scholarly journals Reducing the Dimensionality of Grid Approximations

2021 ◽  
pp. 75-77
Author(s):  
В. Б. Бетелин ◽  
В. А. Галкин
Keyword(s):  
Differential Equations ◽  
Complex Systems ◽  
Exact Solutions ◽  
High Performance ◽  
Computer Applications ◽  
High Dimensional ◽  
Kinetic Methods ◽  
Coarse Meshes ◽  
Systems Simulation

Предложен общий подход к развитию методов математического моделирования сложных систем. Центральной проблемой, связанной с использованием вычислительной техники, являются сеточные аппроксимации большой размерности и суперЭВМ высокой производительности с большим числом параллельно работающих микропроцессоров. В качестве возможных альтернатив сеточным аппроксимациям большой размерности разрабатываются кинетические методы решения дифференциальных уравнений и методы «склейки» точных решений на грубых сетках. A general approach to the development of complex systems simulation is proposed. The key computer applications problem is the high-dimensional grid approximations and high-performance supercomputers with a large number of parallel CPUs. Kinetic methods for solving differential equations and methods for ”gluing” exact solutions produced with coarse meshes are developed as possible alternatives to high-dimensional grid approximations.  

scholarly journals On local fractional operators View of computational complexity: Diffusion and relaxation defined on cantor sets

2016 ◽  
Vol 20 (suppl. 3) ◽  
pp. 755-767 ◽  
Author(s):  
Xiao-Jun Yang ◽  
Zhi-Zhen Zhang ◽  
Tenreiro Machado ◽  
Dumitru Baleanu
Keyword(s):  
Computational Complexity ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Complex Systems ◽  
Exact Solutions ◽  
Fractional Derivatives ◽  
Cantor Sets ◽  
Fractional Operators ◽  
Fractional Partial Differential Equations ◽  
Comparative Results

This paper treats the description of non-differentiable dynamics occurring in complex systems governed by local fractional partial differential equations. The exact solutions of diffusion and relaxation equations with Mittag-Leffler and exponential decay defined on Cantor sets are calculated. Comparative results with other versions of the local fractional derivatives are discussed.

scholarly journals Construction of exact solutions and analysis of stability of complex systems by reduction to ordinary differential equations with power nonlinearities

2019 ◽  
Vol 21 (1) ◽  
pp. 60-69
Author(s):  
Alexander A. Kosov ◽  
Eduard I. Semenov
Keyword(s):  
Differential Equations ◽  
Complex Systems ◽  
Exact Solutions ◽  
Ordinary Differential Equations ◽  
Large Scale ◽  
Nonlinear Partial Differential Equations ◽  
Parabolic Type ◽  
Reaction Diffusion ◽  
Analysis Of Stability ◽  
The Stability

Complex systems described by nonlinear partial differential equations of parabolic type or large-scale systems of ordinary differential equations with switching right-side are considered. The reduction method is applied to the corresponding problem for the system of ordinary differential equations without switching. A parametric family of time-periodic and anisotropic on spatial variables exact solutions of the reaction-diffusion system is constructed. The stability conditions of a large-scale system with switching are obtained, which consist in checking the stability of the reduced system without switching. The conditions for the existence of the first integrals for the reduced system of ordinary differential equations expressed by a combination of power and logarithmic functions are found. For the cases of two-dimensional and three-dimensional reduced systems, these conditions are written in the form of polynomial equations relating the system parameters.

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

Filomat ◽  
2018 ◽  
Vol 32 (9) ◽  
pp. 3347-3354 ◽  
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.

scholarly journals On the Exact Solutions of Two (3+1)-Dimensional Nonlinear Differential Equations

2020 ◽  
Vol 2020 ◽  
pp. 1-8
Author(s):  
Fanning Meng ◽  
Yongyi Gu
Keyword(s):  
Differential Equations ◽  
Exact Solutions ◽  
Shallow Water ◽  
Traveling Wave ◽  
Nonlinear Differential Equations ◽  
Complex Method ◽  
Meromorphic Solutions ◽  
Complex Differential Equations ◽  
Bkp Equation

In this article, exact solutions of two (3+1)-dimensional nonlinear differential equations are derived by using the complex method. We change the (3+1)-dimensional B-type Kadomtsev-Petviashvili (BKP) equation and generalized shallow water (gSW) equation into the complex differential equations by applying traveling wave transform and show that meromorphic solutions of these complex differential equations belong to class W, and then, we get exact solutions of these two (3+1)-dimensional equations.

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