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Recent Advances and New Challenges in Coupled Systems and Networks: Theory, Modelling, and Applications

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A special issue of Mathematical and Computational Applications (ISSN 2297-8747).

Deadline for manuscript submissions: closed (15 October 2024) | Viewed by 15117

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Dr. Sundeep Singh

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Guest Editor

Faculty of Sustainable Design Engineering, University of Prince Edward Island, 550 University Ave, Charlottetown, PE C1A 4P3, Canada
Interests: thermo-fluids; mathematical modeling & simulations; computational fluid dynamics and heat transfer; biomedical engineering
Special Issues, Collections and Topics in MDPI journals

Prof. Dr. Weizhong Dai

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College of Engineering and Science, Louisiana Tech University, 201 Mayfield Ave, Ruston, LA 71272, USA
Interests: numerical methods for partial differential equations (higher-order compact finite difference schemes, stability, convergence); artificial neural network method for solving partial differential equations; simulations for micro/nano scale heat transfer; bio-heat transfer

Special Issue Information

Dear Colleagues,

This Special Issue aims at collecting original research articles related to the recent advancement, development and challenges pertaining to the development of current state-of-the-art models focused on mathematical and computational sciences, with their applications to modelling natural, social, and engineering systems. Review articles aligned to the scope of this Special Issue are also welcome.

Topics of interest for submissions to this Special Issue include, but are not limited to:

Coupled systems and networks models
Mathematics in biology and medicine
Mathematical models for nanoscience and nanotechnology
Modelling complex systems in life sciences
Coupled human–environment systems
Infectious disease models
Mathematical models for renewable and alternative sources of energy
Coupled models for environmentally friendly materials and technologies
Machine learning and multiscale modelling
Computational neuroscience

This Special Issue is in honor of Professor Roderick Melnik (https://m3ai.wlu.ca/), who has pioneered many advanced mathematical models and has significantly contributed to this field over the past few decades. This Special Issue is open to all researchers working in the above areas, including but not limited to the contributors of the Special Symposium on Coupled Complex Systems organized at the International AMMCS Conference Series, held in Waterloo, Canada.

Dr. Sundeep Singh
Prof. Dr. Weizhong Dai
Guest Editors

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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. Mathematical and Computational Applications is an international peer-reviewed open access semimonthly journal published by MDPI.

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Published Papers (10 papers)

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19 pages, 2708 KiB

Open AccessArticle

Modeling of Sedimentation of Particles near Corrugated Surfaces by the Meshless Method of Fundamental Solutions

by Alex Povitsky

Math. Comput. Appl. 2024, 29(5), 90; https://doi.org/10.3390/mca29050090 - 3 Oct 2024

Viewed by 456

Abstract

The velocity and trajectory of particles moving along the corrugated (rough) surface under the action of gravity is obtained by a modified Method of Fundamental Solutions (MFS). This physical situation is found often in biological systems and microfluidic devices. The Stokes equations with no-slip boundary conditions are solved using the Green’s function for Stokeslets. In the present study, the velocity of a moving particle under the action of the gravity force is not known and becomes a part of the MFS solution. This requires an adjustment of the matrix of the MFS linear system to include the unknown particle velocity and incorporate in the MFS the balance of hydrodynamic and gravity forces acting on the particle. The study explores the combination of the regularization of Stokeslets and placement of Stokeslets outside the flow domain to ensure the accuracy and stability of computations for particles moving in proximity to the wall. The MFS results are compared to prior published approximate analytical and experimental results to verify the effectiveness of this methodology to predict the trajectory of particles, including their deviation from the vertical trajectory, and select the optimal set of computational parameters. The developed MFS methodology is then applied to the sedimentation of a pair of two spherical particles in proximity to the corrugated wall, in which case, the analytical solution is not available. The MFS results show that particles in the pair deviate from the trajectory of a single particle: the particle located below moves farther away from vertical wall, and the particle located above shifts closer to the wall. Full article

(This article belongs to the Special Issue Recent Advances and New Challenges in Coupled Systems and Networks: Theory, Modelling, and Applications)

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18 pages, 328 KiB

Open AccessArticle

Loss of the Sturm–Liouville Property of Time-Varying Second-Order Differential Equations in the Presence of Delayed Dynamics

by Manuel De la Sen

Math. Comput. Appl. 2024, 29(5), 89; https://doi.org/10.3390/mca29050089 - 3 Oct 2024

Viewed by 381

Abstract

This paper considers a nominal undelayed and time-varying second-order Sturm–Liouville differential equation on a finite time interval which is a nominal version of another perturbed differential equation subject to a delay in its dynamics. The nominal delay-free differential equation is a Sturm–Liouville system in the sense that it is subject to prescribed two-point boundary conditions. However, the perturbed differential system is not a Sturm–Liouville system, in general, due to the presence of delayed dynamics. The main objective of the paper is to investigate the loss of the boundary values of the Sturm–Liouville nominal undelayed system in the presence of the delayed dynamics. Such a delayed dynamics is considered a perturbation of the nominal dynamics related to the Sturm–Liouville system with given two-point boundary values. In particular, this loss of the Sturm–Liouville exact tracking of the prescribed two-point boundary values might happen because the nominal boundary values may become lost by the state trajectory solution in the presence of delays, related to the undelayed case, due to the presence of the delayed dynamics. The worst-case error description of the deviation of the two-point boundary values of the current perturbed differential with respect to those of the nominal Sturm–Liouville system is characterized in terms of error norms related to the nominal system. Under sufficiently small deviations of the parameterization of the perturbed system with respect to the nominal one, such a worst-error characterization makes the current perturbed system an approximate Sturm–Liouville system of the nominal undelayed one. Full article

(This article belongs to the Special Issue Recent Advances and New Challenges in Coupled Systems and Networks: Theory, Modelling, and Applications)

21 pages, 1461 KiB

Open AccessArticle

DSTree: A Spatio-Temporal Indexing Data Structure for Distributed Networks

by Majid Hojati, Steven Roberts and Colin Robertson

Math. Comput. Appl. 2024, 29(3), 42; https://doi.org/10.3390/mca29030042 - 31 May 2024

Viewed by 1120

Abstract

The widespread availability of tools to collect and share spatial data enables us to produce a large amount of geographic information on a daily basis. This enormous production of spatial data requires scalable data management systems. Geospatial architectures have changed from clusters to cloud architectures and more parallel and distributed processing platforms to be able to tackle these challenges. Peer-to-peer (P2P) systems as a backbone of distributed systems have been established in several application areas such as web3, blockchains, and crypto-currencies. Unlike centralized systems, data storage in P2P networks is distributed across network nodes, providing scalability and no single point of failure. However, managing and processing queries on these networks has always been challenging. In this work, we propose a spatio-temporal indexing data structure, DSTree. DSTree does not require additional Distributed Hash Trees (DHTs) to perform multi-dimensional range queries. Inserting a piece of new geographic information updates only a portion of the tree structure and does not impact the entire graph of the data. For example, for time-series data, such as storing sensor data, the DSTree performs around 40% faster in spatio-temporal queries for small and medium datasets. Despite the advantages of our proposed framework, challenges such as 20% slower insertion speed or semantic query capabilities remain. We conclude that more significant research effort from GIScience and related fields in developing decentralized applications is needed. The need for the standardization of different geographic information when sharing data on the IPFS network is one of the requirements. Full article

(This article belongs to the Special Issue Recent Advances and New Challenges in Coupled Systems and Networks: Theory, Modelling, and Applications)

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13 pages, 660 KiB

Open AccessArticle

Periodic Solutions in a Simple Delay Differential Equation

by Anatoli Ivanov and Sergiy Shelyag

Math. Comput. Appl. 2024, 29(3), 36; https://doi.org/10.3390/mca29030036 - 12 May 2024

Viewed by 1099

Abstract

A simple-form scalar differential equation with delay and nonlinear negative periodic feedback is considered. The existence of several types of slowly oscillating periodic solutions is shown with the same and double periods of the feedback coefficient. The periodic solutions are built explicitly in the case with piecewise constant nonlinearities involved. The periodic dynamics are shown to persist under small perturbations of the equation, which make it smooth. The theoretical results are verified through extensive numerical simulations. Full article

(This article belongs to the Special Issue Recent Advances and New Challenges in Coupled Systems and Networks: Theory, Modelling, and Applications)

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14 pages, 1926 KiB

Open AccessArticle

Modeling of Chemical Vapor Infiltration for Fiber-Reinforced Silicon Carbide Composites Using Meshless Method of Fundamental Solutions

by Patrick Mahoney and Alex Povitsky

Math. Comput. Appl. 2024, 29(2), 27; https://doi.org/10.3390/mca29020027 - 22 Mar 2024

Cited by 1 | Viewed by 1437

Abstract

In this study, the Method of Fundamental Solutions (MFSs) is adopted to model Chemical Vapor Infiltration (CVI) in a fibrous preform. The preparation of dense fiber-reinforced silicon carbide composites is considered. The reaction flux at the solid surface is equal to the diffusion flux towards the surface. The Robin or third-type boundary condition is implemented into the MFS. From the fibers’ surface concentrations obtained by MFS, deposition rates are calculated, and the geometry is updated at each time step, modeling the pore filling over time. The MFS solution is verified by comparing the results to a known analytical solution for a simplified geometry of concentric cylinders with a concentration set at the outer cylinder and a reaction at the inner cylinder. MFS solutions are compared to published experimental data. Porosity transients are obtained by a combination of MFSs with surface deposition to show the relation between the initial and final porosities. Full article

(This article belongs to the Special Issue Recent Advances and New Challenges in Coupled Systems and Networks: Theory, Modelling, and Applications)

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13 pages, 1239 KiB

Open AccessArticle

Magnesium and Calcium Transport along the Male Rat Kidney: Effect of Diuretics

by Pritha Dutta and Anita T. Layton

Math. Comput. Appl. 2024, 29(1), 13; https://doi.org/10.3390/mca29010013 - 7 Feb 2024

Viewed by 1732

Abstract

Calcium (Ca²⁺) and magnesium (Mg²⁺) are essential for cellular function. The kidneys play an important role in maintaining the homeostasis of these cations. Their reabsorption along the nephron is dependent on distinct trans- and paracellular pathways and is coupled to the transport of other electrolytes. Notably, sodium (Na⁺) transport establishes an electrochemical gradient to drive Ca²⁺ and Mg²⁺ reabsorption. Consequently, alterations in renal Na⁺ handling, under pathophysiological conditions or pharmacological manipulations, can have major effects on Ca²⁺ and Mg²⁺ transport. One such condition is the administration of diuretics, which are used to treat a large range of clinical conditions, but most commonly for the management of blood pressure and fluid balance. While the pharmacological targets of diuretics typically directly mediate Na⁺ transport, they also indirectly affect renal Ca²⁺ and Mg²⁺ handling through alterations in the electrochemical gradient. To investigate renal Ca²⁺ and Mg² handling and how those processes are affected by diuretic treatment, we have developed computational models of electrolyte transport along the nephrons. Model simulations indicate that along the proximal tubule and thick ascending limb, the transport of Ca²⁺ and Mg²⁺ occurs in parallel with Na⁺, but those processes are dissociated along the distal convoluted tubule. We also simulated the effects of acute administration of loop, thiazide, and K-sparing diuretics. The model predicted significantly increased Ca²⁺ and Mg²⁺ excretions and significantly decreased Ca²⁺ and Mg²⁺ excretions on treatment with loop and K-sparing diuretics, respectively. Treatment with thiazide diuretics significantly decreased Ca²⁺ excretion, but there was no significant alteration in Mg²⁺ excretion. The present models can be used to conduct in silico studies on how the kidney adapts to alterations in Ca²⁺ and Mg²⁺ homeostasis during various physiological and pathophysiological conditions, such as pregnancy, diabetes, and chronic kidney disease. Full article

(This article belongs to the Special Issue Recent Advances and New Challenges in Coupled Systems and Networks: Theory, Modelling, and Applications)

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15 pages, 645 KiB

Open AccessArticle

A Numerical Method Based on Operator Splitting Collocation Scheme for Nonlinear Schrödinger Equation

by Mengli Yao and Zhifeng Weng

Math. Comput. Appl. 2024, 29(1), 6; https://doi.org/10.3390/mca29010006 - 15 Jan 2024

Cited by 1 | Viewed by 1604

Abstract

In this paper, a second-order operator splitting method combined with the barycentric Lagrange interpolation collocation method is proposed for the nonlinear Schrödinger equation. The equation is split into linear and nonlinear parts: the linear part is solved by the barycentric Lagrange interpolation collocation method in space combined with the Crank–Nicolson scheme in time; the nonlinear part is solved analytically due to the availability of a closed-form solution, which avoids solving the nonlinear algebraic equation. Moreover, the consistency of the fully discretized scheme for the linear subproblem and error estimates of the operator splitting scheme are provided. The proposed numerical scheme is of spectral accuracy in space and of second-order accuracy in time, which greatly improves the computational efficiency. Numerical experiments are presented to confirm the accuracy, mass and energy conservation of the proposed method. Full article

(This article belongs to the Special Issue Recent Advances and New Challenges in Coupled Systems and Networks: Theory, Modelling, and Applications)

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21 pages, 851 KiB

Open AccessArticle

A Finite Difference Method for Solving the Wave Equation with Fractional Damping

by Manruo Cui, Cui-Cui Ji and Weizhong Dai

Math. Comput. Appl. 2024, 29(1), 2; https://doi.org/10.3390/mca29010002 - 29 Dec 2023

Cited by 1 | Viewed by 2317

Abstract

In this paper, we develop a finite difference method for solving the wave equation with fractional damping in 1D and 2D cases, where the fractional damping is given based on the Caputo fractional derivative. Firstly, based on the weighted method, we propose a new numerical approximation for the Caputo fractional derivative and apply it for the 1D case to obtain a time-stepping method. We then develop an alternating direction implicit (ADI) scheme for the 2D case. Using the discrete energy method, we prove that the proposed difference schemes are unconditionally stable and convergent in both 1D and 2D cases. Finally, several numerical examples are given to verify the theoretical results. Full article

(This article belongs to the Special Issue Recent Advances and New Challenges in Coupled Systems and Networks: Theory, Modelling, and Applications)

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21 pages, 4591 KiB

Open AccessArticle

Continuum and Molecular Modeling of Chemical Vapor Deposition at Nano-Scale Fibrous Substrates

by Himel Barua and Alex Povitsky

Math. Comput. Appl. 2023, 28(6), 112; https://doi.org/10.3390/mca28060112 - 6 Dec 2023

Viewed by 1636

Abstract

Chemical vapor deposition (CVD) is a common industrial process that incorporates a complex combination of fluid flow, chemical reactions, and surface deposition. Understanding CVD processes requires rigorous and costly experimentation involving multiple spatial scales, from meters to nanometers. The numerical modeling of deposition over macro-scale substrates has been conducted in the literature and results show compliance with experimental data. For smaller-scale substrates, where the corresponding Knudsen number is larger than zero, continuum modeling does not provide accurate results, which calls for the implementation of molecular-level modeling techniques. In the current study, the finite-volume method (FVM) and Direct Simulation Monte Carlo (DSMC) method were combined to model the reactor-scale flow with CVD around micro- and nano-scale fibers. CVD at fibers with round cross-sections was modeled in the reactor, where fibers were oriented perpendicularly with respect to the feedstock gas flow. The DSMC method was applied to modeling flow around the matrix of nano-scale circular individual fibers. Results show that for smaller diameters of individual fibers with the same filling ratio, the residence time of gas particles inside the fibrous media reduces, and, consequently, the amount of material surface deposition decreases. The sticking coefficient on the fibers’ surface plays an important role; for instance, increasing the sticking coefficient from 20% to 80% will double the deposition rate. Full article

(This article belongs to the Special Issue Recent Advances and New Challenges in Coupled Systems and Networks: Theory, Modelling, and Applications)

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8 pages, 763 KiB

Open AccessArticle

Explicit Integrating Factor Runge–Kutta Method for the Extended Fisher–Kolmogorov Equation

by Yanan Wang and Shuying Zhai

Math. Comput. Appl. 2023, 28(6), 110; https://doi.org/10.3390/mca28060110 - 22 Nov 2023

Viewed by 1607

Abstract

The extended Fisher–Kolmogorov (EFK) equation is an important model for phase transitions and bistable phenomena. This paper presents some fast explicit numerical schemes based on the integrating factor Runge–Kutta method and the Fourier spectral method to solve the EFK equation. The discrete global convergence of these new schemes is analyzed rigorously. Three numerical examples are presented to verify the theoretical analysis and the efficiency of the proposed schemes. Full article

(This article belongs to the Special Issue Recent Advances and New Challenges in Coupled Systems and Networks: Theory, Modelling, and Applications)

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