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In this paper, we study a new kind of nonlocal boundary value problems of nonlinear fractional differential equations supplemented with Erdélyi-Kober type fractional integral conditions. The uniqueness of solutions for the given problem is established by means of contraction mapping principle. Applying nonlinear alternative for contractive maps, we investigate the inclusions case of the problem at hand. Examples illustrating the main results are constructed as well. [ABSTRACT FROM AUTHOR]

In this paper, we develop the theory of hybrid differential equations on time scales. An existence theorem for hybrid differential equations on time scales is given under Lipschitz conditions. Some fun- damental fractional differential inequalities are also established which are utilized to prove the existence of extremal solutions. Necessary tools are considered and the comparison principle is proved which will be useful for further study of qualitative behavior of solutions. Our results in this paper extend and improve some known results. [ABSTRACT FROM AUTHOR]

In this paper, we study a coupled system of nonlinear fractional differential equations with multi-point boundary conditions. The differential operator is taken in the Riemann-Liouville sense. Applying the Schauder fixed-point theorem and the contraction mapping principle, two existence results are obtained for the following system {D0+αx(t) = f (t, y(t), D0+py(t)), t ∈ (0, 1), {D0+βy(t) = g (t, x(t), D0+qx(t)), t ∈ (0, 1), {x(0) = x'(0) = x''(0) = ⋯ = x(m-2)(0) = 0, x(1) = λx(ξ), {y(0) = y'(0) = y''(0) = ⋯ = y(m-2)(0) = 0, y(1) = λy(ξ), where 0 < ξ < 1, m ∈ N; m ≥ 2, α, β ∈ (m-1, m) and α, β, p, q, λ satisfy certain conditions. [ABSTRACT FROM AUTHOR]

Variational methods and critical point theorems are used to discuss existence and multiplicity of solutions for fractional boundary value problem where Riemann-Liouville fractional derivatives and Caputo fractional derivatives are used. Some conditions to determinate nonnegative solutions are presented. An example is given to illustrate our results. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence of solutions for a fractional boundary value problem. By using critical point theory and variational methods, we give some new criteria to guarantee that the problems have at least one solution and infinitely many solutions. [ABSTRACT FROM AUTHOR]

In this paper, we consider a new study about fractional Δ-difference equations. We consider two special classes of Sturm-Liouville problems equipped with fractional Δ-difference operators. In couple of steps, the Lyapunov type inequalities for both classes will be obtained. As application, some qualitative behaviour of mentioned fractional problems such as stability, spectral, disconjugacy and some interesting results about zeros of (oscillatory) solutions will be concluded. [ABSTRACT FROM AUTHOR]

In this paper, operational matrices of Riemann-Liouville fractional integration and Caputo fractional differentiation for shifted Jacobi polynomials are considered. Using the given initial conditions, we trans-form the fractional differential equation (FDE) into a modified fractional differential equation with zero initial conditions. Next, all the existing functions in modified differential equation are approximated by shifted Jacobi polynomials. Then, operational matrices and spectral collocation method are applied to obtain a linear or nonlinear system of algebraic equations. System of algebraic equations can be simultaneously solved (e.g. using MathematicaTM). Main characteristic behind of the this technique is that only a small number of shifted Jacobi polynomials is needed to obtain a satisfactory result which demonstrates the validity and efficiency of the method. Comparison between this method and some other methods confirm the good performance of the presented method. Also, this method is generalized for the multi-point fractional differential equation. [ABSTRACT FROM AUTHOR]

In this paper we apply hybrid functions of general block-pulse functions and Legendre polynomials for solving linear and nonlinear multi-order fractional differential equations (FDEs). Our approach is based on incorporating operational matrices of FDEs with hybrid functions that reduces the FDEs problems to the solution of algebraic systems. Error estimate that verifies a convergence of the approximate solutions is considered. The numerical results obtained by this scheme have been compared with the exact solution to show the efficiency of the method. [ABSTRACT FROM AUTHOR]

It is commonly accepted that fractional differential equations play an important role in the explanation of many physical phenomena. For this reason we need a reliable and efficient technique for the solution of fractional differential equations. This paper deals with the numerical solution of a class of fractional differential equation. The fractional derivatives are described based on the Caputo sense. Our main aim is to generalize the Chebyshev cardinal operational matrix to the fractional calculus. In this work, the Chebyshev cardinal functions together with the Chebyshev cardinal operational matrix of fractional derivatives are used for numerical solution of a class of fractional differential equations. The main advantage of this approach is that it reduces fractional problems to a system of algebraic equations. The method is applied to solve nonlinear fractional differential equations. Illustrative examples are included to demonstrate the validity and applicability of the presented technique. [ABSTRACT FROM AUTHOR]

In this paper, we study the boundary-value problem of frac- tional order dynamic equations on time scales, cΔαu(t) = f(t; u(t)); t 2 [0; 1]Tκ2 := J; 1 < α < 2; u(0) + uΔ(0) = 0; u(1) + uΔ(1) = 0; where 享 is a general time scale with 0; 1 ∈ 享, c?α is the Caputo ?- fractional derivative. We investigate the existence and uniqueness of so- lution for the problem by Banach's fixed point theorem and Schaefer's fixed point theorem. We also discuss the existence of positive solutions of the problem by using the Krasnoselskii theorem. [ABSTRACT FROM AUTHOR]

In this paper, a spectral Tau method for solving fractional Riccati differential equations is considered. This technique describes converting of a given fractional Riccati differential equation to a system of nonlinear algebraic equations by using some simple matrices. We use fractional derivatives in the Caputo form. Convergence analysis of the proposed method is given and rate of convergence is established in the weighted L2-norm. Numerical results are presented to confirm the high accuracy of the method. [ABSTRACT FROM AUTHOR]