Mathematical and Computational Applications

16 pages, 1064 KiB

Open AccessFeature PaperEditor’s ChoiceArticle

Comparison of Two Aspects of a PDE Model for Biological Network Formation

by Clarissa Astuto, Daniele Boffi, Jan Haskovec, Peter Markowich and Giovanni Russo

Math. Comput. Appl. 2022, 27(5), 87; https://doi.org/10.3390/mca27050087 - 17 Oct 2022

Cited by 2 | Viewed by 2022

We compare the solutions of two systems of partial differential equations (PDEs), seen as two different interpretations of the same model which describes the formation of complex biological networks. Both approaches take into account the time evolution of the medium flowing through the [...] Read more.

We compare the solutions of two systems of partial differential equations (PDEs), seen as two different interpretations of the same model which describes the formation of complex biological networks. Both approaches take into account the time evolution of the medium flowing through the network, and we compute the solution of an elliptic–parabolic PDE system for the conductivity vector m, the conductivity tensor

C

and the pressure p. We use finite differences schemes in a uniform Cartesian grid in a spatially two-dimensional setting to solve the two systems, where the parabolic equation is solved using a semi-implicit scheme in time. Since the conductivity vector and tensor also appear in the Poisson equation for the pressure p, the elliptic equation depends implicitly on time. For this reason, we compute the solution of three linear systems in the case of the conductivity vector

m \in R^{2}

and four linear systems in the case of the symmetric conductivity tensor

C \in R^{2 \times 2}

at each time step. To accelerate the simulations, we make use of the Alternating Direction Implicit (ADI) method. The role of the parameters is important for obtaining detailed solutions. We provide numerous tests with various values of the parameters involved to determine the differences in the solutions of the two systems. Full article

(This article belongs to the Special Issue Current Problems and Advances in Computational and Applied Mechanics)

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11 pages, 1524 KiB

Open AccessArticle

Graph Theory Approach to COVID-19 Transmission by Municipalities and Age Groups

by Pedro Machado, Sofia J. Pinheiro, Vera Afreixo, Cristiana J. Silva and Rui Leitão

Math. Comput. Appl. 2022, 27(5), 86; https://doi.org/10.3390/mca27050086 - 13 Oct 2022

Cited by 2 | Viewed by 4020

Abstract

The COVID-19 pandemic remains a global problem that affects the health of millions of people and the world economy. Identifying how the movement of people between regions of the world, countries, and municipalities and how the close contact between individuals of different age [...] Read more.

The COVID-19 pandemic remains a global problem that affects the health of millions of people and the world economy. Identifying how the movement of people between regions of the world, countries, and municipalities and how the close contact between individuals of different age groups promotes the spread of infectious diseases is a pressing concern for society, during epidemic outbreaks and pandemics, such as COVID-19. Networks and Graph Theory provide adequate and powerful tools to study the spread of communicable diseases. In this work, we use Graph Theory to analyze COVID-19 transmission dynamics between municipalities of Aveiro district, in Portugal, and between different age groups, considering data from 2020 and 2021, in order to better understand the spread of this disease, as well as preparing actions for possible future pandemics. We used a digraph structure that models the transmission of SARS-CoV-2 virus between Aveiro’s municipalities and between age groups. To understand how a node fits over the contact digraphs, we studied centrality measures, namely eigencentrality, closeness, degree, and betweenness. Transmission ratios were also considered to determine whether there were certain age groups or municipals that were more responsible for the virus’s spread. According to the results of this research, transmissions mostly occur within the same social groupings, that is, within the same municipalities and age groups. However, the study of centrality measures, eliminating loops, reveals that municipalities such as Aveiro, Estarreja and Ovar are relevant nodes in the transmission network of municipalities as well as the age group of 40–49 in the transmission network of age groups. Furthermore, we conclude that vaccination is effective in reducing the virus. Full article

(This article belongs to the Collection Mathematical Modelling of COVID-19)

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13 pages, 530 KiB

Open AccessArticle

HIV Dynamics with a Trilinear Antibody Growth Function and Saturated Infection Rate

by Fatima Ezzahra Fikri and Karam Allali

Math. Comput. Appl. 2022, 27(5), 85; https://doi.org/10.3390/mca27050085 - 8 Oct 2022

Cited by 1 | Viewed by 1510

Abstract

The objective of this paper is to study a new mathematical model describing the human immunodeficiency virus (HIV). The model incorporates the impacts of cytotoxic T lymphocyte (CTL) immunity and antibodies with trilinear growth functions. The boundedness and positivity of solutions for non-negative [...] Read more.

The objective of this paper is to study a new mathematical model describing the human immunodeficiency virus (HIV). The model incorporates the impacts of cytotoxic T lymphocyte (CTL) immunity and antibodies with trilinear growth functions. The boundedness and positivity of solutions for non-negative initial data are proved, which is consistent with biological studies. The local stability of the equilibrium is established. Finally, numerical simulations are presented to support our theoretical findings. Full article

(This article belongs to the Special Issue Ghana Numerical Analysis Day)

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16 pages, 4179 KiB

Open AccessArticle

Density Peak Clustering Based on Relative Density under Progressive Allocation Strategy

by Yongli Liu, Congcong Zhao and Hao Chao

Math. Comput. Appl. 2022, 27(5), 84; https://doi.org/10.3390/mca27050084 - 6 Oct 2022

Viewed by 1615

Abstract

In traditional density peak clustering, when the density distribution of samples in a dataset is uneven, the density peak points are often concentrated in the region with dense sample distribution, which is easy to affect clustering accuracy. Under the progressive allocation strategy, a [...] Read more.

In traditional density peak clustering, when the density distribution of samples in a dataset is uneven, the density peak points are often concentrated in the region with dense sample distribution, which is easy to affect clustering accuracy. Under the progressive allocation strategy, a density peak clustering algorithm based on relative density is proposed in this paper. This algorithm uses the K-nearest neighbor method to calculate the local density of sample points. In addition, in order to avoid the domino effect during sample allocation, a new similarity calculation method is defined, and a progressive allocation strategy from near to far is used for the allocation of the remaining points. In order to evaluate the effectiveness of this algorithm, comparative experiments with five algorithms were carried out on classical artificial datasets and real datasets. Experimental results show that the proposed algorithm can achieve higher clustering accuracy on datasets with uneven density distribution. Full article

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17 pages, 17738 KiB

Open AccessArticle

Estimation of Pulmonary Arterial Pressure Using Simulated Non-Invasive Measurements and Gradient-Based Optimization Techniques

by Ryno Laubscher, Johan Van Der Merwe, Philip G. Herbst and Jacques Liebenberg

Math. Comput. Appl. 2022, 27(5), 83; https://doi.org/10.3390/mca27050083 - 28 Sep 2022

Cited by 1 | Viewed by 2142

Abstract

Reliable quantification of pulmonary arterial pressure is essential in the diagnostic and prognostic assessment of a range of cardiovascular pathologies, including rheumatic heart disease, yet an accurate and routinely available method for its quantification remains elusive. This work proposes an approach to infer [...] Read more.

Reliable quantification of pulmonary arterial pressure is essential in the diagnostic and prognostic assessment of a range of cardiovascular pathologies, including rheumatic heart disease, yet an accurate and routinely available method for its quantification remains elusive. This work proposes an approach to infer pulmonary arterial pressure based on scientific machine learning techniques and non-invasive, clinically available measurements. A 0D multicompartment model of the cardiovascular system was optimized using several optimization algorithms subject to forward-mode automatic differentiation. Measurement data were synthesized from known parameters to represent the healthy, mitral regurgitant, aortic stenosed, and combined valvular disease situations with and without pulmonary hypertension. Eleven model parameters were selected for optimization based on 95% explained variation in mean pulmonary arterial pressure. A hybrid Adam and limited-memory Broyden–Fletcher–Goldfarb–Shanno optimizer yielded the best results with input data including valvular flow rates, heart chamber volume changes, and systematic arterial pressure. Mean absolute percentage errors ranged from 1.8% to 3.78% over the simulated test cases. The model was able to capture pressure dynamics under hypertensive conditions with pulmonary arterial systole, diastole, and mean pressure average percentage errors of 1.12%, 2.49%, and 2.14%, respectively. The low errors highlight the potential of the proposed model to determine pulmonary pressure for diseased heart valves and pulmonary hypertensive conditions. Full article

(This article belongs to the Special Issue Current Problems and Advances in Computational and Applied Mechanics)

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16 pages, 642 KiB

Open AccessArticle

A SARS-CoV-2 Fractional-Order Mathematical Model via the Modified Euler Method

by Ihtisham Ul Haq, Mehmet Yavuz, Nigar Ali and Ali Akgül

Math. Comput. Appl. 2022, 27(5), 82; https://doi.org/10.3390/mca27050082 - 26 Sep 2022

Cited by 18 | Viewed by 2623

Abstract

This article develops a within-host viral kinetics model of SARS-CoV-2 under the Caputo fractional-order operator. We prove the results of the solution’s existence and uniqueness by using the Banach mapping contraction principle. Using the next-generation matrix method, we obtain the basic reproduction number. [...] Read more.

This article develops a within-host viral kinetics model of SARS-CoV-2 under the Caputo fractional-order operator. We prove the results of the solution’s existence and uniqueness by using the Banach mapping contraction principle. Using the next-generation matrix method, we obtain the basic reproduction number. We analyze the model’s endemic and disease-free equilibrium points for local and global stability. Furthermore, we find approximate solutions for the non-linear fractional model using the Modified Euler Method (MEM). To support analytical findings, numerical simulations are carried out. Full article

(This article belongs to the Special Issue Significance of Mathematical Modelling and Control in Real-World Problems: New Developments and Applications)

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20 pages, 1768 KiB

Open AccessArticle

Comparison of Symbolic Computations for Solving Linear Delay Differential Equations Using the Laplace Transform Method

by Michelle Sherman, Gilbert Kerr and Gilberto González-Parra

Math. Comput. Appl. 2022, 27(5), 81; https://doi.org/10.3390/mca27050081 - 23 Sep 2022

Cited by 4 | Viewed by 2270

Abstract

In this paper, we focus on investigating the performance of the mathematical software program Maple and the programming language MATLAB when using these respective platforms to compute the method of steps (MoS) and the Laplace transform (LT) solutions for neutral and retarded linear [...] Read more.

In this paper, we focus on investigating the performance of the mathematical software program Maple and the programming language MATLAB when using these respective platforms to compute the method of steps (MoS) and the Laplace transform (LT) solutions for neutral and retarded linear delay differential equations (DDEs). We computed the analytical solutions that are obtained by using the Laplace transform method and the method of steps. The accuracy of the Laplace method solutions was determined (or assessed) by comparing them with those obtained by the method of steps. The Laplace transform method requires, among other mathematical tools, the use of the Cauchy residue theorem and the computation of an infinite series. Symbolic computation facilitates the whole process, providing solutions that would be unmanageable by hand. The results obtained here emphasize the fact that symbolic computation is a powerful tool for computing analytical solutions for linear delay differential equations. From a computational viewpoint, we found that the computation time is dependent on the complexity of the history function, the number of terms used in the LT solution, the number of intervals used in the MoS solution, and the parameters of the DDE. Finally, we found that, for linear non-neutral DDEs, MATLAB symbolic computations were faster than Maple. However, for linear neutral DDEs, which are often more complex to solve, Maple was faster. Regarding the accuracy of the LT solutions, Maple was, in a few cases, slightly better than MATLAB, but both were highly reliable. Full article

(This article belongs to the Special Issue Numerical and Symbolic Computation: Developments and Applications 2021)

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13 pages, 693 KiB

Open AccessArticle

Entropy Analysis for Hydromagnetic Darcy–Forchheimer Flow Subject to Soret and Dufour Effects

by Sohail A. Khan and Tasawar Hayat

Math. Comput. Appl. 2022, 27(5), 80; https://doi.org/10.3390/mca27050080 - 19 Sep 2022

Cited by 1 | Viewed by 1822

Abstract

Here, our main aim is to examine the impacts of Dufour and Soret in a radiative Darcy–Forchheimer flow. Ohmic heating and the dissipative features are outlined. The characteristics of the thermo-diffusion and diffusion-thermo effects are addressed. A binary chemical reaction is deliberated. To [...] Read more.

Here, our main aim is to examine the impacts of Dufour and Soret in a radiative Darcy–Forchheimer flow. Ohmic heating and the dissipative features are outlined. The characteristics of the thermo-diffusion and diffusion-thermo effects are addressed. A binary chemical reaction is deliberated. To examine the thermodynamical system performance, we discuss entropy generation. A non-linear differential system is computed by the finite difference technique. Variations in the velocity, concentration, thermal field and entropy rate for the emerging parameters are scrutinized. A decay in velocity is observed for the Forchheimer number. Higher estimation of the magnetic number has the opposite influence for the velocity and temperature. The velocity, concentration and thermal field have a similar effect on the suction variable. The temperature against the Dufour number is augmented. A decay in the concentration is found against the Soret number. A similar trend holds for the entropy rate through the radiation and diffusion variables. An augmentation in the entropy rate is observed for the diffusion variable. Full article

(This article belongs to the Collection Feature Papers in Mathematical and Computational Applications)

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14 pages, 1247 KiB

Open AccessArticle

Area of the Intersection between a Sphere and a Cylindrical Plane

by Charl Gabriël Du Toit

Math. Comput. Appl. 2022, 27(5), 79; https://doi.org/10.3390/mca27050079 - 16 Sep 2022

Cited by 1 | Viewed by 1853

Abstract

A proper understanding of the porous structure of packed beds of spheres is imperative in the analysis and design of the processes involving fluid flow and heat and mass transfer. The radial variation in porosity is of specific interest. When the positions and [...] Read more.

A proper understanding of the porous structure of packed beds of spheres is imperative in the analysis and design of the processes involving fluid flow and heat and mass transfer. The radial variation in porosity is of specific interest. When the positions and sizes of the spheres are known, the radial variation in porosity can be determined using volume-based, area-based, or line-based approaches. Here, the focus is on the area-based methods which employ the intersections between the spheres and selected cylindrical planes to determine the radial variation in porosity, focusing specifically on the calculation of the area of the curved elliptic intersection between a sphere and a cylindrical plane. Using geometrical considerations, analytical integral expressions have been derived based on the axial direction, angular direction, or the radial direction as independent variables. The integral expressions cannot be integrated analytically and have been evaluated using approximations or numerical integration. However, only indirect validation of the calculation of the intersection area has been provided by comparing the radial porosity profiles obtained with experimental data. This study provides direct validation of the calculated area through refined numerical integration of the primary integral expressions and the evaluation of the area employing computer-aided design software. Full article

(This article belongs to the Special Issue Current Problems and Advances in Computational and Applied Mechanics)

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22 pages, 416 KiB

Open AccessArticle

Spectral Analysis of the Finite Element Matrices Approximating 3D Linearly Elastic Structures and Multigrid Proposals

by Quoc Khanh Nguyen, Stefano Serra-Capizzano, Cristina Tablino-Possio and Eddie Wadbro

Math. Comput. Appl. 2022, 27(5), 78; https://doi.org/10.3390/mca27050078 - 14 Sep 2022

Cited by 1 | Viewed by 2271

Abstract

The so-called material distribution methods for topology optimization cast the governing equation as an extended or fictitious domain problem, in which a coefficient field represents the design. In practice, the finite element method is typically used to approximate that kind of governing equations [...] Read more.

The so-called material distribution methods for topology optimization cast the governing equation as an extended or fictitious domain problem, in which a coefficient field represents the design. In practice, the finite element method is typically used to approximate that kind of governing equations by using a large number of elements to discretize the design domain, and an element-wise constant function approximates the coefficient field in that domain. This paper presents a spectral analysis of the coefficient matrices associated with the linear systems stemming from the finite element discretization of a linearly elastic problem for an arbitrary coefficient field in three spatial dimensions. The given theoretical analysis is used for designing and studying an optimal multigrid method in the sense that the (arithmetic) cost for solving the problem, up to a fixed desired accuracy, is linear in the corresponding matrix size. Few selected numerical examples are presented and discussed in connection with the theoretical findings. Full article

(This article belongs to the Collection Feature Papers in Mathematical and Computational Applications)

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24 pages, 1912 KiB

Open AccessEditor’s ChoiceReview

A Review of Artificial Intelligence and Machine Learning for Incident Detectors in Road Transport Systems

by Samuel Olugbade, Stephen Ojo, Agbotiname Lucky Imoize, Joseph Isabona and Mathew O. Alaba

Math. Comput. Appl. 2022, 27(5), 77; https://doi.org/10.3390/mca27050077 - 13 Sep 2022

Cited by 18 | Viewed by 11303

Abstract

Road transport is the most prone to accidents, resulting in significant fatalities and injuries. It also faces a plethora of never-ending problems, such as the frequent loss of lives and valuables during an accident. Appropriate actions need to be taken to address these [...] Read more.

Road transport is the most prone to accidents, resulting in significant fatalities and injuries. It also faces a plethora of never-ending problems, such as the frequent loss of lives and valuables during an accident. Appropriate actions need to be taken to address these problems, such as the establishment of an automatic incident detection system using artificial intelligence and machine learning. This article explores the overview of artificial intelligence and machine learning in facilitating automatic incident detector systems to decrease road accidents. The study examines the critical problems and potential remedies for reducing road traffic accidents and the application of artificial intelligence and machine learning in road transportation systems. More, new, and emerging trends that reduce frequent accidents in the transportation sector are discussed extensively. Specifically, the study organized the following sub-topics: an incident detector with machine learning and artificial intelligence and road management with machine learning and artificial intelligence. Additionally, safety is the primary concern of road transport; the internet of vehicles and vehicle ad hoc networks, including the use of wireless communication technologies such as 5G wireless networks and the use of machine learning and artificial intelligence for road transportation systems planning, are elaborated. Key findings from the review indicate that route optimization, cargo volume forecasting, predictive fleet maintenance, real-time vehicle tracking, and traffic management are critical to safeguarding road transportation systems. Finally, the paper summarizes the challenges facing the application of artificial intelligence in road transport systems, highlights the research trends, identifies the unresolved questions, and highlights the essential research takeaways. The work can serve as reference material for road transport system planning and management. Full article

(This article belongs to the Special Issue Setting the Future of Smart Industry 4.0 with Metaheuristic Algorithms)

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13 pages, 2784 KiB

Open AccessArticle

Furniture Style Compatibility Estimation by Multi-Branch Deep Siamese Network

by Ayumu Taisho, Keiko Ono, Erina Makihara, Naoya Ikushima and Sohei Yamakawa

Math. Comput. Appl. 2022, 27(5), 76; https://doi.org/10.3390/mca27050076 - 4 Sep 2022

Cited by 2 | Viewed by 2118

Abstract

As demands for understanding visual style among interior scenes increase, estimating style compatibility is becoming challenging. In particular, furniture styles are difficult to define due to their various elements, such as color and shape. As a result, furniture style is an ambiguous concept. [...] Read more.

As demands for understanding visual style among interior scenes increase, estimating style compatibility is becoming challenging. In particular, furniture styles are difficult to define due to their various elements, such as color and shape. As a result, furniture style is an ambiguous concept. To reduce ambiguity, Siamese networks have frequently been used to estimate style compatibility by adding various features that represent the style. However, it is still difficult to accurately represent a furniture’s style, even when using alternate features associated with the images. In this paper, we propose a new Siamese model that can learn from several furniture images simultaneously. Specifically, we propose a one-to-many ratio input method to maintain high performance even when inputs are ambiguous. We also propose a new metric for evaluating Siamese networks. The conventional metric, the area under the ROC curve (AUC), does not reveal the actual distance between styles. Therefore, the proposed metric quantitatively evaluates the distance between styles by using the distance between the embedding of each furniture image. Experiments show that the proposed model improved the AUC from 0.672 to 0.721 and outperformed the conventional Siamese model in terms of the proposed metric. Full article

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16 pages, 7339 KiB

Open AccessArticle

A Novel Trajectory Planning Method for Parafoil Airdrop System Based on Geometric Segmentation Strategy

by Haitao Gao and Jin Tao

Math. Comput. Appl. 2022, 27(5), 75; https://doi.org/10.3390/mca27050075 - 29 Aug 2022

Cited by 2 | Viewed by 1942

Abstract

Reasonable trajectory planning is the precondition for the parafoil airdrop system to achieve autonomous accurate homing, and safe landing. To successfully realize the self-homing of the parafoil airdrop system, a new trajectory optimization design scheme is proposed in this paper. The scheme is [...] Read more.

Reasonable trajectory planning is the precondition for the parafoil airdrop system to achieve autonomous accurate homing, and safe landing. To successfully realize the self-homing of the parafoil airdrop system, a new trajectory optimization design scheme is proposed in this paper. The scheme is based on the parafoil’s unique flight and control characteristics and adopts a segmented homing design. The current common trajectory design method faces a problem, whereby straight-line flight distance before landing is limited by the radius of the height-reducing area. The core feature of the proposed design scheme is its avoidance of this problem, thereby ensuring landing accuracy and safety. Firstly, the different starting states of the parafoil airdrop system and the landing requirements were comprehensively considered, and the homing trajectory reasonably segmented. Based on the requirements of energy control, stable flight, and landing accuracy, the optimal objective function of the trajectory was established, and the trajectory parameters, calculation methods, and constraints were given. Secondly, the cuckoo search algorithm was applied to optimize the objective function to obtain the final home trajectory. Finally, the trajectory planning under different airdrop conditions was simulated and verified. The results showed that the planned trajectories could reach the target point accurately and meet the flight direction requirements, proving the proposed scheme’s correctness and feasibility. Full article

(This article belongs to the Section Engineering)

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17 pages, 347 KiB

Open AccessArticle

An Optimal Iterative Technique for Multiple Root Finder of Nonlinear Problems

by Ramandeep Behl, Sonia Bhalla, Fouad Mallawi and Majed Aali Alsulami

Math. Comput. Appl. 2022, 27(5), 74; https://doi.org/10.3390/mca27050074 - 29 Aug 2022

Viewed by 1471

Abstract

In this paper, an optimal higher-order iterative technique to approximate the multiple roots of a nonlinear equation has been presented. The proposed technique has special properties: a two-point method that does not involve any derivatives, has an optimal convergence of fourth-order, is cost-effective, [...] Read more.

In this paper, an optimal higher-order iterative technique to approximate the multiple roots of a nonlinear equation has been presented. The proposed technique has special properties: a two-point method that does not involve any derivatives, has an optimal convergence of fourth-order, is cost-effective, is more stable, and has better numerical results. In addition to this, we adopt the weight function approach at both substeps (which provide us with a more general form of two-point methods). Firstly, the convergence order is studied for multiplicity

m = 2, 3

by Taylor’s series expansion and then general convergence for

m \geq 4

is proved. We have demonstrated the applicability of our methods to six numerical problems. Out of them: the first one is the well-known Van der Waals ideal gas problem, the second one is used to study the blood rheology model, the third one is chosen from the linear algebra (namely, eigenvalue), and the remaining three are academic problems. We concluded on the basis of obtained CPU timing, computational order of convergence, and absolute errors between two consecutive iterations for which our methods illustrate better results as compared to earlier studies. Full article

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Math. Comput. Appl., Volume 27, Issue 5 (October 2022) – 14 articles

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