Special Issue "Advanced Mathematical Methods: Theory and Applications"

A special issue of Mathematics (ISSN 2227-7390).

Deadline for manuscript submissions: closed (15 October 2019).

Special Issue Editors

Prof. Francesco Mainardi
E-Mail Website
Guest Editor
Department of Physics and Astronomy, University of Bologna, Via Irnerio, 46, I-40126 Bologna, Italy
Interests: special functions; fractional calculus complex analysis; asymptotic methods; diffusion and wave propagation problems
Special Issues and Collections in MDPI journals
Dr. Andrea Giusti
E-Mail
Guest Editor
Department of Physics, University of Bologna, Via Irnerio, 46, I-40126 Bologna, Italy
Interests: fractional calculus; special functions; viscoelasticity; theoretical and mathematical physics

Special Issue Information

Dear Colleagues,

The many technical and computational problems that appear to be constantly emerging in various branches of physics and engineering beg for a more detailed understanding of the fundamental mathematics that serves as the cornerstone of our way of understanding natural phenomena. The purpose of this Special Issue is to establish a collection of a restricted number of carefully selected articles authored by promising young scientists and the world’s leading experts in pure and applied mathematics, highlighting the state-of-the-art of the various research lines focussing on the study of analytical and numerical mathematical methods for pure and applied sciences.

We are also interested in methods including non-local operators related to fractional calculus, even though this is not necessarily the main theme of the Special Issue. It is additionally worth stressing that papers involving applications to deterministic and stochastic phenomena are welcome.

Prof. Francesco Mainardi
Dr. Andrea Giusti
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All papers will be peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 1200 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • applied mathematics
  • mathematical modelling
  • mathematical physics
  • mathematical methods for applied and theoretical sciences
  • numerical methods
  • special functions
  • integral transforms
  • non-local operators

Published Papers (13 papers)

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Editorial

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Open AccessEditorial
Advanced Mathematical Methods: Theory and Applications
Mathematics 2020, 8(1), 107; https://doi.org/10.3390/math8010107 - 09 Jan 2020
Abstract
The many technical and computational problems that appear to be constantly emerging in various branches of physics and engineering beg for a more detailed understanding of the fundamental mathematics that serves as the cornerstone of our way of understanding natural phenomena [...] Full article
(This article belongs to the Special Issue Advanced Mathematical Methods: Theory and Applications)

Research

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Open AccessFeature PaperArticle
The Role of the Central Limit Theorem in the Heterogeneous Ensemble of Brownian Particles Approach
Mathematics 2019, 7(12), 1145; https://doi.org/10.3390/math7121145 - 23 Nov 2019
Abstract
The central limit theorem (CLT) and its generalization to stable distributions have been widely described in literature. However, many variations of the theorem have been defined and often their applicability in practical situations is not straightforward. In particular, the applicability of the CLT [...] Read more.
The central limit theorem (CLT) and its generalization to stable distributions have been widely described in literature. However, many variations of the theorem have been defined and often their applicability in practical situations is not straightforward. In particular, the applicability of the CLT is essential for a derivation of heterogeneous ensemble of Brownian particles (HEBP). Here, we analyze the role of the CLT within the HEBP approach in more detail and derive the conditions under which the existing theorems are valid. Full article
(This article belongs to the Special Issue Advanced Mathematical Methods: Theory and Applications)
Open AccessArticle
On the Matrix Mittag–Leffler Function: Theoretical Properties and Numerical Computation
Mathematics 2019, 7(12), 1140; https://doi.org/10.3390/math7121140 - 21 Nov 2019
Abstract
Many situations, as for example within the context of Fractional Calculus theory, require computing the Mittag–Leffler (ML) function with matrix arguments. In this paper, we collect theoretical properties of the matrix ML function. Moreover, we describe the available numerical methods aimed at this [...] Read more.
Many situations, as for example within the context of Fractional Calculus theory, require computing the Mittag–Leffler (ML) function with matrix arguments. In this paper, we collect theoretical properties of the matrix ML function. Moreover, we describe the available numerical methods aimed at this purpose by stressing advantages and weaknesses. Full article
(This article belongs to the Special Issue Advanced Mathematical Methods: Theory and Applications)
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Open AccessArticle
A Note on the Generalized Relativistic Diffusion Equation
Mathematics 2019, 7(11), 1009; https://doi.org/10.3390/math7111009 - 24 Oct 2019
Abstract
We study here a generalization of the time-fractional relativistic diffusion equation based on the application of Caputo fractional derivatives of a function with respect to another function. We find the Fourier transform of the fundamental solution and discuss the probabilistic meaning of the [...] Read more.
We study here a generalization of the time-fractional relativistic diffusion equation based on the application of Caputo fractional derivatives of a function with respect to another function. We find the Fourier transform of the fundamental solution and discuss the probabilistic meaning of the results obtained in relation to the time-scaled fractional relativistic stable process. We briefly consider also the application of fractional derivatives of a function with respect to another function in order to generalize fractional Riesz-Bessel equations, suggesting their stochastic meaning. Full article
(This article belongs to the Special Issue Advanced Mathematical Methods: Theory and Applications)
Open AccessArticle
Modeling Heavy Metal Sorption and Interaction in a Multispecies Biofilm
Mathematics 2019, 7(9), 781; https://doi.org/10.3390/math7090781 - 24 Aug 2019
Abstract
A mathematical model able to simulate the physical, chemical and biological interactions prevailing in multispecies biofilms in the presence of a toxic heavy metal is presented. The free boundary value problem related to biofilm growth and evolution is governed by a nonlinear ordinary [...] Read more.
A mathematical model able to simulate the physical, chemical and biological interactions prevailing in multispecies biofilms in the presence of a toxic heavy metal is presented. The free boundary value problem related to biofilm growth and evolution is governed by a nonlinear ordinary differential equation. The problem requires the integration of a system of nonlinear hyperbolic partial differential equations describing the biofilm components evolution, and a systems of semilinear parabolic partial differential equations accounting for substrates diffusion and reaction within the biofilm. In addition, a semilinear parabolic partial differential equation is introduced to describe heavy metal diffusion and sorption. The biosoption process modeling is completed by the definition and integration of other two systems of nonlinear hyperbolic partial differential equations describing the free and occupied binding sites evolution, respectively. Numerical simulations of the heterotrophic-autotrophic interaction occurring in biofilm reactors devoted to wastewater treatment are presented. The high biosorption ability of bacteria living in a mature biofilm is highlighted, as well as the toxicity effect of heavy metals on autotrophic bacteria, whose growth directly affects the nitrification performance of bioreactors. Full article
(This article belongs to the Special Issue Advanced Mathematical Methods: Theory and Applications)
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Open AccessArticle
Discrete Two-Dimensional Fourier Transform in Polar Coordinates Part I: Theory and Operational Rules
Mathematics 2019, 7(8), 698; https://doi.org/10.3390/math7080698 - 02 Aug 2019
Abstract
The theory of the continuous two-dimensional (2D) Fourier transform in polar coordinates has been recently developed but no discrete counterpart exists to date. In this paper, we propose and evaluate the theory of the 2D discrete Fourier transform (DFT) in polar coordinates. This [...] Read more.
The theory of the continuous two-dimensional (2D) Fourier transform in polar coordinates has been recently developed but no discrete counterpart exists to date. In this paper, we propose and evaluate the theory of the 2D discrete Fourier transform (DFT) in polar coordinates. This discrete theory is shown to arise from discretization schemes that have been previously employed with the 1D DFT and the discrete Hankel transform (DHT). The proposed transform possesses orthogonality properties, which leads to invertibility of the transform. In the first part of this two-part paper, the theory of the actual manipulated quantities is shown, including the standard set of shift, modulation, multiplication, and convolution rules. Parseval and modified Parseval relationships are shown, depending on which choice of kernel is used. Similar to its continuous counterpart, the 2D DFT in polar coordinates is shown to consist of a 1D DFT, DHT and 1D inverse DFT. Full article
(This article belongs to the Special Issue Advanced Mathematical Methods: Theory and Applications)
Open AccessArticle
An Introduction to Space–Time Exterior Calculus
Mathematics 2019, 7(6), 564; https://doi.org/10.3390/math7060564 - 21 Jun 2019
Abstract
The basic concepts of exterior calculus for space–time multivectors are presented: Interior and exterior products, interior and exterior derivatives, oriented integrals over hypersurfaces, circulation and flux of multivector fields. Two Stokes theorems relating the exterior and interior derivatives with circulation and flux, respectively, [...] Read more.
The basic concepts of exterior calculus for space–time multivectors are presented: Interior and exterior products, interior and exterior derivatives, oriented integrals over hypersurfaces, circulation and flux of multivector fields. Two Stokes theorems relating the exterior and interior derivatives with circulation and flux, respectively, are derived. As an application, it is shown how the exterior-calculus space–time formulation of the electromagnetic Maxwell equations and Lorentz force recovers the standard vector-calculus formulations, in both differential and integral forms. Full article
(This article belongs to the Special Issue Advanced Mathematical Methods: Theory and Applications)
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Open AccessFeature PaperArticle
Subordination Approach to Space-Time Fractional Diffusion
Mathematics 2019, 7(5), 415; https://doi.org/10.3390/math7050415 - 09 May 2019
Abstract
The fundamental solution to the multi-dimensional space-time fractional diffusion equation is studied by applying the subordination principle, which provides a relation to the classical Gaussian function. Integral representations in terms of Mittag-Leffler functions are derived for the fundamental solution and the subordination kernel. [...] Read more.
The fundamental solution to the multi-dimensional space-time fractional diffusion equation is studied by applying the subordination principle, which provides a relation to the classical Gaussian function. Integral representations in terms of Mittag-Leffler functions are derived for the fundamental solution and the subordination kernel. The obtained integral representations are used for numerical evaluation of the fundamental solution for different values of the parameters. Full article
(This article belongs to the Special Issue Advanced Mathematical Methods: Theory and Applications)
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Open AccessArticle
Dynamic Keynesian Model of Economic Growth with Memory and Lag
Mathematics 2019, 7(2), 178; https://doi.org/10.3390/math7020178 - 15 Feb 2019
Cited by 7
Abstract
A mathematical model of economic growth with fading memory and continuous distribution of delay time is suggested. This model can be considered as a generalization of the standard Keynesian macroeconomic model. To take into account the memory and gamma-distributed lag we use the [...] Read more.
A mathematical model of economic growth with fading memory and continuous distribution of delay time is suggested. This model can be considered as a generalization of the standard Keynesian macroeconomic model. To take into account the memory and gamma-distributed lag we use the Abel-type integral and integro-differential operators with the confluent hypergeometric Kummer function in the kernel. These operators allow us to propose an economic accelerator, in which the memory and lag are taken into account. The fractional differential equation, which describes the dynamics of national income in this generalized model, is suggested. The solution of this fractional differential equation is obtained in the form of series of the confluent hypergeometric Kummer functions. The asymptotic behavior of national income, which is described by this solution, is considered. Full article
(This article belongs to the Special Issue Advanced Mathematical Methods: Theory and Applications)
Open AccessArticle
On the Fredholm Property of the Trace Operators Associated with the Elastic Layer Potentials
Mathematics 2019, 7(2), 134; https://doi.org/10.3390/math7020134 - 01 Feb 2019
Cited by 1
Abstract
We deal with the system of equations of linear elastostatics, governing the equilibrium configurations of a linearly elastic body. We recall the basics of the theory of the elastic layer potentials and we extend the trace operators associated with the layer potentials to [...] Read more.
We deal with the system of equations of linear elastostatics, governing the equilibrium configurations of a linearly elastic body. We recall the basics of the theory of the elastic layer potentials and we extend the trace operators associated with the layer potentials to suitable sets of singular densities. We prove that the trace operators defined, for example, on W 1 k 1 / q , q ( Ω ) (with k 2 , q ( 1 , + ) and Ω an open connected set of R 3 of class C k ), satisfy the Fredholm property. Full article
(This article belongs to the Special Issue Advanced Mathematical Methods: Theory and Applications)
Open AccessArticle
Probabilistic Interpretation of Solutions of Linear Ultraparabolic Equations
Mathematics 2018, 6(12), 286; https://doi.org/10.3390/math6120286 - 27 Nov 2018
Abstract
We demonstrate the existence, uniqueness and Galerkin approximatation of linear ultraparabolic terminal value/infinite-horizon problems on unbounded spatial domains. Furthermore, we provide a probabilistic interpretation of the solution in terms of the expectation of an associated ultradiffusion process. Full article
(This article belongs to the Special Issue Advanced Mathematical Methods: Theory and Applications)

Review

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Open AccessReview
Evaluation of Fractional Integrals and Derivatives of Elementary Functions: Overview and Tutorial
Mathematics 2019, 7(5), 407; https://doi.org/10.3390/math7050407 - 07 May 2019
Cited by 2
Abstract
Several fractional-order operators are available and an in-depth knowledge of the selected operator is necessary for the evaluation of fractional integrals and derivatives of even simple functions. In this paper, we reviewed some of the most commonly used operators and illustrated two approaches [...] Read more.
Several fractional-order operators are available and an in-depth knowledge of the selected operator is necessary for the evaluation of fractional integrals and derivatives of even simple functions. In this paper, we reviewed some of the most commonly used operators and illustrated two approaches to generalize integer-order derivatives to fractional order; the aim was to provide a tool for a full understanding of the specific features of each fractional derivative and to better highlight their differences. We hence provided a guide to the evaluation of fractional integrals and derivatives of some elementary functions and studied the action of different derivatives on the same function. In particular, we observed how Riemann–Liouville and Caputo’s derivatives converge, on long times, to the Grünwald–Letnikov derivative which appears as an ideal generalization of standard integer-order derivatives although not always useful for practical applications. Full article
(This article belongs to the Special Issue Advanced Mathematical Methods: Theory and Applications)
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Open AccessFeature PaperReview
Some Schemata for Applications of the Integral Transforms of Mathematical Physics
Mathematics 2019, 7(3), 254; https://doi.org/10.3390/math7030254 - 12 Mar 2019
Cited by 1
Abstract
In this survey article, some schemata for applications of the integral transforms of mathematical physics are presented. First, integral transforms of mathematical physics are defined by using the notions of the inverse transforms and generating operators. The convolutions and generating operators of the [...] Read more.
In this survey article, some schemata for applications of the integral transforms of mathematical physics are presented. First, integral transforms of mathematical physics are defined by using the notions of the inverse transforms and generating operators. The convolutions and generating operators of the integral transforms of mathematical physics are closely connected with the integral, differential, and integro-differential equations that can be solved by means of the corresponding integral transforms. Another important technique for applications of the integral transforms is the Mikusinski-type operational calculi that are also discussed in the article. The general schemata for applications of the integral transforms of mathematical physics are illustrated on an example of the Laplace integral transform. Finally, the Mellin integral transform and its basic properties and applications are briefly discussed. Full article
(This article belongs to the Special Issue Advanced Mathematical Methods: Theory and Applications)
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