On Time Fractional Derivatives in Fractional Sobolev Spaces and Applications to Fractional Ordinary Differential Equations

2021 ◽  
pp. 287-308
Author(s):  
Masahiro Yamamoto
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Sobolev Spaces ◽  
Fractional Derivatives ◽  
Fractional Sobolev Spaces

A GENERAL COMPARISON PRINCIPLE FOR CAPUTO FRACTIONAL-ORDER ORDINARY DIFFERENTIAL EQUATIONS

Fractals ◽  
2020 ◽  
Vol 28 (04) ◽  
pp. 2050070 ◽  
Author(s):  
CONG WU
Keyword(s):  
Differential Equations ◽  
Fractional Derivative ◽  
Ordinary Differential Equations ◽  
Fractional Order ◽  
Comparison Principle ◽  
Caputo Fractional Derivative ◽  
Fractional Derivatives ◽  
Fractional Order Systems ◽  
Caputo Fractional Derivatives ◽  
Full Result

In this paper, we work on a general comparison principle for Caputo fractional-order ordinary differential equations. A full result on maximal solutions to Caputo fractional-order systems is given by using continuation of solutions and a newly proven formula of Caputo fractional derivatives. Based on this result and the formula, we prove a general fractional comparison principle under very weak conditions, in which only the Caputo fractional derivative is involved. This work makes up deficiencies of existing results.

On the decomposition of solutions: From fractional diffusion to fractional Laplacian

2021 ◽  
Vol 24 (5) ◽  
pp. 1571-1600
Author(s):  
Yulong Li
Keyword(s):  
Differential Equations ◽  
Diffusion Equation ◽  
Ordinary Differential Equations ◽  
Laplace Equation ◽  
Fractional Derivatives ◽  
Fractional Laplacian ◽  
Fractional Diffusion ◽  
Variable Coefficients ◽  
Deep Insight ◽  
Structure Of Solutions

Abstract This paper investigates the structure of solutions to the BVP of a class of fractional ordinary differential equations involving both fractional derivatives (R-L or Caputo) and fractional Laplacian with variable coefficients. This family of equations generalize the usual fractional diffusion equation and fractional Laplace equation. We provide a deep insight to the structure of the solutions shared by this family of equations. The specific decomposition of the solution is obtained, which consists of the “good” part and the “bad” part that precisely control the regularity and singularity, respectively. Other associated properties of the solution will be characterized as well.

scholarly journals On a new class of fractional partial differential equations

2015 ◽  
Vol 8 (4) ◽  
Author(s):  
Tien-Tsan Shieh ◽  
Daniel E. Spector
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Sobolev Spaces ◽  
Variational Problems ◽  
Fractional Sobolev Spaces ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
New Class ◽  
Natural Way

AbstractIn this paper, we study a new class of fractional partial differential equations which are obtained by minimizing variational problems in fractional Sobolev spaces. We introduce a notion of fractional gradient which has the potential to extend many classical results in the Sobolev spaces to the nonlocal and fractional setting in a natural way.

Initialization of Fractional-Order Operators and Fractional Differential Equations

2008 ◽  
Vol 3 (2) ◽  
Author(s):  
Carl F. Lorenzo ◽  
Tom T. Hartley
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Fractional Differential Equations ◽  
Fractional Derivatives ◽  
Fractional Integral ◽  
Laplace Transforms ◽  
Time Varying ◽  
Fractional Operators ◽  
Fractional Differential ◽  
Derivatives Of

It has been known that the initialization of fractional operators requires time-varying functions, a complicating factor. This paper simplifies the process of initialization of fractional differential equations by deriving Laplace transforms for the initialized fractional integral and derivative that generalize those for the integer-order operators. The new transforms unify the initialization of systems of fractional and ordinary differential equations. The paper provides background on past work in the area and determines the Laplace transforms for the initialized fractional integral and fractional derivatives of any (real) order. An application provides insight and demonstrates the theory.

scholarly journals Fractional Complex Transform and exp-Function Methods for Fractional Differential Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Ahmet Bekir ◽  
Özkan Güner ◽  
Adem C. Cevikel
Keyword(s):  
Differential Equations ◽  
Exact Solutions ◽  
Ordinary Differential Equations ◽  
Fractional Differential Equations ◽  
Function Method ◽  
Fractional Derivatives ◽  
Fractional Complex Transform ◽  
Fractional Equations ◽  
New Exact Solutions ◽  
Fractional Differential

The exp-function method is presented for finding the exact solutions of nonlinear fractional equations. New solutions are constructed in fractional complex transform to convert fractional differential equations into ordinary differential equations. The fractional derivatives are described in Jumarie's modified Riemann-Liouville sense. We apply the exp-function method to both the nonlinear time and space fractional differential equations. As a result, some new exact solutions for them are successfully established.

scholarly journals Fractional Sobolev Spaces via Riemann-Liouville Derivatives

2013 ◽  
Vol 2013 ◽  
pp. 1-15 ◽  
Author(s):  
Dariusz Idczak ◽  
Stanisław Walczak
Keyword(s):  
Boundary Value Problems ◽  
Sobolev Spaces ◽  
Fractional Derivatives ◽  
Boundary Value ◽  
Fractional Sobolev Spaces

Using Riemann-Liouville derivatives, we introduce fractional Sobolev spaces, characterize them, define weak fractional derivatives, and show that they coincide with the Riemann-Liouville ones. Next, we prove equivalence of some norms in the introduced spaces and derive their completeness, reflexivity, separability, and compactness of some imbeddings. An application to boundary value problems is given as well.

scholarly journals The Analytical Solution of Some Fractional Ordinary Differential Equations by the Sumudu Transform Method

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Hasan Bulut ◽  
Haci Mehmet Baskonus ◽  
Fethi Bin Muhammad Belgacem
Keyword(s):  
Analytical Solution ◽  
Differential Equations ◽  
Fractional Calculus ◽  
Ordinary Differential Equations ◽  
Fractional Derivatives ◽  
General Setting ◽  
Two Dimensional ◽  
Transform Method ◽  
Symbolic Algebra ◽  
Sumudu Transform

We introduce the rudiments of fractional calculus and the consequent applications of the Sumudu transform on fractional derivatives. Once this connection is firmly established in the general setting, we turn to the application of the Sumudu transform method (STM) to some interesting nonhomogeneous fractional ordinary differential equations (FODEs). Finally, we use the solutions to form two-dimensional (2D) graphs, by using the symbolic algebra package Mathematica Program 7.

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