scholarly journals Special Functions as Solutions to the Euler–Poisson–Darboux Equation with a Fractional Power of the Bessel Operator

Mathematics ◽  
2021 ◽  
Vol 9 (13) ◽  
pp. 1484
Author(s):  
Azamat Dzarakhohov ◽  
Yuri Luchko ◽  
Elina Shishkina
Keyword(s):  
Initial Value Problems ◽  
Fractional Derivatives ◽  
Integral Transform ◽  
Special Functions ◽  
Fractional Power ◽  
Fundamental Solutions ◽  
Initial Value ◽  
Explicit Formulas ◽  
Bessel Differential Operator ◽  
Darboux Equation

In this paper, we consider fractional ordinary differential equations and the fractional Euler–Poisson–Darboux equation with fractional derivatives in the form of a power of the Bessel differential operator. Using the technique of the Meijer integral transform and its modification, fundamental solutions to these equations are derived in terms of the Fox–Wright function, the Fox H-function, and their particular cases. We also provide some explicit formulas for the solutions to the corresponding initial-value problems in terms of the generalized convolutions introduced in this paper.

scholarly journals Intrinsic Discontinuities in Solutions of Evolution Equations Involving Fractional Caputo–Fabrizio and Atangana–Baleanu Operators

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 2023
Author(s):  
Christopher Nicholas Angstmann ◽  
Byron Alexander Jacobs ◽  
Bruce Ian Henry ◽  
Zhuang Xu
Keyword(s):  
Initial Value Problems ◽  
Fractional Derivatives ◽  
Evolution Equations ◽  
Integral Transform ◽  
Initial Value ◽  
Intrinsic Nature ◽  
Recent Interest ◽  
Special Cases ◽  
Singular Kernels

There has been considerable recent interest in certain integral transform operators with non-singular kernels and their ability to be considered as fractional derivatives. Two such operators are the Caputo–Fabrizio operator and the Atangana–Baleanu operator. Here we present solutions to simple initial value problems involving these two operators and show that, apart from some special cases, the solutions have an intrinsic discontinuity at the origin. The intrinsic nature of the discontinuity in the solution raises concerns about using such operators in modelling. Solutions to initial value problems involving the traditional Caputo operator, which has a singularity inits kernel, do not have these intrinsic discontinuities.

Combined effects in some initial value problems involving Riemann–Liouville fractional derivatives in bounded domains

2016 ◽  
Vol 13 (6) ◽  
pp. 5135-5146
Author(s):  
Sonia Ben Makhlouf ◽  
Majda Chaieb ◽  
Malek Zribi
Keyword(s):  
Initial Value Problems ◽  
Fractional Derivatives ◽  
Bounded Domains ◽  
Combined Effects ◽  
Initial Value

On generalized fractional operators and a gronwall type inequality with applications

Filomat ◽  
2017 ◽  
Vol 31 (17) ◽  
pp. 5457-5473 ◽  
Author(s):  
Yassine Adjabi ◽  
Fahd Jarad ◽  
Thabet Abdeljawad
Keyword(s):  
Differential Equations ◽  
Initial Value Problems ◽  
Fractional Derivatives ◽  
Initial Conditions ◽  
Type Inequality ◽  
Weighted Spaces ◽  
Fractional Operators ◽  
Initial Value ◽  
Generalized Fractional Derivatives

In this paper, we obtain the Gronwall type inequality for generalized fractional operators unifying Riemann-Liouville and Hadamard fractional operators. We apply this inequality to the dependence of the solution of differential equations, involving generalized fractional derivatives, on both the order and the initial conditions. More properties for the generalized fractional operators are formulated and the solutions of initial value problems in certain new weighted spaces of functions are established as well.

A family of the incomplete hypergeometric functions and associated integral transform and fractional derivative formulas

Filomat ◽  
2017 ◽  
Vol 31 (1) ◽  
pp. 125-140 ◽  
Author(s):  
Rekha Srivastava ◽  
Ritu Agarwal ◽  
Sonal Jain
Keyword(s):  
Hypergeometric Functions ◽  
Fractional Derivatives ◽  
Integral Transform ◽  
Special Functions ◽  
Applied Mathematics ◽  
Integral Transforms ◽  
Mathematical Physics ◽  
Natural Generalization ◽  
Derivative Formulas ◽  
Derivatives Of

Recently, Srivastava et al. [Integral Transforms Spec. Funct. 23 (2012), 659-683] introduced the incomplete Pochhammer symbols that led to a natural generalization and decomposition of a class of hypergeometric and other related functions as well as to certain potentially useful closed-form representations of definite and improper integrals of various special functions of applied mathematics and mathematical physics. In the present paper, our aim is to establish several formulas involving integral transforms and fractional derivatives of this family of incomplete hypergeometric functions. As corollaries and consequences, many interesting results are shown to follow from our main results.

scholarly journals Generalization of Fuzzy Laplace Transforms of Fuzzy Fractional Derivatives about the General Fractional Ordern-1<β

2016 ◽  
Vol 2016 ◽  
pp. 1-13
Author(s):  
Amal Khalaf Haydar ◽  
Ruaa Hameed Hassan
Keyword(s):  
Fractional Order ◽  
Initial Value Problems ◽  
Fractional Derivatives ◽  
Laplace Transforms ◽  
Initial Value ◽  
Caputo Fractional Derivatives

The main aim in this paper is to use all the possible arrangements of objects such thatr1of them are equal to 1 andr2(the others) of them are equal to 2, in order to generalize the definitions of Riemann-Liouville and Caputo fractional derivatives (about order0<β<n) for a fuzzy-valued function. Also, we find fuzzy Laplace transforms for Riemann-Liouville and Caputo fractional derivatives about the general fractional ordern-1<β<nunder H-differentiability. Some fuzzy fractional initial value problems (FFIVPs) are solved using the above two generalizations.

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