scholarly journals ON A SECOND-ORDER DIFFERENTIAL PROBLEM WITH FRACTIONAL DERIVATIVES OF ORDER GREATER THAN ONE

2013 ◽  
Vol 18 (1) ◽  
pp. 53-65
Author(s):  
Nasser-eddine Tatar
Keyword(s):  
Existence And Uniqueness ◽  
Present Author ◽  
Fractional Derivatives ◽  
Second Order ◽  
Classical Solutions ◽  
Differential Problem ◽  
Abstract Problem ◽  
Integer Order ◽  
First Order ◽  
Derivatives Of

A second-order abstract problem with derivatives of non-integer order is investigated. The nonlinearity involves fractional derivatives between 1 and 2. Existence and uniqueness of mild and classical solutions are established in appropriate spaces. This work extends similar works with or without a derivative of first order and also a work of the present author, where the order of the derivatives were between 0 and 1.

scholarly journals Semilinear Volterra Integrodifferential Problems with Fractional Derivatives in the Nonlinearities

2011 ◽  
Vol 2011 ◽  
pp. 1-11 ◽  
Author(s):  
Mokhtar Kirane ◽  
Milan Medveď ◽  
Nasser-eddine Tatar
Keyword(s):  
Integrodifferential Equation ◽  
Existence And Uniqueness ◽  
Fractional Derivatives ◽  
Mild Solutions ◽  
Second Order ◽  
Classical Solutions ◽  
Volterra Integrodifferential Equation

A second-order semilinear Volterra integrodifferential equation involving fractional time derivatives is considered. We prove existence and uniqueness of mild solutions and classical solutions in appropriate spaces.

scholarly journals A new second-order midpoint approximation formula for Riemann–Liouville derivative: algorithm and its application

2017 ◽  
Vol 82 (5) ◽  
pp. 909-944 ◽  
Author(s):  
Hengfei Ding ◽  
Changpin Li ◽  
Qian Yi
Keyword(s):  
Fractional Derivatives ◽  
Second Order ◽  
Stability And Convergence ◽  
Approximation Formula ◽  
Finite Difference Schemes ◽  
First Order ◽  
Liouville Derivative ◽  
Two Space Dimensions ◽  
Fractional Differential ◽  
Nicolson Scheme

Abstract Compared to the classical first-order Grünwald–Letnikov formula at time $t_{k+1}\; (\text{or}\; t_{k})$, we firstly propose a second-order numerical approximate formula for discretizing the Riemann–Liouvile derivative at time $t_{k+\frac{1}{2}}$, which is very suitable for constructing the Crank–Nicolson scheme for the fractional differential equations with time fractional derivatives. The established formula has the following form RLD0,tαu(t)| t=tk+12=τ−α∑ℓ=0kϖℓ(α)u(tk−ℓτ)+O(τ2),k=0,1,…,α∈(0,1), where the coefficients $\varpi_{\ell}^{(\alpha)}$$(\ell=0,1,\ldots,k)$ can be determined via the following generating function G(z)=(3α+12α−2α+1αz+α+12αz2)α,|z|<1. Next, applying the formula to the time fractional Cable equations with Riemann–Liouville derivative in one and two space dimensions. Then the high-order compact finite difference schemes are obtained. The solvability, stability and convergence with orders $\mathcal{O}(\tau^2+h^4)$ and $\mathcal{O}(\tau^2+h_x^4+h_y^4)$ are shown, where $\tau$ is the temporal stepsize and $h$, $h_x$, $h_y$ are the spatial stepsizes, respectively. Finally, numerical experiments are provided to support the theoretical analysis.

scholarly journals Existence of Solutions ofα∈(2,3]Order Fractional Three-Point Boundary Value Problems with Integral Conditions

2014 ◽  
Vol 2014 ◽  
pp. 1-12 ◽  
Author(s):  
N. I. Mahmudov ◽  
S. Unul
Keyword(s):  
Fixed Point ◽  
Existence And Uniqueness ◽  
Fractional Derivatives ◽  
Fixed Point Theorems ◽  
Existence Of Solutions ◽  
Point Boundary ◽  
Integral Conditions ◽  
Uniqueness Of Solutions ◽  
Fractional Differential ◽  
Derivatives Of

Existence and uniqueness of solutions forα∈(2,3]order fractional differential equations with three-point fractional boundary and integral conditions involving the nonlinearity depending on the fractional derivatives of the unknown function are discussed. The results are obtained by using fixed point theorems. Two examples are given to illustrate the results.

scholarly journals Method for Evaluating Fractional Derivatives of Fractional Functions

2020 ◽  
pp. 286-290
Author(s):  
Chii-Huei Yu
Keyword(s):  
Power Series ◽  
Fractional Derivatives ◽  
Fractional Power ◽  
Series Method ◽  
Power Series Method ◽  
Differential Problem ◽  
Fractional Differential ◽  
Derivatives Of ◽  
Fractional Power Series

This paper studies the fractional differential problem of fractional functions, regarding the modified Riemann-Liouvellie (R-L) fractional derivatives. A new multiplication and the fractional power series method are used to obtain any order fractional derivatives of some elementary fractional functions.

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