Related papers: A new Laplace-type fractional derivative
The purpose of this work is to study a finite element method for finding solutions to the eigenvalue problem for the fractional Laplacian. We prove that the discrete eigenvalue problem converges to the continuous one and we show the order…
The L-fractional derivative is defined as a certain normalization of the well-known Caputo derivative, so alternative properties hold: smoothness and finite slope at the origin for the solution, velocity units for the vector field, and a…
The aim of this paper is to study certain problems of calculus of variations, that are dependent upon a Lagrange function on a Caputo-type fractional derivative. This type of fractional operator is a generalization of the Caputo and the…
Main results and techniques of the fractional calculus of variations are surveyed. We consider variational problems containing Caputo derivatives and study them using both indirect and direct methods. In particular, we provide necessary…
In this work, we propose an efficient finite element method for solving fractional Sturm-Liouville problems involving either the Caputo or Riemann-Liouville derivative of order $\alpha\in(1,2)$ on the unit interval $(0,1)$. It is based on…
A fractional diffusion equation based on Riemann-Liouville fractional derivatives is solved exactly. The initial values are given as fractional integrals. The solution is obtained in terms of $H$-functions. It differs from the known…
We study the inverse problem for the fractional Laplace equation with multiple nonlinear lower order terms. We show that the direct problem is well-posed and the inverse problem is uniquely solvable. More specifically, the unknown…
Recently, it was found that a new set of simple techniques allow one to conveniently express ordinary integrals through differentiation. These techniques add to the general toolbox for integration and integral transforms such as the Fourier…
We introduce a new fractional derivative that generalizes the so-called alternative fractional derivative recently proposed by Katugampola. We denote this new differential operator by $\mathscr{D}_{M}^{\alpha,\beta }$, where the parameter…
This paper is in continuation of the authors' recently published paper (Journal of Mathematical Physics 55(2014)083519) in which computational solutions of an unified reaction-diffusion equation of distributed order associated with Caputo…
A fractional Hamiltonian formalism is introduced for the recent combined fractional calculus of variations. The Hamilton-Jacobi partial differential equation is generalized to be applicable for systems containing combined Caputo fractional…
Using the new conformable fractional derivative, which differs from the Riemann-Liouville and Caputo fractional derivatives, we reformulate the second-order conjugate boundary value problem in this new setting. Utilizing the corresponding…
This paper deals with the investigation of the computational solutions of an unified fractional reaction-diffusion equation, which is obtained from the standard diffusion equation by replacing the time derivative of first order by the…
We establish a Liouville type theorem for the fractional Lane-Emden system: \begin{eqnarray*} \left\{\begin{array}{l@{\quad }l} (-\Delta)^\alpha u=v^q&{\rm in}\,\,\R^N,\\ (-\Delta)^\alpha v=u^p&{\rm in}\,\,\R^N, \end{array} \right.…
We establish pointwise formulas for the shape derivative of solutions to the Dirichlet problem associated with the fractional Laplacian. Specifically, we consider the equation $(-\Delta)^s u = h$ in $\Omega$ and $u=0$ in $\Omega^c$, where…
We introduce a discrete-time fractional calculus of variations on the time scales $\mathbb{Z}$ and $(h\mathbb{Z})_a$. First and second order necessary optimality conditions are established. Some numerical examples illustrating the use of…
A recent development in the theory of fractional differential equations with variable coefficients has been a method for obtaining an exact solution in the form of an infinite series involving nested fractional integral operators. This…
In this paper, the fractal calculus of fractal sets and fractal curves are compared. The analogues of the Riemann-Liouville and the Caputo integrals and derivatives are defined for the fractal curves which are non-local derivatives. The…
This paper is a continuation of our earlier paper in which we have derived the solution of an unified fractional reaction-diffusion equation associated with the Caputo derivative as the time-derivative and the Riesz-Feller fractional…
We present a new discretization for the Hadamard fractional derivative, that simplifies the computations. We then apply the method to solve a fractional differential equation and a fractional variational problem with dependence on the…