scholarly journals A Note on Fractional Sumudu Transform

2010 ◽  
Vol 2010 ◽  
pp. 1-9 ◽  
Author(s):  
V. G. Gupta ◽  
Bhavna Shrama ◽  
Adem Kiliçman
Keyword(s):  
Differential Equations ◽  
Fractional Order ◽  
Fractional Differential Equations ◽  
Inversion Formula ◽  
Differentiable Functions ◽  
Liouville Derivative ◽  
Fractional Analysis ◽  
Sumudu Transform ◽  
Fractional Differential ◽  
Definition Of

We propose a new definition of a fractional-order Sumudu transform for fractional differentiable functions. In the development of the definition we use fractional analysis based on the modified Riemann-Liouville derivative that we name the fractional Sumudu transform. We also established a relationship between fractional Laplace and Sumudu duality with complex inversion formula for fractional Sumudu transform and apply new definition to solve fractional differential equations.

New spectral techniques for systems of fractional differential equations using fractional-order generalized Laguerre orthogonal functions

2014 ◽  
Vol 17 (4) ◽  
Author(s):  
Ali Bhrawy ◽  
Yahia Alhamed ◽  
Dumitru Baleanu ◽  
Abdulrahim Al-Zahrani
Keyword(s):  
Differential Equations ◽  
Fractional Order ◽  
Fractional Differential Equations ◽  
Laguerre Polynomials ◽  
Orthogonal Functions ◽  
Spectral Techniques ◽  
Fractional Differential ◽  
Definition Of ◽  
Pseudo Spectral Method ◽  
Pseudo Spectral

AbstractFractional-order generalized Laguerre functions (FGLFs) are proposed depends on the definition of generalized Laguerre polynomials. In addition, we derive a new formula expressing explicitly any Caputo fractional-order derivatives of FGLFs in terms of FGLFs themselves. We also propose a fractional-order generalized Laguerre tau technique in conjunction with the derived fractional-order derivative formula of FGLFs for solving Caputo type fractional differential equations (FDEs) of order ν (0 < ν < 1). The fractional-order generalized Laguerre pseudo-spectral approximation is investigated for solving nonlinear initial value problem of fractional order ν. The extension of the fractional-order generalized Laguerre pseudo-spectral method is given to solve systems of FDEs. We present the advantages of using the spectral schemes based on FGLFs and compare them with other methods. Several numerical example are implemented for FDEs and systems of FDEs including linear and nonlinear terms. We demonstrate the high accuracy and the efficiency of the proposed techniques.

scholarly journals Converting fractional differential equations into partial differential equations

2012 ◽  
Vol 16 (2) ◽  
pp. 331-334 ◽  
Author(s):  
Ji-Huan He ◽  
Zheng-Biao Li
Keyword(s):  
Fractal Dimension ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Fractional Order ◽  
Fractional Differential Equations ◽  
Partial Differential ◽  
Liouville Derivative ◽  
Fractional Differential

A transform is suggested in this paper to convert fractional differential equations with the modified Riemann-Liouville derivative into partial differential equations, and it is concluded that the fractional order in fractional differential equations is equivalent to the fractal dimension.

scholarly journals Lyapunov Functions and Lipschitz Stability for Riemann–Liouville Non-Instantaneous Impulsive Fractional Differential Equations

Symmetry ◽  
2021 ◽  
Vol 13 (4) ◽  
pp. 730
Author(s):  
Ravi Agarwal ◽  
Snezhana Hristova ◽  
Donal O’Regan
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Lyapunov Functions ◽  
Sufficient Conditions ◽  
Initial Time ◽  
Lipschitz Stability ◽  
Comparison Results ◽  
Liouville Fractional Derivative ◽  
Fractional Differential ◽  
Definition Of

In this paper a system of nonlinear Riemann–Liouville fractional differential equations with non-instantaneous impulses is studied. We consider a Riemann–Liouville fractional derivative with a changeable lower limit at each stop point of the action of the impulses. In this case the solution has a singularity at the initial time and any stop time point of the impulses. This leads to an appropriate definition of both the initial condition and the non-instantaneous impulsive conditions. A generalization of the classical Lipschitz stability is defined and studied for the given system. Two types of derivatives of the applied Lyapunov functions among the Riemann–Liouville fractional differential equations with non-instantaneous impulses are applied. Several sufficient conditions for the defined stability are obtained. Some comparison results are obtained. Several examples illustrate the theoretical results.

Analysis of a New Class of Impulsive Implicit Sequential Fractional Differential Equations

2020 ◽  
Vol 21 (6) ◽  
pp. 571-587 ◽  
Author(s):  
Akbar Zada ◽  
Sartaj Ali ◽  
Tongxing Li
Keyword(s):  
Differential Equations ◽  
Fractional Order ◽  
Fractional Differential Equations ◽  
Sufficient Conditions ◽  
Uniqueness Of Solutions ◽  
Integer Order ◽  
New Class ◽  
Proposed Model ◽  
Fractional Differential ◽  
Sequential Fractional Differential Equations

AbstractIn this paper, we study an implicit sequential fractional order differential equation with non-instantaneous impulses and multi-point boundary conditions. The article comprehensively elaborate four different types of Ulam’s stability in the lights of generalized Diaz Margolis’s fixed point theorem. Moreover, some sufficient conditions are constructed to observe the existence and uniqueness of solutions for the proposed model. The proposed model contains both the integer order and fractional order derivatives. Thus, the exponential function appearers in the solution of the proposed model which will lead researchers to study fractional differential equations with well known methods of integer order differential equations. In the last, few examples are provided to show the applicability of our main results.

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