scholarly journals On Laplace transforms with respect to functions and their applications to fractional differential equations

2021 ◽  
Author(s):  
Hafiz Muhammad Fahad ◽  
Mujeeb ur Rehman ◽  
Arran Fernandez
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Laplace Transforms ◽  
Fractional Differential

The Solution of Linear Fractional Differential Equations Using the Fractional Meta-Trigonometric Functions

2011 ◽  
Author(s):  
Carl F. Lorenzo ◽  
Rachid Malti ◽  
Tom T. Hartley
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Constant Coefficient ◽  
Laplace Transforms ◽  
New Method ◽  
Trigonometric Functions ◽  
Fractional Differential ◽  
Linear Fractional Differential Equations

A new method for the solution of linear constant coefficient fractional differential equations of any commensurate order based on the Laplace transforms of the fractional meta-trigonometric functions and the R-function is presented. The new method simplifies the solution of such equations. A simplifying characterization that reduces the number of parameters in the fractional meta-trigonometric functions is introduced.

The Fractional Hyperbolic Functions: With Application to Fractional Differential Equations

2005 ◽  
Author(s):  
Carl F. Lorenzo ◽  
Tom T. Hartley
Keyword(s):  
Differential Equations ◽  
Exponential Function ◽  
Fractional Differential Equations ◽  
Laplace Transforms ◽  
Hyperbolic Functions ◽  
Solution Sets ◽  
Generate Solution ◽  
Fractional Differential

This paper develops the fractional hyperbolic functions based on fractional generalization of the exponential function, the R-function. The fractional hyperbolic functions contain the traditional hyperbolic functions as proper subsets. Laplace transforms and various identities are derived for the new functions and are used to generate solution sets for various classes of fractional differential equations. Also developed are relationships between the R-function and the fractional hyperbolic functions.

scholarly journals Sinc Based Inverse Laplace Transforms, Mittag-Leffler Functions and Their Approximation for Fractional Calculus

2021 ◽  
Vol 5 (2) ◽  
pp. 43
Author(s):  
Gerd Baumann
Keyword(s):  
Differential Equations ◽  
Fractional Calculus ◽  
Laplace Transform ◽  
Fractional Differential Equations ◽  
Numerical Approximation ◽  
Laplace Transforms ◽  
Inverse Laplace Transform ◽  
Exact Methods ◽  
Sinc Methods ◽  
Fractional Differential

We shall discuss three methods of inverse Laplace transforms. A Sinc-Thiele approximation, a pure Sinc, and a Sinc-Gaussian based method. The two last Sinc related methods are exact methods of inverse Laplace transforms which allow us a numerical approximation using Sinc methods. The inverse Laplace transform converges exponentially and does not use Bromwich contours for computations. We apply the three methods to Mittag-Leffler functions incorporating one, two, and three parameters. The three parameter Mittag-Leffler function represents Prabhakar’s function. The exact Sinc methods are used to solve fractional differential equations of constant and variable differentiation order.

Initialization of Fractional Differential Equations: Background and Theory

2007 ◽  
Author(s):  
Carl F. Lorenzo ◽  
Tom T. Hartley
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Fractional Derivatives ◽  
Companion Paper ◽  
Laplace Transforms ◽  
Fractional Integrals ◽  
Fractional Operators ◽  
Higher Order Derivative ◽  
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. This paper provides background on past work in the area and determines the Laplace transforms for initialized fractional integrals of any order and fractional derivatives of order less than one. A companion paper in this conference extends the theory to higher order derivative operators and provides application insight.

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.

A New Method for Laplace Transforms of Multi-Term Fractional Differential Equations of the Caputo Type

2021 ◽  
Author(s):  
Masataka Fukunaga
Keyword(s):  
Differential Equations ◽  
Laplace Transform ◽  
Fractional Differential Equations ◽  
Initial Conditions ◽  
Expansion Method ◽  
Laplace Transforms ◽  
Transform Method ◽  
Change Of Variables ◽  
The Laplace Transform ◽  
Fractional Differential

Abstract The Laplace transform method is one of the powerful tools in studying the frac- tional differential equations (FDEs). In this paper, it is shown that the Heaviside expansion method for integer order differential equations is also applicable to the Laplace transforms of multi-term Caputo fractional differential equations (FDEs) of zero initial conditions if the orders of Caputo derivatives are integer multiples of a common real number. The particular solution of a linear multi-term Caputo FDE is obtained by its Laplace transform and the Heaviside expansion method. A Caputo FDE of non zero initial conditions is transformed to an Caputo FDE of zero initial conditions by an appropriate change of variables. In the latter, the terms originated from the initial conditions appear as nonhomogeneous terms. Thus, the Caputo FDE of nonzero initial conditions is obtained as the particular solutions to the equivalent Caputo FDE of zero initial conditions. The solutions of a linear multi-term Caputo FDEs of nonzero initial conditions are expressed through the two parameter Mittag-Leffler functions.

Initialization of Fractional Differential Equations: Theory and Application

2007 ◽  
Author(s):  
Carl F. Lorenzo ◽  
Tom T. Hartley
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Fractional Derivatives ◽  
Companion Paper ◽  
Laplace Transforms ◽  
Fractional Integrals ◽  
Fractional Operators ◽  
The Laplace Transform ◽  
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. A companion paper in this conference determines the Laplace transforms for initialized fractional integrals of any order and fractional derivatives of order less than one. This paper extends the theory for the Laplace transform of the derivative to higher order and provides applications.

Sign in / Sign up

Export Citation Format

Share Document