We study calculus of variations problems, where the Lagrange function depends on the Caputo-Katugampola fractional derivative. This type of fractional operator is a generalization of the Caputo and the Caputo–Hadamard fractional derivatives, with dependence on a real parameter ρ. We present sufficient and necessary conditions of first and second order to determine the extremizers of a functiona...

In this paper, we concerned with positive solutions for higher order m-point nonlinear fractional boundary value problems with integral boundary conditions. We establish the criteria for the existence of at least one, two and three positive solutions for higher order m-point nonlinear fractional boundary value problems with integral boundary conditions by using a result from the theory of fixed...

and Applied Analysis 3 Subject to the initial condition D α−k 0 U (x, 0) = f k (x) , (k = 0, . . . , n − 1) , D α−n 0 U (x, 0) = 0, n = [α] , D k 0 U (x, 0) = g k (x) , (k = 0, . . . , n − 1) , D n 0 U (x, 0) = 0, n = [α] , (11) where ∂α/∂tα denotes the Caputo or Riemann-Liouville fraction derivative operator, f is a known function, N is the general nonlinear fractional differential operator, a...

Keywords: Nonlocal boundary value problem Caputo's fractional derivative Fractional integral Fixed point theorems a b s t r a c t In the light of the fixed point theorems, we analytically establish the conditions for the uniqueness of solutions as well as the existence of at least one solution in the nonlocal boundary value problem for a specific kind of nonlinear fractional differential equati...

In this article, the electro-osmotic flow of Oldroyd-B fluid in a circular micro-channel with slip boundary condition is considered. The corresponding fractional system is represented by using a newly defined time-fractional Caputo-Fabrizio derivative without singular kernel. Closed form solutions for the velocity field are acquired by means of Laplace and finite Hankel transforms. Additionally...

This paper builds on the notion of the so-called orthogonal derivative, where an n-th order derivative is approximated by an integral involving an orthogonal polynomial of degree n. This notion was reviewed in great detail in a paper by the author and Koornwinder in 2012. Here, an approximation of the Weyl or Riemann–Liouville fractional derivative is considered by replacing the n-th derivative...

The aim of this paper is to extend the split-step idea for the solution of fractional partial differential equations. We consider the multidimensional nonlinear Schr"{o}dinger equation with the Riesz space fractional derivative and propose an efficient numerical algorithm to obtain it's approximate solutions. To this end, we first discretize the Riesz fractional derivative then apply the Crank-...

Abstract-To the authors’ knowledge, previous derivations of the fractional diffusion equation are based on stochastic principles [l], with the result that physical interpretation of the resulting fractional derivatives has been elusive [2]. Herein, we develop a fmctional J%Z Iaw relating solute flux at a given point to what might be called the complete (twosided) fractional derivative of the co...

a method for solving a class of weakly singular volterra integral equations is given by using the fractional differential transform method. the approximate solution of these equa­tions is calculated in the form of a finite series with easily computable terms. while in some examples this series solution increased up to the exact closed solution, in some other examples, we can see the accuracy an...

In this paper, a new fractional sub-equation method is proposed for finding exact solutions of fractional partial differential equations (FPDEs) in the sense of modified Riemann-Liouville derivative. With the aid of symbolic computation, we choose the space-time fractional Zakharov-Kuznetsov-Benjamin-Bona-Mahony (ZKBBM) equation in mathematical physics with a source to illustrate the validity a...