scholarly journals A finite difference approach to find exact solution of differential equations

2015 ◽  
Vol 38 ◽  
pp. 1560080
Author(s):  
Sergei Zuev
Keyword(s):  
Dynamical System ◽  
Finite Difference ◽  
Differential Equations ◽  
Characteristic Equation ◽  
Linear Partial Differential Equation ◽  
Finite Difference Equation ◽  
Wide Range ◽  
Constant Coefficients ◽  
The Given ◽  
Autonomous Dynamical System

This paper contains the background and samples of an approach to construct exact solutions of a wide range of differential equations (DEs). This approach is based on the finite difference equation which corresponds to the given DE. There are three cases considered: linear partial differential equation (PDE) with constant coefficients and at least one non-zero root of characteristic equation, linear PDE with constant coefficients and completely zero roots of the characteristic equation, and a case of nonlinear autonomous dynamical system of second order. Each of these cases is illustrated by an example.

Self-Excited Oscillations in Dynamical Systems Possessing Retarded Actions

1942 ◽  
Vol 9 (2) ◽  
pp. A65-A71 ◽  
Author(s):  
Nicholas Minorsky
Keyword(s):  
Differential Equation ◽  
Dynamical Systems ◽  
Differential Equations ◽  
Linear Equation ◽  
Finite Order ◽  
Characteristic Equation ◽  
High Derivative ◽  
Time Lags ◽  
Constant Coefficients ◽  
The Subject

Abstract There exists a variety of dynamical systems, possessing retarded actions, which are not entirely describable in terms of differential equations of a finite order. The differential equations of such systems are sometimes designated as hysterodifferential equations. An important particular case of such equations, encountered in practice, is when the original differential equation for unretarded quantities is a linear equation with constant coefficients and the time lags are constant. The characteristic equation, corresponding to the hysterodifferential equation for retarded quantities in such a case, has a series of subsequent high-derivative terms which generally converge. It is possible to develop a simple graphical interpretation for this equation. Such systems with retarded actions are capable of self-excitation. Self-excited oscillations of this character are generally undesirable in practice and it is to this phase of the subject that the present paper is devoted.

scholarly journals Exact Finite-Difference Schemes ford-Dimensional Linear Stochastic Systems with Constant Coefficients

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Peng Jiang ◽  
Xiaofeng Ju ◽  
Dan Liu ◽  
Shaoqun Fan
Keyword(s):  
Finite Difference ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Symplectic Structure ◽  
Stochastic Systems ◽  
First Integral ◽  
Difference Schemes ◽  
Finite Difference Schemes ◽  
Linear Stochastic Systems ◽  
Constant Coefficients

The authors attempt to construct the exact finite-difference schemes for linear stochastic differential equations with constant coefficients. The explicit solutions to Itô and Stratonovich linear stochastic differential equations with constant coefficients are adopted with the view of providing exact finite-difference schemes to solve them. In particular, the authors utilize the exact finite-difference schemes of Stratonovich type linear stochastic differential equations to solve the Kubo oscillator that is widely used in physics. Further, the authors prove that the exact finite-difference schemes can preserve the symplectic structure and first integral of the Kubo oscillator. The authors also use numerical examples to prove the validity of the numerical methods proposed in this paper.

A 3D Autonomous System with Infinitely Many Chaotic Attractors

2019 ◽  
Vol 29 (12) ◽  
pp. 1950166 ◽  
Author(s):  
Ting Yang ◽  
Qigui Yang
Keyword(s):  
Dynamical System ◽  
Differential Equations ◽  
Autonomous System ◽  
Three Dimensional ◽  
Chaotic Attractors ◽  
Double Zero ◽  
Finite Dimensional ◽  
Periodic Attractors ◽  
The Stability ◽  
Autonomous Dynamical System

Intuitively, a finite-dimensional autonomous system of ordinary differential equations can only generate finitely many chaotic attractors. Amazingly, however, this paper finds a three-dimensional autonomous dynamical system that can generate infinitely many chaotic attractors. Specifically, this system can generate infinitely many coexisting chaotic attractors and infinitely many coexisting periodic attractors in the following three cases: (i) no equilibria, (ii) only infinitely many nonhyperbolic double-zero equilibria, and (iii) both infinitely many hyperbolic saddles and nonhyperbolic pure-imaginary equilibria. By analyzing the stability of double-zero and pure-imaginary equilibria, it is shown that the classic Shil’nikov criteria fail in verifying the existence of chaos in the above three cases.

ON SOLVING NONLINEAR FUNCTIONAL, FINITE DIFFERENCE, COMPOSITION, AND ITERATED EQUATIONS

Fractals ◽  
1999 ◽  
Vol 07 (03) ◽  
pp. 277-282 ◽  
Author(s):  
RAY BROWN
Keyword(s):  
Finite Difference ◽  
Difference Equation ◽  
Difference Equations ◽  
Finite Difference Equation ◽  
Euler Approximation ◽  
Nonlinear Functional ◽  
Finite Difference Equations ◽  
Wide Range ◽  
Iterated Equations ◽  
General Method

In this letter, we present a general method for solving a wide range of nonlinear functional and finite difference equations, as well as iterated equations such as the Hénon and Mandelbrot equations. The method extends to differential equations using an Euler approximation to obtain a finite difference equation.

SIMPLE METHODS OF ENGINEERING CALCULATION FOR SOLVING HEAT TRANSFER PROBLEMS

2003 ◽  
Vol 8 (1) ◽  
pp. 33-42 ◽  
Author(s):  
H. Kalis ◽  
I. Kangro
Keyword(s):  
Heat Transfer ◽  
Finite Difference ◽  
Differential Equations ◽  
Transfer Problem ◽  
Initial Boundary ◽  
Engineering Calculation ◽  
Transfer Coefficients ◽  
Partial Differential ◽  
The Given ◽  
Element Method

There are well-known numerical methods for solving the initial‐boundary value problems for partial differential equations. We mention only some of them: finite difference method (FDM), finite element method (FEM), boundary element method (BEM), Galerkin type methods and others. In the given work FDM and BEM are considered for determination a distribution of heat in the multilayer media. These methods were used for the reduction of the 1D heat transfer problem described by a partial differential equation to an initial‐value problem for a system of ordinary differential equations (ODEs). Such a procedure allows us to obtain a simple engineering algorithm for solving heat transfer equation in multilayered domain. In a stationary case the exact finite difference scheme is obtained. An inverse problem is also solved. The heat transfer coefficients are found and temperatures in the interior layers depending on the given temperatures inside and outside of a domain are obtained.

A Posterior Model Reduction Method Based on Weighted Residue for Linear Partial Differential Equations

2014 ◽  
Vol 684 ◽  
pp. 34-40
Author(s):  
Jie Sha ◽  
Li Xiang Zhang ◽  
Chui Jie Wu
Keyword(s):  
Dynamical System ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Linear Partial Differential Equation ◽  
Galerkin Projection ◽  
Lagrangian Multiplier ◽  
Partial Differential ◽  
Projection Equation ◽  
Dimensional Dynamical System ◽  
Low Dimensional

This paper is concerned with a new model reduced method based on optimal large truncated low-dimensional dynamical system, by which the solution of linear partial differential equation (PDE) is able to be approximate with highly accuracy. The method proposed is based on the weighted residue of PDE under consideration, and the weighted residue is used as an alternative optimal control condition (POT-WR) while solving the PDE. A set of bases is constructed to describe a dynamical system required in case. The Lagrangian multiplier is introduced to eliminate the constraints of the Galerkin projection equation, and the penalty function is used to remove the orthogonal constraint. According to the extreme principle, a set of the ordinary differential equations is obtained by taking the variational operation on generalized optimal function. A conjugate gradients algorithm on FORTRAN code is developed to solve these ordinary differential equations with Fourier polynomials as the initial bases for iterations. The heat transfer equation under a potential initial condition is used to verify the method proposed. Good agreement between the simulations and the analytical solutions of example was obtained, indicating that the POT-WR method presented in this paper provides the most effective posterior way of capturing the dominant characteristics of an infinite-dimensional dynamical system with only finitely few bases.

scholarly journals THE EXACT FINITE-DIFFERENCE SCHEME FOR VECTOR BOUNDARY‐VALUE PROBLEMS WITH PIECE‐WISE CONSTANT COEFFICIENTS

1998 ◽  
Vol 3 (1) ◽  
pp. 114-123
Author(s):  
H. Kalis
Keyword(s):  
Finite Difference ◽  
Differential Equations ◽  
Difference Scheme ◽  
Finite Difference Scheme ◽  
Boundary Value ◽  
Parabolic Type ◽  
Second Order ◽  
Mhd Equations ◽  
System Of Differential Equations ◽  
Constant Coefficients

We will consider the exact finite‐difference scheme for solving the system of differential equations of second order with piece‐wise constant coefficients. It is well‐known, that the presence of large parameters at first order derivatives or small parameters at second order derivatives in the system of hydrodynamics and magnetohydrodynamics (MHD) equations (large Reynolds, Hartmann and others numbers) causes additional difficulties for the applications of general classical numerical methods. Thus, important to work out special methods of solution, the so‐called uniform converging computational methods. This gives a basis for the development of special monotone finite vector‐difference schemes with perturbation coefficient of function‐matrix for solving the system of differential equations. Special finite‐difference approximations are constructed for a steady‐state boundary‐value problem, systems of parabolic type partial differential equations, a system of two MHD equations, 2‐D flows and MHD‐flows equations in curvilinear orthogonal coordinates.

Mapping of non-autonomous dynamical systems to autonomous ones

2019 ◽  
Vol 16 (06) ◽  
pp. 1950089
Author(s):  
Davood Momeni ◽  
Phongpichit Channuie ◽  
Mudhahir Al Ajmi
Keyword(s):  
Dynamical System ◽  
Partial Differential Equations ◽  
Dynamical Systems ◽  
Differential Equations ◽  
Coordinate Transformations ◽  
Partial Differential ◽  
First Order ◽  
Special Cases ◽  
Autonomous Dynamical Systems ◽  
Autonomous Dynamical System

Using a proper choice of the dynamical variables, we show that a non-autonomous dynamical system transforming to an autonomous dynamical system with a certain coordinate transformations can be obtained by solving a general nonlinear first-order partial differential equations. We examine some special cases and provide particular physical examples. Our framework constitutes general machineries in investigating other non-autonomous dynamical systems.

scholarly journals Asymptotic Behavior and Stability in Linear Impulsive Delay Differential Equations with Periodic Coefficients

Mathematics ◽  
2020 ◽  
Vol 8 (10) ◽  
pp. 1802
Author(s):  
Ali Fuat Yeniçerioğlu ◽  
Vildan Yazıcı ◽  
Cüneyt Yazıcı
Keyword(s):  
Asymptotic Behavior ◽  
Differential Equations ◽  
Characteristic Equation ◽  
Delay Differential Equations ◽  
Periodic Coefficients ◽  
Special Cases ◽  
Impulsive Delay Differential Equations ◽  
Constant Coefficients ◽  
Impulsive Delay ◽  
Delay Differential

We study first order linear impulsive delay differential equations with periodic coefficients and constant delays. This study presents some new results on the asymptotic behavior and stability. Thus, a proper real root was used for a representative characteristic equation. Applications to special cases, such as linear impulsive delay differential equations with constant coefficients, were also presented. In this study, we gave three different cases (stable, asymptotic stable and unstable) in one example. The findings suggest that an equation that is in a way that characteristic equation plays a crucial role in establishing the results in this study.

Computing Model for Vibratory Digging-Out of Sugar Beet Roots

2014 ◽  
Vol 1 (2) ◽  
pp. 52-60
Author(s):  
V. Bulgakov ◽  
V. Adamchuk ◽  
H. Kaletnyk
Keyword(s):  
Differential Equations ◽  
Sugar Beet ◽  
Vertical Plane ◽  
Forward Motion ◽  
Admissible Velocity ◽  
Beet Root ◽  
Optimum Speed ◽  
Analytical Research ◽  
Sugar Beet Root ◽  
The Given

The new design mathematical model of the sugar beet roots vibration digging-out process with the plowshare vibration digging working part has been created. In this case the sugar beet root is simulated as a solid body , while the plowshare vibration digging working part accomplishes fl uctuations in the longitudinal - vertical plane with the given amplitude and frequency in the process of work . The aim of the current research has been to determine the dependences between the design and kinematic parameters of the sugar beet roots vibra- tion digging-out technological process from soil , which provide the ir non-damage. Methods . For the aim ac- complishment, the methods of design mathematical models constructing based on the classical laws of me- chanics are applied. The solution of the obtained differential equations is accomplished with the PC involve- ment. Results . The differential equations of the sugar beet root’s motion in course of the vibration digging-out have been comprised . They allow to determine the admissible velocity of the vibration digging working part’s forward motion depending on the angular parameters of the latter. In the result of the computational simula- tion i.e., the solution of the obtained analytical dependence by PC, the graphic dependences of the admissible velocity of plowshare v ibration digging working part’s forward motion providing the extraction of the sugar beet root from soil without the breaking-off of its tail section have been determined. Conclusions . Due to the performed analytical research , it has been established that γ = 13 ... 16 ° , β = 20 ... 30 ° should be considered as the most reasonable values of γ and β angles of the vibration digging working part providing both its forward motion optimum speed and sugar beet root digging-out from the soil without damage . On the ground of the data obtained from the analytical rese arch, the new vibration digging working parts for the sugar beet roots have been designed; also the patents of Ukraine for the inventions have been obtained for them.

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