the theory of catalytic reactions, etc. An analytical solution to these problems is rarely obtained, and numerical procedure is related with significant difficulties associated with ill-conditionality in the neighborhoods of the boundary and interior layers. The aim of the paper is the scope analysis of traditional numerical methods for solving this class problems and approbation of alternative solution methods. Materials and methods. The traditional explicit Euler and fourth-order Runge-Kutta methods, as well as the implicit Euler method with constant and variable step sizes are used for the numerical solution of the Cauchy problem. The method of solution continuation with respect to the best argument is suggested as an alternative to use. The solution continuation method consists in replacing the original argument of the problem with a new one, measured along the integral curve of the problem. The transformation to the best argument allows obtaining the best conditioned Cauchy problem. Results. The computational difficulties arising when solving the equations with contrast structures by traditional explicit and implicit methods are shown on the example of the test problem solution. These difficulties are expressed in a significant decrease of the step size in the neighborhood of the boundary and interior layers. It leads to the increase of the computational time, as well as to the complication of the solving process for super stiff problems. The authenticity of the obtained results is confirmed by the comparison with the analytical solution and the works of other authors. Conclusions. The results of the computational experiment demonstrate the applicability of the traditional methods for solving the Cauchy problem for equations with contrast structures only at low stiffness. In other cases these methods are ineffective. It is shown that the method of solution continuation with respect to the best argument allows eliminating most of the disadvantages inherent to the original problem. It is reflected in decreasing the computational time and in increasing the solution accuracy. Keywords: contrast structures, method of solution continuation, the best argument, illconditionality, the Cauchy problem, ordinary differential equation For citation: Kuznetsov E. B., Leonov S. S., Tsapko E. D. The Parametrization of the Cauchy Problem for Nonlinear Differential Equations with Contrast Structures. Vestnik Mordovskogo universiteta = Mordovia University Bulletin. 2018; 28(4):486–510. DOI: https://doi.org/10.15507/0236-2910.028.201804.486-510 Acknowledgements: This work was supported by the Russian Science Foundation, project no. 18-19-00474.

scholarly journals On the solvability of the modified Cauchy problem for linear systems of generalized ordinary differential equations with singularities

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Malkhaz Ashordia ◽  
Inga Gabisonia ◽  
Mzia Talakhadze
Keyword(s):  
Cauchy Problem ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Linear Systems ◽  
Unique Solvability ◽  
Sufficient Conditions ◽  
The Cauchy Problem ◽  
Generalized Ordinary Differential Equations

AbstractEffective sufficient conditions are given for the unique solvability of the Cauchy problem for linear systems of generalized ordinary differential equations with singularities.

Old and New Results for First Order Periodic ODEs without Uniqueness: a Comprehensive Study by Lower and Upper Solutions

2004 ◽  
Vol 4 (3) ◽  
Author(s):  
Franco Obersnel ◽  
Pierpaolo Omari
Keyword(s):  
Cauchy Problem ◽  
Differential Equations ◽  
Lower And Upper Solutions ◽  
Elementary Approach ◽  
Qualitative Properties ◽  
First Order ◽  
The Past ◽  
The Cauchy Problem ◽  
Qualitative Properties Of Solutions ◽  
Comprehensive Study

AbstractAn elementary approach, based on a systematic use of lower and upper solutions, is employed to detect the qualitative properties of solutions of first order scalar periodic ordinary differential equations. This study is carried out in the Carathéodory setting, avoiding any uniqueness assumption, in the future or in the past, for the Cauchy problem. Various classical and recent results are recovered and generalized.

Hodograph Method for Solving the Problem of Shallow Water Under a Solid Lid

2021 ◽  
pp. 15-24
Author(s):  
Tatiana F. Dolgikh
Keyword(s):  
Cauchy Problem ◽  
Green Function ◽  
Differential Equations ◽  
Shallow Water ◽  
Ordinary Differential Equations ◽  
Riemann Invariants ◽  
Hodograph Method ◽  
First Order ◽  
The Cauchy Problem ◽  
Derivatives Of

One of the mathematical models describing the behavior of two horizontally infinite adjoining layers of an ideal incompressible liquid under a solid cover moving at different speeds is investigated. At a large difference in the layer velocities, the Kelvin-Helmholtz instability occurs, which leads to a distortion of the interface. At the initial point in time, the interface is not necessarily flat. From a mathematical point of view, the behavior of the liquid layers is described by a system of four quasilinear equations, either hyperbolic or elliptic, in partial derivatives of the first order. Some type shallow water equations are used to construct the model. In the simple version of the model considered in this paper, in the spatially one-dimensional case, the unknowns are the boundary between the liquid layers h(x,t) and the difference in their velocities γ(x,t). The main attention is paid to the case of elliptic equations when |h|<1 and γ>1. An evolutionary Cauchy problem with arbitrary sufficiently smooth initial data is set for the system of equations. The explicit dependence of the Riemann invariants on the initial variables of the problem is indicated. To solve the Cauchy problem formulated in terms of Riemann invariants, a variant of the hodograph method based on a certain conservation law is used. This method allows us to convert a system of two quasilinear partial differential equations of the first order to a single linear partial differential equation of the second order with variable coefficients. For a linear equation, the Riemann-Green function is specified, which is used to construct a two-parameter implicit solution to the original problem. The explicit solution of the problem is constructed on the level lines (isochrons) of the implicit solution by solving a certain Cauchy problem for a system of ordinary differential equations. As a result, the original Cauchy problem in partial derivatives of the first order is transformed to the Cauchy problem for a system of ordinary differential equations, which is solved by numerical methods. Due to the bulkiness of the expression for the Riemann-Green function, some asymptotic approximation of the problem is considered, and the results of calculations, and their analysis are presented.

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