Advanced Methods in Computational Mathematical Physics

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Computational and Applied Mathematics".

Deadline for manuscript submissions: closed (28 February 2022) | Viewed by 15021

Special Issue Editors

Institute for Theoretical Physics, São Paulo State University, São Paulo 01049-010, Brazil
Interests: complex systems; mathematical biology; time series analysis
Engineering School (DEIM), University of Tuscia, Largo dell'Università, 01100 Viterbo, Italy
Interests: wavelets; fractals; fractional and stochastic equations; numerical and computational methods; mathematical physics; nonlinear systems; artificial intelligence
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

This Special Issue aims to focus on advanced computational methods for the solution of mathematical physics problems and those problems arising in all areas of science, engineering application, finance and natural science whose solution is inspired by mathematical physics methods.

Computational and mathematical modeling of biological, engineering, and physical systems are becoming more and more attractive and suitable for the solution of daily complex nonlinear problems.

These problems can be easily solved using methods of modern advanced calculus and original, recently discovered theories and algorithms.

Many mathematical physical problems follow patterns described by systems of ordinary or partial differential equations, either integer or fractional order derivatives, or described by some special functions. Some of those patterns are self-similar (fractal); at the same time, they follow the power law and have a fading memory. There are many physical problems in nature and engineering applications following such patterns, such as fluidodynamics, mechanics, elasticity, fracture mechanics, rheology, avionics, biology, relativity, geohydrology, and finance.

In the analysis of these problems, advanced calculus and computational–numerical methods are combined with the analysis of the physical nature of these problems.

Many complex phenomena have been formulated in nonlinear evolution equations and systems with an integer or fractional order. Consequently, many researchers have begun focusing on deriving novel computational schemes to evaluate the exact traveling and solitary wave or solutions, while in more complex problems, advanced numerical schemes are employed.

Prof. Dr. Brenno Caetano Troca Cabella
Prof. Dr. Carlo Cattani
Guest Editors

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Keywords

  • Numerical methods
  • Computational methods
  • Mathematical physics
  • Nonlinear problems
  • Dynamical systems
  • Fractional problems
  • Travelling waves
  • Stochastic equations
  • Fractals
  • Wavelets
  • Stability
  • Chaos
  • Special functions

Published Papers (9 papers)

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Research

11 pages, 299 KiB  
Article
Traveling Waves for the Generalized Sinh-Gordon Equation with Variable Coefficients
Mathematics 2022, 10(5), 822; https://doi.org/10.3390/math10050822 - 04 Mar 2022
Cited by 3 | Viewed by 1643
Abstract
The sinh-Gordon equation is simply the classical wave equation with a nonlinear sinh source term. It arises in diverse scientific applications including differential geometry theory, integrable quantum field theory, fluid dynamics, kink dynamics, and statistical mechanics. It can be used to describe generic [...] Read more.
The sinh-Gordon equation is simply the classical wave equation with a nonlinear sinh source term. It arises in diverse scientific applications including differential geometry theory, integrable quantum field theory, fluid dynamics, kink dynamics, and statistical mechanics. It can be used to describe generic properties of string dynamics for strings and multi-strings in constant curvature space. In the present paper, we study a generalized sinh-Gordon equation with variable coefficients with the goal of obtaining analytical traveling wave solutions. Our results show that the traveling waves of the variable coefficient sinh-Gordon equation can be derived from the known solutions of the standard sinh-Gordon equation under a specific selection of a choice of the variable coefficients. These solutions include some real single and multi-solitons, periodic waves, breaking kink waves, singular waves, periodic singular waves, and compactons. These solutions might be valuable when scientists model some real-life phenomena using the sinh-Gordon equation where the balance between dispersion and nonlinearity is perturbed. Full article
(This article belongs to the Special Issue Advanced Methods in Computational Mathematical Physics)
20 pages, 1295 KiB  
Article
A Cuboid Registers Topic, Activity and Competency Data to Exude Feedforward and Continuous Assessment of Competencies
Mathematics 2022, 10(3), 415; https://doi.org/10.3390/math10030415 - 28 Jan 2022
Cited by 4 | Viewed by 1330
Abstract
Evaluating competencies achieved by students within a subject and its different topics is a multivariable and complex task whose outcome should provide actual information on their evolution. A relevant feature when a continuous assessment (CA) rules this evaluation is to track their learning [...] Read more.
Evaluating competencies achieved by students within a subject and its different topics is a multivariable and complex task whose outcome should provide actual information on their evolution. A relevant feature when a continuous assessment (CA) rules this evaluation is to track their learning process so that pertinent feedforward may be harnessed to proactively promote improvement when required. As this process is performed via a number of activities, such as lectures, problem solving, and lab practice, different competencies are developed, depending on the recurrence and type of conducted activity. Measuring and registering their achievement is the leitmotif of competency-based assessment. In this paper, we assemble topic, activity and competency data into a 3D matrix array to form what we call a TAC cuboid. This cuboid showcases a detailed account of each student evolution, aiding instructors and students to design and follow, respectively, an individualized curricular strategy within a continuous and aligned assessment methodology, which facilitates each student to adequately modify his/her level of development of each competency. In addition, we compare the TAC cuboids’ usage in grading a mathematics subject versus a traditional CA method as well as when a dynamical continuous assessment approach is considered to measure the achievement of mathematical competencies. Full article
(This article belongs to the Special Issue Advanced Methods in Computational Mathematical Physics)
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14 pages, 804 KiB  
Article
Additive Noise Effects on the Stabilization of Fractional-Space Diffusion Equation Solutions
Mathematics 2022, 10(1), 130; https://doi.org/10.3390/math10010130 - 02 Jan 2022
Cited by 18 | Viewed by 1045
Abstract
This paper considers a class of stochastic fractional-space diffusion equations with polynomials. We establish a limiting equation that specifies the critical dynamics in a rigorous way. After this, we use the limiting equation, which is an ordinary differential equation, to approximate the solution [...] Read more.
This paper considers a class of stochastic fractional-space diffusion equations with polynomials. We establish a limiting equation that specifies the critical dynamics in a rigorous way. After this, we use the limiting equation, which is an ordinary differential equation, to approximate the solution of the stochastic fractional-space diffusion equation. This equation has never been studied before using a combination of additive noise and fractional-space, therefore we generalize some previously obtained results as special cases. Furthermore, we use Fisher’s and Ginzburg–Landau equations to illustrate our results. Finally, we look at how additive noise affects the stabilization of the solutions. Full article
(This article belongs to the Special Issue Advanced Methods in Computational Mathematical Physics)
16 pages, 7844 KiB  
Article
Resolutions of the Jerk and Snap Vectors for a Quasi Curve in Euclidean 3-Space
Mathematics 2021, 9(23), 3128; https://doi.org/10.3390/math9233128 - 04 Dec 2021
Cited by 4 | Viewed by 1249
Abstract
This work aims at studying resolutions of the jerk and snap vectors of a point particle moving along a quasi curve in Euclidean 3-space E3. In particular, we obtain the resolution of the jerk and snap vectors along the quasi vectors [...] Read more.
This work aims at studying resolutions of the jerk and snap vectors of a point particle moving along a quasi curve in Euclidean 3-space E3. In particular, we obtain the resolution of the jerk and snap vectors along the quasi vectors and offer an alternative resolution of the jerk and snap vectors along the tangential direction and two special radial directions that lie in the osculating and rectifying planes. This alternative resolution for a quasi plane curve in Euclidean 3-space E3 is given as corollary. Moreover, our results are illustrated via some examples. Full article
(This article belongs to the Special Issue Advanced Methods in Computational Mathematical Physics)
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15 pages, 1437 KiB  
Article
Numerical Simulation for a Multidimensional Fourth-Order Nonlinear Fractional Subdiffusion Model with Time Delay
Mathematics 2021, 9(23), 3050; https://doi.org/10.3390/math9233050 - 27 Nov 2021
Cited by 6 | Viewed by 1462
Abstract
The purpose of this paper is to develop a numerical scheme for the two-dimensional fourth-order fractional subdiffusion equation with variable coefficients and delay. Using the L21σ approximation of the time Caputo derivative, a finite difference method with second-order accuracy [...] Read more.
The purpose of this paper is to develop a numerical scheme for the two-dimensional fourth-order fractional subdiffusion equation with variable coefficients and delay. Using the L21σ approximation of the time Caputo derivative, a finite difference method with second-order accuracy in the temporal direction is achieved. The novelty of this paper is to introduce a numerical scheme for the problem under consideration with variable coefficients, nonlinear source term, and delay time constant. The numerical results show that the global convergence orders for spatial and time dimensions are approximately fourth order in space and second-order in time. Full article
(This article belongs to the Special Issue Advanced Methods in Computational Mathematical Physics)
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25 pages, 83137 KiB  
Article
A New Fractional-Order Chaotic System with Its Analysis, Synchronization, and Circuit Realization for Secure Communication Applications
Mathematics 2021, 9(20), 2593; https://doi.org/10.3390/math9202593 - 15 Oct 2021
Cited by 23 | Viewed by 1907
Abstract
This article presents a novel four-dimensional autonomous fractional-order chaotic system (FOCS) with multi-nonlinearity terms. Several dynamics, such as the chaotic attractors, equilibrium points, fractal dimension, Lyapunov exponent, and bifurcation diagrams of this new FOCS, are studied analytically and numerically. Adaptive control laws are [...] Read more.
This article presents a novel four-dimensional autonomous fractional-order chaotic system (FOCS) with multi-nonlinearity terms. Several dynamics, such as the chaotic attractors, equilibrium points, fractal dimension, Lyapunov exponent, and bifurcation diagrams of this new FOCS, are studied analytically and numerically. Adaptive control laws are derived based on Lyapunov theory to achieve chaos synchronization between two identical new FOCSs with an uncertain parameter. For these two identical FOCSs, one represents the master and the other is the slave. The uncertain parameter in the slave side was estimated corresponding to the equivalent master parameter. Next, this FOCS and its synchronization were realized by a feasible electronic circuit and tested using Multisim software. In addition, a microcontroller (Arduino Due) was used to implement the suggested system and the developed synchronization technique to demonstrate its digital applicability in real-world applications. Furthermore, based on the developed synchronization mechanism, a secure communication scheme was constructed. Finally, the security analysis metric tests were investigated through histograms and spectrograms analysis to confirm the security strength of the employed communication system. Numerical simulations demonstrate the validity and possibility of using this new FOCS in high-level security communication systems. Furthermore, the secure communication system is highly resistant to pirate attacks. A good agreement between simulation and experimental results is obtained, showing that the new FOCS can be used in real-world applications. Full article
(This article belongs to the Special Issue Advanced Methods in Computational Mathematical Physics)
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16 pages, 3938 KiB  
Article
Dynamical Continuous Discrete Assessment of Competencies Achievement: An Approach to Continuous Assessment
Mathematics 2021, 9(17), 2082; https://doi.org/10.3390/math9172082 - 28 Aug 2021
Cited by 12 | Viewed by 1898
Abstract
Learning is a non-deterministic complex dynamical system where students transform inputs (classes, assignments, personal work, gamification activities, etc.) into outcomes (acquired knowledge, skills, and competencies). In the process, students generate outputs in a variety of ways (exams, tests, portfolios, etc.). The result of [...] Read more.
Learning is a non-deterministic complex dynamical system where students transform inputs (classes, assignments, personal work, gamification activities, etc.) into outcomes (acquired knowledge, skills, and competencies). In the process, students generate outputs in a variety of ways (exams, tests, portfolios, etc.). The result of these outputs is a grade aimed at measuring the (level of) competencies achieved by each student. We revisit the relevance of continuous assessment to obtain this grading. We simultaneously investigate the generated outputs in different moments as modifiers of the system itself, since they may reveal a variation of the level of competencies achievement previously assessed. This is a novelty in the literature, and a cornerstone of our methodology. This process is called a Dynamical Continuous Discrete assessment, which is a form of blended assessment that may be used under traditional or blended learning environments. This article provides an 11-year perspective of applying this Dynamical Continuous Discrete assessment in a Mathematics class for aerospace engineering students, as well as the students’ perception of continuous assessments. Full article
(This article belongs to the Special Issue Advanced Methods in Computational Mathematical Physics)
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12 pages, 784 KiB  
Article
Study of the Boundary Value Problems for Nonlinear Wave Equations on Domains with a Complex Structure of the Boundary and Prehistory
Mathematics 2021, 9(16), 1888; https://doi.org/10.3390/math9161888 - 09 Aug 2021
Cited by 1 | Viewed by 1291
Abstract
We study a boundary value problem for nonlinear partial differential equations of the hyperbolic type on the plain in a domain with a complex boundary. To find the missing data for the given boundary constraints, we solve a supplementary nonlinear problem. For the [...] Read more.
We study a boundary value problem for nonlinear partial differential equations of the hyperbolic type on the plain in a domain with a complex boundary. To find the missing data for the given boundary constraints, we solve a supplementary nonlinear problem. For the approximation of solutions, one constructive method is built. Full article
(This article belongs to the Special Issue Advanced Methods in Computational Mathematical Physics)
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12 pages, 2106 KiB  
Article
Discontinuous Galerkin Isogeometric Analysis of Convection Problem on Surface
Mathematics 2021, 9(5), 497; https://doi.org/10.3390/math9050497 - 28 Feb 2021
Cited by 3 | Viewed by 1593
Abstract
The objective of this work is to study finite element methods for approximating the solution of convection equations on surfaces embedded in R3. We propose the discontinuous Galerkin (DG) isogeometric analysis (IgA) formulation to solve convection problems on implicitly defined surfaces. [...] Read more.
The objective of this work is to study finite element methods for approximating the solution of convection equations on surfaces embedded in R3. We propose the discontinuous Galerkin (DG) isogeometric analysis (IgA) formulation to solve convection problems on implicitly defined surfaces. Three numerical experiments shows that the numerical scheme converges with the optimal convergence order. Full article
(This article belongs to the Special Issue Advanced Methods in Computational Mathematical Physics)
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