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A Geometric Based Connection between Fractional Calculus and Fractal Functions
Establishing the accurate relationship between fractional calculus and fractals is an important research content of fractional calculus theory. In...
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Abstract Algebraic Construction in Fractional Calculus: Parametrised Families with Semigroup Properties
What structure can be placed on the burgeoning field of fractional calculus with assorted kernel functions? This question has been addressed by the...
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Elements of Fractional Calculus
This chapter is devoted to introducing the elements of fractional calculus, Fractional calculusemphasizing some aspects of the historical development...
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Introduction to Fractional Calculus and Modelling
This chapter aims to familiarize the reader with the new field of mathematics known as ‘fractional calculus’, as well as the operators, techniques,...
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On univariate fractional calculus with general bivariate analytic kernels
Several fractional integral and derivative operators have been defined recently with a bivariate structure, acting on functions of a single variable...
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A survey of fractional calculus applications in artificial neural networks
Artificial neural network (ANN) is the backbone of machine learning, specifically deep learning. The interpolating and learning ability of an ANN...
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An unified formulation of strong non-local elasticity with fractional order calculus
The research of a formulation to model non-local interactions in the mechanical behavior of matter is currently an open problem. In this context, a...
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Calculus of variations with higher order Caputo fractional derivatives
In this work, we consider fractional variational problems depending on higher order fractional derivatives. We obtain optimality conditions for such...
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Refinements of Local Fractional Hilbert-Type Inequalities
We study the refinements of several well-known local fractional Hilbert-type inequalities obtained by interpolating the Lebesgue norms of local...
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From Radiation and Space Exploration to the Fractional Calculus
We start with some basic mathematical statements about the roots of the Fractional Calculus, with a historical touch. At the same time, we describe...
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Advanced Analysis of Local Fractional Calculus Applied to the Rice Theory in Fractal Fracture Mechanics
In this chapter, the recent results for the analysis of local fractional calculus are considered for the first time. The local fractional derivative...
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A vacuum solution of modified Einstein equations based on fractional calculus
In this work, we construct a modified version of the Einstein field equations for a vacuum and spherically symmetric spacetime in terms of the...
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Connections between nonlocal operators: from vector calculus identities to a fractional Helmholtz decomposition
Nonlocal vector calculus, which is based on the nonlocal forms of gradient, divergence, and Laplace operators in multiple dimensions, has shown...
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Mathematical Foundation of Fractional Calculus
In the long history of the development of fractional calculus, a variety of definitions have been proposed by researchers from different...
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Fractal-view analysis of local fractional Fokker–Planck equation occurring in modelling of particle’s Brownian motion
In this paper, the solution and behaviour of local fractional Fokker–Planck equation (LFFPE) is investigated in fractal media. For this purpose, the...
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Solution of space–time fractional diffusion equation involving fractional Laplacian with a local radial basis function approximation
Radial basis function-based finite difference (RBF-FD) schemes generalize finite difference methods, providing flexibility in node distribution as...
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Towards a Unified theory of Fractional and Nonlocal Vector Calculus
Nonlocal and fractional-order models capture effects that classical partial differential equations cannot describe; for this reason, they are...
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Modeling and analysis of an implicit fractional order differential equation with multiple first-order fractional derivatives and non-local boundary conditions
In recent years, researchers have investigated boundary value problems of single terms and, in rare circumstances, multi-term fractional differential...
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Generalized Mellin transform and its applications in fractional calculus
In this paper, we introduce a generalized Mellin transform in the framework of fractional operators with respect to functions. The generalized Mellin...
