On one class of differential equations of fractional order

We consider the Darboux problem for a differential equation of fractional order that contains a regularized mixed derivative. Sufficient conditions for the existence and uniqueness of a solution of this problem are obtained in the class of continuous functions. We also propose a method for finding an approximate solution of this problem and prove the convergence of this method.     

Darboux problem for a differential equation with fractional derivative

We find conditions for the unique solvability of the problem u _xy (x, y) = f(x, y, u(x, y), (D ₀ ^r u)(x, y)), u(x, 0) = u(0, y) = 0, x ∈ [0, a], y ∈ [0, b], where (D ₀ ^r u)(x, y) is the mixed Riemann-Liouville derivative of order r = (r ₁, r ₂), 0 < r ₁, r ₂ < 1, in the class of functions that have the continuous derivatives u _xy (x, y) and (D ₀ ^r u)(x, y). We propose a numerical method for solving this problem and prove the convergence of the method. __________ Translated from Neliniini Kolyvannya, Vol. 8, No. 4, pp. 456–467, October–December, 2005.     

Stability analysis of linear fractional differential system with multiple time delays

In this paper, we study the stability of n-dimensional linear fractional differential equation with time delays, where the delay matrix is defined in (R ⁺)^n×n. By using the Laplace transform, we introduce a characteristic equation for the above system with multiple time delays. We discover that if all roots of the characteristic equation have negative parts, then the equilibrium of the above linear system with fractional order is Lyapunov globally asymptotical stable if the equilibrium exist that is almost the same as that of classical differential equations. As its an application, we apply our theorem to the delayed system in one spatial dimension studied by Chen and Moore [Nonlinear Dynamics 29, 2002, 191] and determine the asymptotically stable region of the system. We also deal with synchronization between the coupled Duffing oscillators with time delays by the linear feedback control method and the aid of our theorem, where the domain of the control-synchronization parameters is determined.     

Existence of solutions of systems of partial differential equations of fractional order

We obtain sufficient conditions for the existence and uniqueness of a solution of a system of partial differential equations of fractional order in spaces of integrable functions.Translated from Neliniini Kolyvannya, Vol. 7, No. 3, pp. 328–335, July–September, 2004.     

A differential equation for the minors of the state-transition matrix of linear systems

Summary A differential equation for the minors of the state transition matrix of linear time-invariant systems is established.An example is then presented, where the differential equation is employed in order to check the existence of the extremal control of a linear system with a performance index given by the integral of a quadratic form.

---

Sommario Si stabilisce un'equazione differenziale per i minori della matrice di transizione dei sistemi lineari stazionari.Si illustra quindi un esempio, in cui tale equazione differenziale viene impiegata per verificare l'esistenza del controllo estremale di un sistema lineare con indice di comportamento dato dall'integrale di una forma quadratica.

     

Discrete fractional diffusion equation

Nonlinear Dynamics - The tool of the discrete fractional calculus is introduced to discrete modeling of diffusion problem. A fractional time discretization diffusion model is presented in the...     

Comments on time-varying fractional order

Nonlinear Dynamics - The calculus of arbitrary order, known as the fractional calculus, allows for real- and complex-valued orders. Interest in time-varying order is growing in order to study...     

A fractional model for robust fractional order Smith predictor

In this paper, we propose to use a fractional order model to predict the process output in Smith predictor. The parameters of the model are determined by minimizing the error between its output and one of the processes using a genetic algorithm. After determining the model’s parameters, a fractional PID controller is proposed to improve the controlled system performances. The parameters of the controller are also determined in an optimal way by minimizing the position error taking into account the sensitivity and the complementary sensitivity conditions. Applications on a dead time and multiple lags processes have been performed, where the simulation results show that the proposed Smith predictor enhance the closed loop control system.     

Periodic solutions of linear degenerate systems of ordinary differential equations of the second order

We propose sufficient conditions for the existence of a periodic solution of a system of linear ordinary differential equations of the second order with a degenerate symmetric matrix in the coefficient of the second-order derivative in the case of an arbitrary periodic inhomogeneity.     

On the initial value problems of first order impulsive differential systems

The purpose of this paper is to study the existence and iterative approximation of minimax quasi-solutions for a class of initial value problems of first order impulsive differential systems by using monotone iterative methods. Project supported by the National Natural Science Foundation of China     

Theory of fractional order in electro-thermoelasticity

A new mathematical model of electro-thermoelasticity has been constructed in the context of a new consideration of heat conduction with fractional order. The state space approach developed in Ezzat (2008) is adopted for the solution of one-dimensional problem in the presence of heat sources. The Laplace transform technique is used. A numerical method is employed for the inversion of the Laplace transforms. According to the numerical results and its graphs, conclusion about the new theory has been constructed. Some comparisons have been shown in figures to estimate the effects of the fractional order parameter on all the studied fields.