Optimal boundary controls for the systems described by diffusion-type equation with fractional-order time derivative

2016 ◽  
Author(s):  
Victor Kubyshkin ◽  
Sergey Postnov
Keyword(s):  
Fractional Order ◽  
Time Derivative ◽  
Type Equation ◽  
Diffusion Type ◽  
Optimal Boundary ◽  
Boundary Controls

scholarly journals Mathematical modelling of the mass-spring-damper system - A fractional calculus approach

2012 ◽  
Vol 22 (5) ◽  
pp. 5-11 ◽  
Author(s):  
José Francisco Gómez Aguilar ◽  
Juan Rosales García ◽  
Jesus Bernal Alvarado ◽  
Manuel Guía
Keyword(s):  
Differential Equation ◽  
Fractional Calculus ◽  
Mathematical Modelling ◽  
Fractional Differential Equation ◽  
Fractional Order ◽  
Time Derivative ◽  
System A ◽  
Fractional Differential ◽  
Mass Spring ◽  
Derivatives Of

In this paper the fractional differential equation for the mass-spring-damper system in terms of the fractional time derivatives of the Caputo type is considered. In order to be consistent with the physical equation, a new parameter is introduced. This parameter char­acterizes the existence of fractional components in the system. A relation between the fractional order time derivative and the new parameter is found. Different particular cases are analyzed

Mode Analysis on Onset of Turing Instability in Time-Fractional Reaction-Subdiffusion Equations by Two-Dimensional Numerical Simulations

2019 ◽  
Vol 14 (6) ◽  
Author(s):  
Masataka Fukunaga
Keyword(s):  
Critical Point ◽  
Fractional Order ◽  
Time Derivative ◽  
Turing Instability ◽  
Reaction Diffusion Equations ◽  
Mode Analysis ◽  
Two Dimensional ◽  
Linearized Equations ◽  
Special Cases ◽  
Fractional Reaction

There are two types of time-fractional reaction-subdiffusion equations for two species. One of them generalizes the time derivative of species to fractional order, while in the other type, the diffusion term is differentiated with respect to time of fractional order. In the latter equation, the Turing instability appears as oscillation of concentration of species. In this paper, it is shown by the mode analysis that the critical point for the Turing instability is the standing oscillation of the concentrations of the species that does neither decays nor increases with time. In special cases in which the fractional order is a rational number, the critical point is derived analytically by mode analysis of linearized equations. However, in most cases, the critical point is derived numerically by the linearized equations and two-dimensional (2D) simulations. As a by-product of mode analysis, a method of checking the accuracy of numerical fractional reaction-subdiffusion equation is found. The solutions of the linearized equation at the critical points are used to check accuracy of discretized model of one-dimensional (1D) and 2D fractional reaction–diffusion equations.

Workspace Generation As a Diffusion Process

2000 ◽  
Author(s):  
Yunfeng Wang ◽  
Gregory S. Chirikjian
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Fourier Transform ◽  
Density Function ◽  
Type Equation ◽  
Diffusion Type ◽  
Solvable In Closed Form ◽  
Rigid Body Motions ◽  
The Fourier Transform ◽  
Kinematic Properties

Abstract In this paper we show that the workspace of a highly articulated manipulator can be found by solving a partial differential equation. This diffusion-type equation describes the evolution of the workspace density function depending on manipulator length and kinematic properties. The support of the workspace density function is the workspace of the manipulator. The PDE governing workspace density evolution is solvable in closed form using the Fourier transform on the group of rigid-body motions. We present numerical results that use this technique.

scholarly journals Analytical Method for Solving the Fractional Order Generalized KdV Equation by a Beta-Fractional Derivative

2020 ◽  
Vol 2020 ◽  
pp. 1-18
Author(s):  
Majid Bagheri ◽  
Ali Khani
Keyword(s):  
Fractional Derivative ◽  
Exact Solutions ◽  
Fractional Order ◽  
Kdv Equation ◽  
Time Derivative ◽  
Hyperbolic Function ◽  
Nonlinear Phenomena ◽  
Generalized Kdv Equation ◽  
Wave Solutions ◽  
Fractional Time Derivative

The present work is related to solving the fractional generalized Korteweg-de Vries (gKdV) equation in fractional time derivative form of order α . Some exact solutions of the fractional-order gKdV equation are attained by employing the new powerful expansion approach by using the beta-fractional derivative which is used to get many solitary wave solutions by changing various parameters. The obtained solutions include three classes of soliton wave solutions in terms of hyperbolic function, trigonometric function, and rational function solutions. The obtained solutions and the exact solutions are shown graphically, highlighting the effects of nonlinearity. Some of the nonlinear equations arise in fluid dynamics and nonlinear phenomena.

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