The Error Incurred in Using the Caputo-Derivative Laplace-Transform

2009 ◽  
Author(s):  
Tom T. Hartley ◽  
Carl F. Lorenzo
Keyword(s):  
Differential Equations ◽  
Laplace Transform ◽  
Fractional Derivative ◽  
Fractional Order ◽  
Caputo Derivative ◽  
Fractional Order Differential Equations ◽  
Definition Of

This paper considers the initialization of fractional-order differential equations. The initialization responses obtained using the Caputo derivative are compared with the exact initialization responses from the Riemann-Liouville definition of the fractional derivative. The error incurred in using the Caputo derivative for initialization problems in fractionalorder differential equations is presented.

scholarly journals REGARDING NEW NUMERICAL RESULTS FOR THE DYNAMICAL MODEL OF ROMANTIC RELATIONSHIPS WITH FRACTIONAL DERIVATIVE

Fractals ◽  
2021 ◽  
pp. 2240009
Author(s):  
WEI GAO ◽  
P. VEERESHA ◽  
D. G. PRAKASHA ◽  
HACI MEHMET BASKONUS
Keyword(s):  
Romantic Relationships ◽  
Laplace Transform ◽  
Fractional Derivative ◽  
Fractional Order ◽  
Real Word ◽  
Transform Method ◽  
Coupled Equations ◽  
Fractional Order Differential Equations ◽  
Analysis Technique ◽  
Homotopy Analysis

The main purpose of the present investigation is to find the solution of fractional coupled equations describing the romantic relationships using [Formula: see text]-homotopy analysis transform method ([Formula: see text]-HATM). The considered scheme is a unification of [Formula: see text]-homotopy analysis technique with Laplace transform (LT). More preciously, we scrutinized the behavior of the obtained solution for the considered model with fractional-order, in order to elucidate the effectiveness of the proposed algorithm. Further, for the different fractional-order and parameters offered by the considered method, the physical natures have been apprehended. The obtained consequences evidence that the proposed method is very effective and highly methodical to study and examine the nature and its corresponding consequences of the system of fractional order differential equations describing the real word problems.

scholarly journals On the Generalized Laplace Transform

Symmetry ◽  
2021 ◽  
Vol 13 (4) ◽  
pp. 669
Author(s):  
Paul Bosch ◽  
Héctor José Carmenate García ◽  
José Manuel Rodríguez ◽  
José María Sigarreta
Keyword(s):  
Differential Equations ◽  
Laplace Transform ◽  
Fractional Derivative ◽  
Harmonic Oscillator ◽  
Fractional Differential Equations ◽  
Fractional Differential ◽  
Definition Of ◽  
Generalized Harmonic Oscillator

In this paper we introduce a generalized Laplace transform in order to work with a very general fractional derivative, and we obtain the properties of this new transform. We also include the corresponding convolution and inverse formula. In particular, the definition of convolution for this generalized Laplace transform improves previous results. Additionally, we deal with the generalized harmonic oscillator equation, showing that this transform and its properties allow one to solve fractional differential equations.

Equivalence of History-Function Based and Infinite-Dimensional-State Initializations for Fractional-Order Operators

2013 ◽  
Vol 8 (4) ◽  
Author(s):  
Tom T. Hartley ◽  
Carl F. Lorenzo ◽  
Jean-Claude Trigeassou ◽  
Nezha Maamri
Keyword(s):  
Differential Equations ◽  
Laplace Transform ◽  
State Space ◽  
Fractional Order ◽  
Laplace Transforms ◽  
Space Representation ◽  
Infinite Dimensional ◽  
State Space Representation ◽  
Fractional Order Differential Equations ◽  
Dimensional State Space

Proper initialization of fractional-order operators has been an ongoing problem, particularly in the application of Laplace transforms with correct initialization terms. In the last few years, a history-function-based initialization along with its corresponding Laplace transform has been presented. Alternatively, an infinite-dimensional state-space representation along with its corresponding Laplace transform has also been presented. The purpose of this paper is to demonstrate that these two approaches to the initialization problem for fractional-order operators are equivalent and that the associated Laplace transforms yield the correct initialization terms and can be used in the solution of fractional-order differential equations.

scholarly journals On the Stability of Fractional Differential Equations Involving Generalized Caputo Fractional Derivative

2020 ◽  
Vol 2020 ◽  
pp. 1-14
Author(s):  
Minh Duc Tran ◽  
Vu Ho ◽  
Hoa Ngo Van
Keyword(s):  
Global Existence ◽  
Differential Equations ◽  
Fractional Derivative ◽  
Fractional Order ◽  
Fractional Differential Equations ◽  
Caputo Fractional Derivative ◽  
Caputo Derivative ◽  
Fractional Differential ◽  
The Stability ◽  
The Given

This work presents the results of the global existence for fractional differential equations involving generalized Caputo derivative with the case of the fractional order derivative α∈1,2. In addition, the Ulam–Hyers–Mittag-Leffler stability of the given problems is also established.

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