Solving FDEs with Caputo‐Fabrizio derivative by operational matrix based on Genocchi polynomials

Abstract

We introduce a new approach to solve a type of fractional order differential equations without singularity. For fractional integration, we obtain the operational matrix through Genocchi polynomials. Some examples are presented to test the applicability and efficiency of the technique.

A Galerkin FEM for Riesz space-fractional CNLS

“…The test results indicate the soliton shapes are greatly changed with α, while the conserved quantities are well preserved. It is noteworthy that the problem discussed here can possibly be solved by other numerical methods like the lattice fractional equation methods in [28,29], operational matrix methods in [57,58], or the fourth-order nonstandard compact and sixth-order weighted essentially non-oscillatory (WENO) finite difference methods in [59,60], which can reduce computing effort by converting the original problem to a system of differential/algebraic equations, or achieving high-order convergence without spurious oscillations. In future works, we will use the Riesz fractional differences and operational matrix techniques to deal with Riesz derivative or apply the ideas of the high-order nonstandard compact and WENO schemes in the above citations to derive more efficient numerical methods for Eqs.…”

Distributed Control for Time-Fractional Differential System Involving Schrödinger Operator

“…Now, with the help of Lemmas 6 and 7, and Lax-Milgram lemma, we state and prove the existence and uniqueness theorems for the solution of problem (8).…”

“…Using the general theory of Lions [14], the following result gives the existence and uniqueness of optimal control. Moreover it gives the first order necessary and sufficient optimality conditions for the distributed optimal control problem (1) and (2), formulated to the time-fractional differential system (8).…”

“…In [7], the numerical solution for the time-fractional advectiondiffusion problem in one-dimension with the initial boundary condition is studied. A new approach to solving a type of fractional order differential equations without singularity is introduced in [8]. In [9], a type of Fokker-Planck equation (FPE) with Caputo-Fabrizio fractional derivative is considered; they present a numerical approach which is based on the Ritz method with known basis functions to transform this equation into an optimization problem.…”

Analytical solutions of linear inhomogeneous fractional differential equation with continuous variable coefficients

“…Currently, methods for solving FDEs with initial conditions can be classified into two classes, namely, approximative method and analytical methods. Typical approximative methods include the operational matrix method based on orthogonal functions, the predictor-corrector method, fractional Euler method, and so on ( [11][12][13][14][15][16]). The most practical analytical methods are the Adomian decomposition method, the homotopy analysis method, the homotopy perturbation method, the Laplace transform method, and the variational iteration method ( [17][18][19][20][21]).…”