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Keywords = generalized tempered fractional laplace operator

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14 pages, 312 KB  
Article
Maximum Principle for Variable-Order Fractional Conformable Differential Equation with a Generalized Tempered Fractional Laplace Operator
by Tingting Guan and Lihong Zhang
Fractal Fract. 2023, 7(11), 798; https://doi.org/10.3390/fractalfract7110798 - 1 Nov 2023
Cited by 1 | Viewed by 1865
Abstract
In this paper, we investigate properties of solutions to a space-time fractional variable-order conformable nonlinear differential equation with a generalized tempered fractional Laplace operatorby using the maximum principle. We first establish some new important fractional various-order conformable inequalities. With these inequalities, we prove [...] Read more.
In this paper, we investigate properties of solutions to a space-time fractional variable-order conformable nonlinear differential equation with a generalized tempered fractional Laplace operatorby using the maximum principle. We first establish some new important fractional various-order conformable inequalities. With these inequalities, we prove a new maximum principle with space-time fractional variable-order conformable derivatives and a generalized tempered fractional Laplace operator. Moreover, we discuss some results about comparison principles and properties of solutions for a family of space-time fractional variable-order conformable nonlinear differential equations with a generalized tempered fractional Laplace operator by maximum principle. Full article
(This article belongs to the Special Issue Symmetry and Solutions of Fractional Differential Equations with Their Developments)
25 pages, 407 KB  
Article
Weighted Fractional Calculus: A General Class of Operators
by Arran Fernandez and Hafiz Muhammad Fahad
Fractal Fract. 2022, 6(4), 208; https://doi.org/10.3390/fractalfract6040208 - 7 Apr 2022
Cited by 35 | Viewed by 4036
Abstract
We conduct a formal study of a particular class of fractional operators, namely weighted fractional calculus, and its extension to the more general class known as weighted fractional calculus with respect to functions. We emphasise the importance of the conjugation relationships with the [...] Read more.
We conduct a formal study of a particular class of fractional operators, namely weighted fractional calculus, and its extension to the more general class known as weighted fractional calculus with respect to functions. We emphasise the importance of the conjugation relationships with the classical Riemann–Liouville fractional calculus, and use them to prove many fundamental properties of these operators. As examples, we consider special cases such as tempered, Hadamard-type, and Erdélyi–Kober operators. We also define appropriate modifications of the Laplace transform and convolution operations, and solve some ordinary differential equations in the setting of these general classes of operators. Full article
(This article belongs to the Special Issue Numerical and Analytical Methods for Differential Equations and Systems)
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