Analysis of differential equations of fractional order

This paper provides a robust convergence checking method for nonlinear differential equations of fractional order with consideration of homotopy perturbation technique. The differential operators are taken in the Caputo sense. Some theorems to prove the existence and uniqueness of the series solutions are presented. Results show that the proposed theoretical analysis is accurate.     

Solvability for p-Laplacian generalized fractional coupled systems with two-sided memory effects

This paper determines the solvability of multipoint boundary value problems for p-Laplacian generalized fractional differential systems with Riesz–Caputo derivative, which exhibits two-sided nonlocal memory effects. An equivalent integral form for the generalized fractional differential system is deduced by transformation. First, we obtain the existence of solutions on the basis of the upper–lower solutions method, in which an explicit iterative approach for approximating the solution is established. Second, we deal with a special case of our fractional differential system; in order to obtain novel results, an abstract sum-type operator equation A(x,x)+Bx+e=x on ordered Banach space is discussed. Without requiring the existence of upper–lower solutions or compactness conditions, we get several unique results of solutions for this operator equation, which provide new inspiration for the study of boundary value problems. Then, we apply these abstract results to get the uniqueness of solutions for our differential system.     

Existence theorems and Hyers-Ulam stability for a class of Hybrid fractional differential equations with $p$-Laplacian operator

In this paper, we prove necessary conditions for existence and uniqueness of solution (EUS) as well Hyers-Ulam stability for a class of hybrid fractional differential equations (HFDEs) with $p$-Laplacian operator. For these aims, we take help from topological degree theory and Leray Schauder-type fixed point theorem. An example is provided to illustrate the results.     

Analysis of fractional order differential coupled systems

In this paper, we investigate the existence of solutions to nonlinear fractional order differential coupled systemswith the classical nonlocal initial conditions.We introduce a useful vector norm, named β·B‐vector norm,which is not only a novelty but also provides another way to deal with a large number of problems not limit to integer and noninteger differential systems and singular integral systems. Copyright © 2014 John Wiley & Sons, Ltd.     

Stability by linear approximation and the relation between the stability of difference and differential fractional systems

The paper is devoted to study the stability of nonlinear fractional order difference systems by their linear approximation. Additionally, we show the relation between the stability of linear fractional order differential systems and their discretizations. Copyright © 2016 John Wiley & Sons, Ltd.     

Stability analysis of impulsive fractional differential systems with delay

In this paper, a class of impulsive fractional differential systems with finite delay is considered. Some sufficient conditions for the finite-time stability of above systems are obtained by using generalized Bellman–Gronwall’s inequality, which extend some known results.     

On solvability of differential equations with the Riesz fractional derivative

We consider the solvability of fractional differential equations involving the Riesz fractional derivative. Our approach basically relies on the reduction of the problem considered to the equivalent nonlinear mixed Volterra and Cauchy-type singular integral equation and on the theory of fractional calculus. By establishing a compactness property of the Riemann–Liouville fractional integral operator on Lebesgue spaces and using the well-known Krasnoselskii's fixed point theorem, an existence of at least one solution is gleaned. An example is finally included to show the applicability of the theory.     

New predictor-corrector approach for nonlinear fractional differential equations: error analysis and stability

In this paper, the predictor-corrector approach is used to propose two algorithms for the numerical solution of linear and non-linear fractional differential equations (FDE). The fractional order derivative is taken to be in the sense of Caputo and its properties are used to transform FDE into a Volterra-type integral equation. Simpson''s 3/8 rule is used to develop new numerical schemes to obtain the approximate solution of the integral equation associated with the given FDE. The error and stability analysis for the two methods are presented. The proposed methods are compared with the ones available in the literature. Numerical simulation is performed to demonstrate the validity and applicability of both the proposed techniques. As an application, the problem of dynamics of the new fractional order non-linear chaotic system introduced by Bhalekar and Daftardar-Gejji is investigated by means of the obtained numerical algorithms.     

On stability of equilibrium points in nonlinear fractional differential equations and fractional Hamiltonian systems

In this article, a brief stability analysis of equilibrium points in nonlinear fractional order dynamical systems is given. Then, based on the first integral concept, a definition of planar Hamiltonian systems with fractional order introduced. Some interesting properties of these fractional Hamiltonian systems are also presented. Finally, we illustrate two examples to see the differences between fractional Hamiltonian systems with their classical order counterparts.© 2014 Wiley Periodicals, Inc. Complexity 21: 93–99, 2015     

On a nonlinear distributed order fractional differential equation

We study the existence and the uniqueness of mild and classical solutions for a class of equations of the form . Such equations arise in distributed derivatives models of viscoelasticity and system identification theory. We also formulate a variational principle for a more general equation based on a method of doubling of variables for such equations.     

On the concept and existence of solution for impulsive fractional differential equations

This paper is motivated from some recent papers treating the problem of the existence of a solution for impulsive differential equations with fractional derivative. We firstly show that the formula of solutions in cited papers are incorrect. Secondly, we reconsider a class of impulsive fractional differential equations and introduce a correct formula of solutions for a impulsive Cauchy problem with Caputo fractional derivative. Further, some sufficient conditions for existence of the solutions are established by applying fixed point methods. Some examples are given to illustrate the results.     

Existence and Ulam's type stability results for a class of fractional boundary value problems on a star graph

This paper is concerned with a nonlinear fractional boundary value problem on a star graph. By using a transformation, the suggested problem is converted into an equivalent system of fractional boundary value problem. Schaefer's fixed point theorem and Banach's contraction principle is used to establish its existence and uniqueness results. Further, different kinds of Ulam's type stability results for the proposed problem have been discussed. Finally, two examples are presented to illustrate the application of the obtained results.     

Stability results and existence theorems for nonlinear delay-fractional differential equations with $\varphi^*_p$-operator

The study of delay-fractional differential equations (fractional DEs) have recently attracted a lot of attention from scientists working on many different subjects dealing with mathematically modeling. In the study of fractional DEs the first question one might raise is whether the problem has a solution or not. Also, whether the problem is stable or not? In order to ensure the answer to these questions, we discuss the existence and uniqueness of solutions (EUS) and Hyers-Ulam stability (HUS) for our proposed problem, a nonlinear fractional DE with $p$-Laplacian operator and a non zero delay $\tau>0$ of order $n-1<\nu^*,\,\epsilon相似文献   

A short note on the solvability of impulsive fractional differential equations with Caputo derivatives

In this paper, we prove the existence and non-existence of solutions to two impulsive fractional differential equations with strong or weak Caputo derivatives in Euclidean space, respectively.     

Stability analysis of a coupled system of nonlinear implicit fractional anti‐periodic boundary value problem

In this article, we study the existence and uniqueness of solution for a coupled system of nonlinear implicit fractional anti‐periodic boundary value problem. Further, we investigate different kinds of stability such as Ulam‐Hyers stability, generalized Ulam‐Hyers stability, Ulam‐Hyers‐Rassias stability, and generalized Ulam‐Hyers‐Rassias stability. We develop conditions for existence and uniqueness by using the classical fixed point theorem. Also, two examples are provided to illustrate the obtained results.     

A computational matrix method for solving systems of high order fractional differential equations

In this paper, we introduced an accurate computational matrix method for solving systems of high order fractional differential equations. The proposed method is based on the derived relation between the Chebyshev coefficient matrix A of the truncated Chebyshev solution u(t)$\mathbf{u}(\mathbf{t})$ and the Chebyshev coefficient matrix A^(ν)${\mathbf{A}}^{(\nu )}$ of the fractional derivative u^(ν)${\mathbf{u}}^{(\nu )}$. The fractional derivatives are presented in terms of Caputo sense. The matrix method for the approximate solution for the systems of high order fractional differential equations (FDEs) in terms of Chebyshev collocation points is presented. The systems of FDEs and their conditions (initial or boundary) are transformed to matrix equations, which corresponds to system of algebraic equations with unknown Chebyshev coefficients. The remaining set of algebraic equations is solved numerically to yield the Chebyshev coefficients. Several numerical examples for real problems are provided to confirm the accuracy and effectiveness of the present method.     

A reliable algorithm of homotopy analysis method for solving nonlinear fractional differential equations

In this paper, based on the homotopy analysis method (HAM), a powerful algorithm is developed for the solution of nonlinear ordinary differential equations of fractional order. The proposed algorithm presents the procedure of constructing the set of base functions and gives the high-order deformation equation in a simple form. Different from all other analytic methods, it provides us with a simple way to adjust and control the convergence region of solution series by introducing an auxiliary parameter ?$?$. The analysis is accompanied by numerical examples. The algorithm described in this paper is expected to be further employed to solve similar nonlinear problems in fractional calculus.     

The convergence and stability analysis of the Jacobi collocation method for solving nonlinear fractional differential equations with integral boundary conditions

In this paper, we apply the Jacobi collocation method for solving nonlinear fractional differential equations with integral boundary conditions. Due to existence of integral boundary conditions, after reformulation of this equation in the integral form, the method is proposed for solving the obtained integral equation. Also, the convergence and stability analysis of the proposed method are studied in two main theorems. Furthermore, the optimum degree of convergence in the L₂ norm is obtained for this method. Furthermore, some numerical examples are presented in order to illustrate the performance of the presented method. Finally, an application of the model in control theory is introduced. Copyright © 2015 John Wiley & Sons, Ltd.     

Existence of solutions of initial value problems for nonlinear fractional differential equations

In this paper, by using the Schauder fixed point theorem, we study the existence and uniqueness of solutions of initial value problems for nonlinear fractional differential equations and obtain some new results.