On the global existence of solutions to a class of fractional differential equations

We present two global existence results for an initial value problem associated to a large class of fractional differential equations. Our approach differs substantially from the techniques employed in the recent literature. By introducing an easily verifiable hypothesis, we allow for immediate applications of a general comparison type result from [V. Lakshmikantham, A.S. Vatsala, Basic theory of fractional differential equations, Nonlinear Anal. TMA 69 (2008), 2677–2682].     

Existence of a solution for the fractional differential equation with nonlinear boundary conditions

Using the method of upper and lower solutions and its associated monotone iterative, we present an existence theorem for a nonlinear fractional differential equation with nonlinear boundary conditions.     

The existence of a solution for a fractional differential equation with nonlinear boundary conditions considered using upper and lower solutions in reverse order

Using the method of upper and lower solutions in reverse order, we present an existence theorem for a linear fractional differential equation with nonlinear boundary conditions.     

New matrix bounds,an existence uniqueness and a fixed-point iterative algorithm for the solution of the unified coupled algebraic Riccati equation

In this paper, combining the equivalent form of the unified coupled algebraic Riccati equation (UCARE) with the eigenvalue inequalities of a matrix's sum and product, using the properties of an M-matrix and its inverse matrix, we offer new lower and upper matrix bounds for the solution of the UCARE. Furthermore, applying the derived lower and upper matrix bounds and a fixed-point theorem, an existence uniqueness condition of the solution of the UCARE is proposed. Then, we propose a new fixed-point iterative algorithm for the solution of the UCARE. Finally, we present a corresponding numerical example to demonstrate the effectiveness of our results.     

On some qualitative properties for solutions of a certain two-dimensional fractional differential systems

By means of the modification of Medve?’s de-singular method and a result of two-dimensional linear integral inequalities, components-wise (not on some norm) upper bounds are obtained for solutions of a class of nonlinear two-dimensional systems of fractional differential equations. The uniqueness and continuous dependence of the solutions are also discussed here.     

Approximate solution of the fractional advection-dispersion equation

In this paper, we consider practical numerical method to solve a space-time fractional advection-dispersion equation with variable coefficients on a finite domain. The equation is obtained from the standard advection-dispersion equation by replacing the first-order time derivative by the Caputo fractional derivative, and the first-order and second-order space derivatives by the Riemann-Liouville fractional derivative, respectively. Here, a new method for solving this equation is proposed in the reproducing kernel space. The representation of solution is given by the form of series and the n-term approximation solution is obtained by truncating the series. The method is easy to implement and the numerical results show the accuracy of the method.     

On the topological structure of attraction basins for differential inclusions

We show that when a compact set is globally asymptotically stable under the action of a differential inclusion satisfying certain regularity properties, there exists a smooth differential equation rendering the same compact set globally asymptotically stable. The regularity properties assumed in this work stem from the consideration of Krasovskii/Filippov solutions to discontinuous differential equations and the robustness of asymptotic stability under perturbation. In particular, the results in this work show that when a compact set cannot be globally asymptotically stabilized by continuous feedback due to topological obstructions, it cannot be robustly globally asymptotically stabilized by discontinuous feedback either. The results follow from converse Lyapunov theory and parallel what is known for the local stabilization problem.     

A new stochastic approach for solution of Riccati differential equation of fractional order

In this article, a stochastic technique has been developed for the solution of nonlinear Riccati differential equation of fractional order. Feed-forward artificial neural network is employed for accurate mathematical modeling and learning of its weights is made with heuristic computational algorithm based on swarm intelligence. In this scheme, particle swarm optimization is used as a tool for the rapid global search method, and simulating annealing for efficient local search. The scheme is equally capable of solving the integer order or fractional order Riccati differential equations. Comparison of results was made with standard approximate analytic, as well as, stochastic numerical solvers and exact solutions.     

Toward the existence and uniqueness of solutions of second-order fuzzy differential equations

In this paper the existence and uniqueness of solutions for second-order fuzzy differential equations with initial conditions under generalized H-differentiability is proved. To this end, the concept of second-order generalized differential equation is defined, which is based on an enlargement of the class of differentiable fuzzy mappings.     

On the existence and uniqueness of solutions in adaptive controlsystems

The issue of existence and uniqueness of solutions, which arises in the analysis of adaptive control systems, is investigated. Sufficient conditions that guarantee existence and uniqueness in positive time of solutions to differential equations describing a wide class of adaptive schemes are given. The concept of Filippov's solution is introduced as a method of dealing with discontinuous adaptive systems     

On the numerical solution of a nonlinear partial differential equation related to the optimal control of a noisy oscillator

This paper deals with the optimal control of a linear oscillator with parametric excitation. It is shown that, in order to implement the optimal feedback control law, a nonlinear partial differential equation has to be solved. A finite difference algorithm for the solution of this equation is proposed, and its efficiency and applicability are demonstrated with examples.     

Existence and uniqueness of mild solutions for abstract delay fractional differential equations

In this paper, by means of solution operator approach and contraction mapping theorem, the existence and uniqueness of mild solutions for a class of abstract delay fractional differential equations are obtained.     

Existence, uniqueness and asymptotic behavior of the solutions of a fuzzy differential equation with piecewise constant argument

In this paper we study the existence, the uniqueness and the asymptotic behavior of the solutions of the fuzzy differential equation of the form:

     

On uniqueness of solution to the multi-peg towers of hanoi

The multi-peg Towers of Hanoi problem is still open. No provably optimal constructive algorithm to solve the problem is known. The minimum number of moves required is also unknown. Though optimal solutions are observed to be non-unique, the exact uniqueness criteria of an optimal solution are little known to the literature. This paper revisits a deterministic algorithm conjectured to optimally solve the problem and addresses the uniqueness characteristics of an optimal solution.