Editorial

2 pages, 133 KiB

Open AccessEditorial

Advanced Mathematical Methods: Theory and Applications

by Andrea Giusti and Francesco Mainardi

Mathematics 2020, 8(1), 107; https://doi.org/10.3390/math8010107 - 09 Jan 2020

Viewed by 2465

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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9 pages, 253 KiB

Open AccessFeature PaperArticle

The Role of the Central Limit Theorem in the Heterogeneous Ensemble of Brownian Particles Approach

by Silvia Vitali, Iva Budimir, Claudio Runfola and Gastone Castellani

Mathematics 2019, 7(12), 1145; https://doi.org/10.3390/math7121145 - 23 Nov 2019

Cited by 4 | Viewed by 2749

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)

12 pages, 318 KiB

Open AccessArticle

On the Matrix Mittag–Leffler Function: Theoretical Properties and Numerical Computation

by Marina Popolizio

Mathematics 2019, 7(12), 1140; https://doi.org/10.3390/math7121140 - 21 Nov 2019

Cited by 20 | Viewed by 3520

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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9 pages, 257 KiB

Open AccessArticle

A Note on the Generalized Relativistic Diffusion Equation

by Luisa Beghin and Roberto Garra

Mathematics 2019, 7(11), 1009; https://doi.org/10.3390/math7111009 - 24 Oct 2019

Cited by 3 | Viewed by 1817

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)

12 pages, 1979 KiB

Open AccessArticle

Modeling Heavy Metal Sorption and Interaction in a Multispecies Biofilm

by Berardino D’Acunto, Luigi Frunzo, Vincenzo Luongo and Maria Rosaria Mattei

Mathematics 2019, 7(9), 781; https://doi.org/10.3390/math7090781 - 24 Aug 2019

Cited by 7 | Viewed by 2852

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

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28 pages, 338 KiB

Open AccessArticle

Discrete Two-Dimensional Fourier Transform in Polar Coordinates Part I: Theory and Operational Rules

by Natalie Baddour

Mathematics 2019, 7(8), 698; https://doi.org/10.3390/math7080698 - 02 Aug 2019

Cited by 6 | Viewed by 4872

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

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20 pages, 356 KiB

Open AccessArticle

An Introduction to Space–Time Exterior Calculus

by Ivano Colombaro, Josep Font-Segura and Alfonso Martinez

Mathematics 2019, 7(6), 564; https://doi.org/10.3390/math7060564 - 21 Jun 2019

Cited by 7 | Viewed by 3705

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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12 pages, 312 KiB

Open AccessFeature PaperArticle

Subordination Approach to Space-Time Fractional Diffusion

by Emilia Bazhlekova and Ivan Bazhlekov

Mathematics 2019, 7(5), 415; https://doi.org/10.3390/math7050415 - 09 May 2019

Cited by 11 | Viewed by 2360

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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17 pages, 307 KiB

Open AccessEditor’s ChoiceArticle

Dynamic Keynesian Model of Economic Growth with Memory and Lag

by Vasily E. Tarasov and Valentina V. Tarasova

Mathematics 2019, 7(2), 178; https://doi.org/10.3390/math7020178 - 15 Feb 2019

Cited by 20 | Viewed by 5357

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)

12 pages, 284 KiB

Open AccessArticle

On the Fredholm Property of the Trace Operators Associated with the Elastic Layer Potentials

by Giulio Starita and Alfonsina Tartaglione

Mathematics 2019, 7(2), 134; https://doi.org/10.3390/math7020134 - 01 Feb 2019

Cited by 6 | Viewed by 1730

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} (\partial Ω)

(with

k \geq 2, q \in (1, + \infty)

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)

16 pages, 271 KiB

Open AccessArticle

Probabilistic Interpretation of Solutions of Linear Ultraparabolic Equations

by Michael D. Marcozzi

Mathematics 2018, 6(12), 286; https://doi.org/10.3390/math6120286 - 27 Nov 2018

Cited by 2 | Viewed by 2097

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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21 pages, 425 KiB

Open AccessEditor’s ChoiceReview

Evaluation of Fractional Integrals and Derivatives of Elementary Functions: Overview and Tutorial

by Roberto Garrappa, Eva Kaslik and Marina Popolizio

Mathematics 2019, 7(5), 407; https://doi.org/10.3390/math7050407 - 07 May 2019

Cited by 88 | Viewed by 6661

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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18 pages, 275 KiB

Open AccessFeature PaperReview

Some Schemata for Applications of the Integral Transforms of Mathematical Physics

by Yuri Luchko

Mathematics 2019, 7(3), 254; https://doi.org/10.3390/math7030254 - 12 Mar 2019

Cited by 9 | Viewed by 2478

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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