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].     

On the existence and stability of fuzzy CF variable fractional differential equation for COVID-19 epidemic

In this paper, we convert the recent COVID-19 model with the use of the most influential theories, such as variable fractional calculus and fuzzy theory. We propose the fuzzy variable fractional differential equation for the COVID-19 model in which the variable fractional-order derivative is described using the Caputo-Fabrizio in the Caputo sense. Furthermore, we provide the results on the existence and uniqueness using Lipschitz conditions. Also, discuss the stability analysis of the present new COVID-19 model by employing Hyers-Ulam stability.

     

广义Maxwell流体分数阶微分方程的数值解法

针对广义Maxwell粘弹性流体分数阶微分方程,建立了一种隐式差分格式,给出了数值解的求解公式,证明了隐式差分格式稳定性与收敛性。     

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.     

On the existence of solutions of the Riccati differential equation

We generalize a technique given in C. Martin [1], to obtain a characterization of finite escape times for time-varying Riccati equations which also apply to the non-definite case.     

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.     

Numerical solution of fractional delay differential equation by shifted Jacobi polynomials

In this paper, the fractional delay differential equation (FDDE) is considered for the purpose to develop an approximate scheme for its numerical solutions. The shifted Jacobi polynomial scheme is used to solve the results by deriving operational matrix for the fractional differentiation and integration in the Caputo and Riemann–Liouville sense, respectively. In addition to it, the Jacobi delay coefficient matrix is developed to solve the linear and nonlinear FDDE numerically. The error of the approximate solution of proposed method is discussed by applying the piecewise orthogonal technique. The applicability of this technique is shown by several examples like a mathematical model of houseflies and a model based on the effect of noise on light that reflected from laser to mirror. The obtained numerical results are tabulated and displayed graphically.     

The existence uniqueness and the fixed iterative algorithm of the solution for the discrete coupled algebraic Riccati equation

In this article, applying the properties of M-matrix and non-negative matrix, utilising eigenvalue inequalities of matrix's sum and product, we firstly develop new upper and lower matrix bounds of the solution for discrete coupled algebraic Riccati equation (DCARE). Secondly, we discuss the solution existence uniqueness condition of the DCARE using the developed upper and lower matrix bounds and a fixed point theorem. Thirdly, a new fixed iterative algorithm of the solution for the DCARE is shown. Finally, the corresponding numerical examples are given to illustrate the effectiveness of the developed 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.     

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.     

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.     

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