Recent Advances and New Challenges in Coupled Systems and Networks: Theory, Modelling, and Applications (Special Issue in Honor of Professor Roderick Melnik)

Coupled systems and networks are ubiquitous across all branches of science and engineering, while mathematical and computational models play a fundamental role in their studies. This Special Issue of Mathematical and Computational Applications is in honor of Professor Roderick Melnik, who has pioneered many advanced models, allowing for significant breakthroughs in such studies, and has made influential contributions to this field over the past few decades. His professional activities during this time, covering three continents, exemplify ceaseless service to the interdisciplinary scientific community, noted for remarkable achievements and contributions across a wide range of topics.

1. Introduction

Diverse as they may appear, all of Professor Roderick Melnik’s research activities stem from his boundless curiosity about our world, its multiscale nature, and the spatio-temporal relationships among the systems, processes, and phenomena within it.

Roderick has always been fascinated by the breathtaking capabilities of the human brain. With some 86 billion neurons forming around 100 trillion connections among themselves, and supported by approximately 85 billion nonneuronal cells, the brain is a remarkable organ that typically fits within an average volume of around 1300 cubic centimetres. But how does this complex structure interact with our vast universe, which is estimated to contain 2 trillion galaxies and 10²⁴ stars? Albert Einstein famously said, “There are only two ways to live your life. One is as though nothing is a miracle. The other is as though everything is a miracle”. Roderick has always chosen the latter perspective. He has dedicated his career to advancing our understanding of complex systems, processes, and phenomena, as well as their dynamic interactions with the environment. As a distinguished expert in applied and computational mathematics, he also possesses extensive knowledge and expertise across various other disciplines. His groundbreaking contributions are truly interdisciplinary, covering fields such as computational and network sciences, engineering and informatics, and natural and social sciences.

Professor Melnik has held the Tier I Canada Research Chair in Mathematical Modelling for the past two decades. He is an inspiring educator who instils a passion for the power of mathematical models in his students and young scholars. His active engagement with graduate students, postgraduate students, and postdoctoral fellows has earned him a reputation as an enthusiastic and motivational teacher. He has been a founding director of several research institutes, including the MS2Discovery Interdisciplinary Research Institute for Mathematical and Statistical Modelling in Scientific Discovery, Innovation and Sustainability. Additionally, he has been a driving force behind international interdisciplinary initiatives, such as the Applied Mathematics, Modelling and Computational Science (AMMCS) framework, Banff International Research Station workshops, and Fields Institute research programs. These initiatives have fostered interdisciplinary activities, and have played a significant role in producing exceptional graduates who make a profound impact in the fields of interdisciplinary science, mathematics, engineering, and industry.

Born to a family of teachers, books about scientists and mathematicians appeared early on the table of Dr. Melnik. His father was a mathematics and physics teacher, while his mother was a linguist, teaching English and Ukrainian. One of Dr. Melnik’s favourite books during high school was “Norbert Wiener: A Life in Cybernetics”, which significantly influenced his career choice. In 1989, he earned his PhD in Computational Mathematics from Kyiv State University. He began his academic career as an Assistant Professor at the National Technical University KPI, where he was promoted to Associate Professor in 1991, before moving to the University of South Australia in Adelaide.

Dr. Melnik’s initial interest in coupled complex systems, particularly in smart materials and structures, as well as in information theory and control, dates back to the early years of his career. After a brief stint at a university in Queensland, his family relocated to Sydney, where he worked as a senior mathematician at the Commonwealth Scientific and Industrial Research Organization, based at the Macquarie University Campus. During this time, he expanded his research interests and significantly contributed to various applied fields, including applied mathematics in electronic and mechanical engineering, climate modelling, and information and network sciences. His work also involved developing innovative models and computational methodologies to describe complex physical processes, such as phase transformations in complex materials with memory, and applications in the life sciences.

At 37, Dr. Melnik became a full professor at the University of Southern Denmark, where he served as the Head of Mathematical Modelling at the Mads Clausen Institute. There, he led multidisciplinary teams of researchers, and actively collaborated with colleagues in both the European Union and the United States. During this time, Dr. Melnik and his teams made significant contributions to coupled multiscale problems and developed innovative models for nanoscience and nanotechnology. After moving to North America, he held full professorships in the USA and Canada, actively maintaining collaborations with his European colleagues. His exceptional achievements have earned him numerous distinctions and honors across Denmark, Spain, Italy, and the United Kingdom, in addition to his Tier I Canada Research Chair role. In North America, Dr. Melnik has made remarkable contributions across a wide range of fields where mathematical, statistical, and computational sciences intersect with natural sciences and engineering, as well as with the humanities and social sciences, particularly in the areas of cognitive systems, networks, and neuroscience.

Professor Roderick Melnik has an extensive body of work, including publications in books, book chapters, and over 500 journal and conference articles. Summarizing his contributions in a brief introduction is a daunting task. However, it is essential to highlight Dr. Melnik’s remarkable ability to initiate and advance research in various fields, making significant contributions to each one. He is an endless source of inspiration and innovative ideas. Several key themes consistently persist throughout Professor Melnik’s scientific career. He has a deep and lasting interest in complex systems and networks, particularly how they couple and interact with their environment.

The pervasiveness of coupled systems and networks in all branches of science, engineering, human society, and the world surrounding us leads to many forefront challenges. These systems and their networks span a wide range of scales, from the nanoscale to planetary and cosmic scales, often exhibiting complex spatiotemporal interactions. Below, we highlight some representative contributions of Prof. Melnik within this rapidly evolving field. Following this, we provide an overview of the current Special Issue, and summarize the contributions from several esteemed research groups around the globe.

2. Physical and Engineering Coupled Systems and Their Networks

One of the areas of coupled phenomena that attracted the attention of Dr. Melnik at an early age was thermoelasticity, which is based on the principles of thermodynamics, elasticity, and heat conduction. Coupled thermoelastic phenomena are crucial in modern technology, from semiconductor engineering [1] to manufacturing and materials science. His contributions to this area are vast and insightful. Among others, they include essential works with his collaborators on the theory and applications of thermoelasticity with thermal relaxation, also known as hyperbolic thermoelasticity. Working on these problems, he expanded his interest to extended thermodynamics where non-equilibrium states and processes, particularly those involving fast and steep changes, may occur.

In describing the time evolution of a statistical ensemble, relaxation-time approximations and their generalizations play a fundamental role by connecting the distributions of the variables of interest at different stages of their temporal evolution [2]. A hierarchy of these approximations was explored by Dr. Melnik, which uncovered essential features of nonlocal and nonequilibrium phenomena in semiconductors.

Complex nonlinear dynamics, including those exhibited during various phase transformations, have been an ongoing topic in Dr. Melnik’s works. Some such topics and, in particular, their applications, have their origins in smart materials and structures. Among them is a class of materials with memory, such as shape memory alloys. In a quite general setting, many related problems can be formulated as dynamic problems of nonlinear thermoelasticity, as presented for the first time in Dr. Melnik’s research.

Further generalizations and systematic methodologies for constructing computational models for coupled multiscale dynamic problems were developed in [3]. The early works by Dr. Melnik continue to stand out today. There is a growing recognition within the scientific and engineering communities that many real-world problems are inherently interconnected. This interconnectedness or coupling can arise from various sources, such as the interaction of different physical fields. Examples of such coupled problems include electroelasticity, thermoelasticity, viscoelasticity, and magnetoelasticity. Additionally, analyzing different states of matter together leads to other coupled issues such as fluid–structure interactions, aeroelasticity, and various phase transitions. These problems are often referred to as multi-physics problems. However, this term does not encompass many other significant coupled issues where factors such as chemical and biological fields play a crucial role. For instance, in climate modeling, it is essential to consider the physical, chemical, biochemical, geological, and many other characteristics of the Earth and its surroundings, and how they interact with one another. Scientific fields and engineering disciplines such as biomechanics, climate modeling, geomechanics, and nanotechnology share a common thread: advancements in these areas are closely linked to our ability to effectively solve coupled problems [3].

In the context of smart materials, such as shape memory alloys (SMA), it is essential to use an extension of the standard Ginzburg–Landau theory by incorporating the full coupling between mechanical and thermal fields, leading to non-isothermal Ginzburg–Landau models, as shown in works of Dr. Melnik and his collaborators. Dealing with phase transitions in these materials requires the development of advanced approaches, such as those based on the phase-field theory, allowing them to handle sharp interfaces. Furthermore, at the numerical level, efficient procedures need to be developed, which were put in the framework of isogeometric analysis by Dr. Melnik and his team. Expanding the phase-field approach, they also developed new models and numerical procedures for SMA nanostructures with enhanced properties and other areas covering sensor and actuator technology, biomedical applications, and aerospace.

The field of smart materials and structures is abounded in coupled systems and networks. Another group from this field where Dr. Melnik made significant contributions is the development of mathematical models for piezoelectricity. Piezoelectric materials, ranging from crystals and ceramics to polymers and biological materials, have a wide range of applications, including sensor and actuator technology and energy harvesting. Additionally, piezoelectricity extends beyond the physical and engineering fields, as it is a dynamic phenomenon also observed in biological systems. Dr. Melnik was the first to rigorously prove the well-posedness of a large class of piezoelectricity models in the dynamic case. He also worked and contributed extensively to interactions of coupled piezoelectric systems with various media, such as acoustic and fluid, as well as accounting for the coupling to other physical fields such as thermal, focusing on the dynamic case.

Prof. Melnik’s quest for a better understanding of coupled systems and networks at the smallest scales, such as nano, has continued with his outstanding contributions in many different areas of nanoscience and nanotechnology. One such area is low-dimensional nanostructures such as quantum wells, nanorods, quantum dots, and superlattices where, with his doctoral students and colleagues, they made several important discoveries, including predicting barrier localization in modulated nanowires and quantum confinement phenomena in nanowire superlattices. These tiny objects exhibit unique properties and functionalities unseen in the bulk case. Leveraging these unique properties and potential for advanced technologies, they have diverse applications across various fields, including energy storage, catalysis, sensors, optoelectronics, and biomedicine, to name just a few.

In advancing the mathematical theory of these nanostructures, Dr. Melnik’s team developed a systematic methodology for incorporating deformation effects, nonlinear strain theory, as well as higher-order nonlinear electromechanical effects. They also proposed an innovative solution to a long-standing problem in the field. One of the prominent features of this research was accounting systematically for the coupling of physical fields that was also carried out to the general multiband case, and revealing new attributes of complex dynamics, as well as new phenomena and effects, particularly in the context of nanowire superlattices. Part of the research in this direction was linked to their studies of SMA nanostructures mentioned earlier.

Two-dimensional materials such as graphene and other layered materials have a special place among low-dimensional nanostructures. With its exceptional properties, including high strength, flexibility, and conductivity, the range of graphene applications continues to grow. A special focus of Dr. Melnik’s team in this context was strain engineering for graphene nanoribbons, where they provided new critical insight into the properties of these objects by using state-of-the-art first-principle ab initio calculations.

The other critical class of problems addressed in Dr. Melnik’s research was related to spintronics, where we attempt to control or manipulate electron spin for advanced technological applications, including high-density data storage, sensor and actuator technology, as well as emerging technologies pertinent to alternative types of computing such as quantum or neuromorphic. Much of the research in this direction, carried out by Dr. Melnik’s group, focused on spintronics in quantum dots and the applications of geometric phases (such as the Berry phase), as well as the Feynman disentangling technique (e.g., [4]). While advancing a general theory of spin–orbit coupling, critical new insight was also obtained for other classes of low-dimensional nanostructures, such as nanowire superlattices and graphene nanoribbons, as well for the theory and potential applications of quantum computing, quantum information processing, and next-generation solid-state quantum devices. The next generation of nanoelectronic devices is poised to benefit significantly from this development by reducing their power consumption and increasing their memory and processing capabilities.

3. Biological and Ecological Coupled Systems and Their Networks

As the importance of coupled biological systems continues to grow, the role of modelling and analysis in this field becomes critical. In many cases, information has to be transferred between scales to develop predictive models of complex behaviour, such as biocell dynamics, tissue evolution, or the progression of neurodegenerative diseases. To extract meaningful information from large datasets or complex networks, adequate mathematical approaches and computational tools, including multiscale techniques, are essential. Dr. Melnik has significantly contributed to these areas, with several breakthrough discoveries made due to the concerted interdisciplinary efforts of his team. Among the diverse areas he contributed to, we mention the analysis of complex nonlinear cell dynamics and the dynamics of molecules that carry the genetic instructions for the development and functioning of an organism, such as DNA. He and his team looked deeper into coupled processes and phenomena within biocells, revealing their importance and evaluating the role of microtubules, contributions of nonlocal effects, and exploring auxeticity in biosystems. Some of this research led them to uncover the role of cell structures and organelles, and to our better understanding of piezoelectricity and flexoelectricity in biological cells.

The research along these directions is just one example that discoveries can often be made at the interface between different disciplines, and such interfaces are never rigid. For example, the research in coupled physical and biological systems substantially overlap through their applications in bioengineering and biomedicine, representing other areas where we can find significant contributions from Dr. Melnik and his team. Some of them include the analysis of coupled effects in biological tissues, while others analyze the emergence of carbon nanotubes and their arrays as promising materials for various biomedical applications, ranging from tissue engineering to bioimaging.

Significant contributions were also made to nucleic architectonics, where the design and construction of complex structures and devices are carried out using nucleic acids as building blocks. Some of the key contributions were made to the development of RNA nanostructures, where Dr. Melnik and his team discovered the new phenomenon of self-stabilization and revealed their new properties through studies by molecular dynamics and atomistic-to-continuum modelling approaches.

Moreover, new nonlinear and nonlocal models were developed to better describe the vibrations of piezoelectric nanowire resonators. Such resonators hold a high potential in detecting tiny bio-particles such as DNA, RNA, proteins, viruses, and bacteria. Many of the models developed in Dr. Melnik’s group and described here were designed with data-driven environments in mind.

The intrinsic coupling between mechanical and thermal fields is essential in describing processes and phenomena connected with ultrashort pulsed lasers with their applications in diverse fields, from material processing to biomedicine. Amplified by full thermo–electro–mechanical interactions, it also plays a critical role in thermal ablation of biological tissues where heat relaxation time effects should also be considered [5]. When developing effective therapies that utilize radiofrequency, thermal ablation, and emerging technologies, it is crucial to consider the various domain heterogeneities involved. Additionally, distinct characteristics, such as the non-Fourier heat transfer, play a significant role in the thermal ablation of biological tissues, the laser ablation of brain tissues, and cardiac ablation. These topics were thoroughly examined in the studies of Dr. Melnik’s team.

A significant part of Dr. Melnik’s group research over recent years has been devoted to studies of the brain, brain networks, and neurodegenerative diseases, where new multiscale approaches have been developed by his group (e.g., [6]). Some of the problems at the forefront of the current knowledge in this field have been confronted by Dr. Melnik’s group with their developed nonlocal models (e.g., [7]), while others required in-depth analyses of complex non-Markovian dynamics and the application of nonequilibrium approaches.

Social and biosocial interactions and complex dynamics of knowledge creation have been the areas of interest for Dr. Melnik’s research for a long time. Together with his collaborators in Europe, they developed and analyzed the processes of meaningful social interactions and the emergence of collective knowledge through knowledge networks (e.g., [8]), as well as underlying key mechanisms of self-organized criticality in these social processes that led to several important discoveries pertinent to the intrinsic features of social dynamics. This research has underpinned the critical importance of coupled biosocial dynamics in the refined modelling of infection transmissions, including the recent pandemic. Furthermore, modelling human dynamics, including human crowds, has also led to valuable findings by Dr. Melnik’s group. The applications of the latter work span diverse fields, ranging from efficient evacuation procedures in emergencies to our better understanding of complex social and urban systems.

Developing new ecological models, including the areas of population dynamics and pattern formations, has been another area of Dr. Melnik’s contributions over the years. He and his team discovered that for addressing many problems in this area, accounting for nonlocal interactions is essential for an adequate description of the underlying processes where, in some cases, we observe the emergence of nonequilibrium dynamics. His group has also pioneered the models that allow for the integration of psychological effects and collective decision-making in cooperative behaviour.

4. Intelligent Systems and AI Technologies in Data-Driven Applications

As we mentioned, data-driven applications, coupled systems, and networks have been a driving force for much of Dr. Melnik’s research described in the previous sections, including the directions pertinent to complex dynamics and the emergent behaviour. Once human factors are incorporated in the modelling framework for control, as occurred for the first time in [9], a realm of intelligent systems emerge. Such systems are known as technologically advanced devices, computers, or machines that perceive and respond to the world around them through gathering, analyzing, and responding to data, potentially learning and adapting based on experience, mimicking humans. Artificial Intelligence (AI) is a technology behind this development that enables such devices, computers, or machines to simulate human learning, comprehension, problem solving, decision-making, creativity, and autonomy. Using predictive modelling, generative AI is putting this development at a new level, while underlying technologies on which it is built remain machine learning (ML) and deep learning (DL). These two latter technologies have been developed by Dr. Melnik’s group in various contexts, ranging from shape memory graphene nanoribbons to bio-inspired robot propulsion.

When AI is integrated with model-based design, reduced-order models can be particularly useful. Conversely, AI technologies can also enhance the development of reduced-order models. However, creating reduced-order models that effectively simplify complex, high-fidelity models while accurately capturing their behavior is a challenging task. Dr. Melnik’s group has addressed this issue in various data-driven applications ranging from materials science to biomimetic systems.

Bayesian methods serve as fundamental tools in many AI learning algorithms, helping to extract important information from relatively small datasets and manage missing data effectively. One notable Bayesian technique is Approximate Bayesian Computation, which allows for likelihood-free inference in AI applications. The development of this method, along with other Bayesian approaches, played a crucial role in Dr. Melnik’s group’s research, contributing to our improved understanding of neurodegenerative diseases such as Alzheimer’s and Parkinson’s.

The original paper on coupling control with human factors in complex systems was published in Engineering Applications of Artificial Intelligence [9]. It used intelligent transportation systems as a prototype example. The theoretical foundations of this control approach can be found in earlier papers (e.g., [10] and others).

The intersection of network science and AI provides powerful tools for understanding and managing complex systems. AI utilizes network structures to analyze data, make predictions, and automate tasks. In its turn, network science offers the framework for modeling and comprehending these intricate relationships. Dr. Melnik and his collaborators on this topic contributed significantly to this field, especially to higher-order networks revealing new intrinsic features of human connectomes and brain hubs (e.g., [11]).

5. Other Applications of Coupled Systems and Networks

There are many other fields pertinent to the research on coupled systems and networks where one can find remarkable contributions from Prof. Melnik. In this section, we collected only some of them.

The development of novel multiscale models and accounting for size effects, which can be captured with nonlocal models, have been one of the highly visible threads in his research. At smaller scales, particularly in nanotechnological applications, such effects can be of great consequence. At large scales, e.g., in climate modelling, their influence can be crucial, requiring far more detailed analyses than currently given in the literature. Both of these scenarios were examined in Dr. Melnik’s research.

Coupled systems and their networks are commonly encountered in manufacturing processes and materials science. Complex materials, such as polymers and composites, necessitate the development of efficient methods to enhance our understanding of the relationship between their structure and properties, even at the initiation of the computational process. In addressing these challenges, the results obtained by Dr. Melnik and his collaborators have significant implications, including in drug design and delivery, a field where he has made contributions in various contexts. Furthermore, when considering the dynamics of these complex materials, relaxation effects may become crucial. A significant portion of the research in this field involved the development and application of advanced computational methods, including molecular dynamics (MD) and first-principles studies. Dr. Melnik’s group also utilized these methodologies to uncover the properties of complex materials that are important to geosciences and seismology.

Green and alternative energy is a field of ongoing research where Dr. Melnik and his interdisciplinary team have made significant contributions. They developed innovative models for photovoltaic systems that convert sunlight into electricity using semiconductor materials. Additionally, they designed and analyzed energy systems to enhance efficiency. Dr. Melnik’s contributions also extend to other renewable energy sources, such as hydrogen production, which offers clean and sustainable energy solutions. Furthermore, he has worked on models to improve the efficiency of electric vehicles by accounting for stick-skid phenomena, enabling better control of their dynamics.

The development of novel models for energy harvesting and environmentally friendly, sustainable technologies continues to be an important avenue of research for Dr. Melnik’s team. This development also included research into photosynthetic systems and light-harvesting complexes, as well as lead-free piezocomposites, with detailed analyses of the roles of polycrystallinity, nanoadditives, and anisotropy, nonlinear, and nonlocal effects (e.g., [12]). Based on this research, they proposed several innovative multiscale designs of nanoengineered matrices for lead-free piezocomposites, including those where improved performance can be achieved via controlling anisotropy and auxeticity. Auxeticity is a phenomenon observed in single molecules, crystals, or specific macroscopic structures, and it has been actively explored across various fields [13], leading to an expanding range of applications. Formally, auxetic materials have negative Poisson’s ratios, with a counter-intuitive property of expanding rather than contracting perpendicular to an applied stretch. Another compelling property that needs to be considered when modelling lead-free piezoelectric composites is related to flexoelectricity. It is a universal electromechanical coupling effect in which electric polarization occurs in a material due to a gradient of mechanical strain or, conversely, mechanical stress occurs due to a gradient of the electric field. Dr. Melnik’s team made essential contributions to the flexoelectric enhancement of green composites, which were further expanded to the development of fully coupled models for such composites, accounting for the influence of thermo–electro–mechanical interactions.

We also note that Dr. Melnik’s team made many cutting-edge contributions to developing more efficient industrial processes and innovative technologies in other core parts of the energy sector. Among their notable achievements is their work on altering the wetting state of materials, a promising area of research with potential applications across different fields of engineering and science.

Nonperturbative approaches are becoming increasingly important in various fields of physics and related disciplines. These include equilibrium and nonequilibrium dynamics, quantum mechanics, particle physics, nanoscience, nanotechnology, and quantum gravity. For example, the nonequilibrium Green’s function (NEGF) method is a well-known nonperturbative approach that is widely utilized in condensed matter physics and nanoscience. Developing nonperturbative methods and computational algorithms is vital for effectively describing various complex systems. The significance of nonperturbative effects is well-known, particularly in systems such as spin glasses. Such effects often emerge in critical phenomena, where the renormalization group (RG) serves as the most widely used general method. This is the area where Dr. Melnik and his collaborators from the EU made significant advancements. It is crucial to develop nonperturbative approaches, including functional renormalization, Monte Carlo (MC) methods, and Monte Carlo RG (MCRG) methods, on which they have successfully worked.

Notable models in this field, such as the Wetterich and Polchinski equations, are recognized as nonperturbative or exact RG equations. However, these equations can often only be solved approximately. Thus, nonperturbative simulation methods, such as various MC and MCRG algorithms, play a crucial role in studying systems of interacting particles and critical phenomena, as they offer a reliable foundation for comparing and verifying results. Dr. Melnik’s team proposed a new and more efficient method for solving the Wetterich equation than existing counterparts. The Wetterich equation is a fundamental concept in the Functional Renormalization Group used to analyze the flow of effective actions in quantum field theories. It has applications in various fields, including statistical physics and high-energy particle physics, especially quantum chromodynamics and quantum gravity, particularly within the framework of the asymptotic safety scenario.

The development and applications of stochastic mathematical models have been a significant part of Dr. Melnik’s research portfolio. Some of them we have already mentioned in the context of modelling biosocial systems, while others are more related to business, economics, and finance. Stochastic processes are often used as mathematical models to represent systems and phenomena that appear to change randomly. These models incorporate elements of randomness and uncertainty, allowing for analyzing systems with unpredictable outcomes. By doing so, they enable the estimation of probability distributions for potential results. One of the key contributions of Dr. Melnik’s team in this area, pertinent to finance, was related to first passage time problems and jump-diffusion stochastic processes. The first passage refers to the time it takes for a stochastic process to reach a specific state or boundary. This concept is vital and has applications across various fields, including finance, engineering, and biology, as it aids in analyzing and predicting system behaviour.

Materials science explores the properties and behaviour of complex multiscale materials, presenting numerous challenging problems in network science. In its turn, network science offers tools and frameworks for analyzing complex systems, including those encountered in materials science. By combining these two fields, we can gain a deeper understanding of material structures and properties, which includes the ability to predict synthesis routes and optimize material behaviour. Some of the contributions of Dr. Melnik to these fields have already been mentioned, including his works on materials with memory, polymers, and nanostructured materials. Among others, we should also mention here his impressive contributions, made in collaboration with other scientists, on intermetallic compounds, carbon allotropes such as diamond, graphene and graphyne, polymorphic systems and perovskites, as well as special types of nanostructures such as ZnO and diamond nanowires. Diamond nanowires have a variety of potential applications, including quantum technologies, sensing, and biomedical devices. ZnO nanowires also possess a wide range of properties and applications, such as semiconducting, piezoelectric, and photocatalytic capabilities. Beyond these uses, ZnO nanostructures can also be utilized in solar cells, energy storage devices, and energy harvesting devices. Additionally, they play a role in environmental remediation by facilitating the photocatalytic degradation of pollutants in both water and air. Some of this research required the development of new computational methods to handle the problems at hand, ranging from innovative MD procedures and first-principles calculations to efficient combinations of nonequilibrium Green’s function and DFT. It is also important to highlight that Dr. Melnik’s research in these areas faced significant challenges related to complex phase transformations, requiring determining phase stability limits, as well as to complex interfaces and defects, including those induced by dislocation nucleation.

Gyroscopic systems are utilized in a variety of applications, ranging from everyday devices to advanced technologies in space, such as spacecraft and satellites that maintain their pointing direction. They are crucial in astronomy as well, helping to ensure the orientation and stability of telescopes such as Hubble. This stability is essential for making precise observations of the universe and advancing our understanding of cosmology. Dr. Melnik’s team has also made significant contributions to this field.

Plasmonics, electrochemical systems, and the applications of 2D materials remain crucial directions of Dr. Melnik’s research, where he continues working with his students and colleagues, discovering new properties of coupled systems in these fields and developing efficient methods for their studies.

The list of areas Professor Roderick Melnik has contributed to is extensive, and we cannot exhaust it here. However, it is important to highlight that while there are many distinguished researchers in mathematical modelling, few have made significant contributions across such a diverse range of fields as he has. One might question how one individual could impact so many disciplines. A key aspect of this is his research’s integration through a groundbreaking paradigm shift towards the study of coupled complex systems and networks that he initiated at the beginning of his career. However, there is also another key aspect. Those who know Dr. Melnik well often attribute his success to his broad interdisciplinary knowledge, deep insights, and warm and humble personality. He treats everyone with respect, whether they are distinguished professors or students. It is, therefore, not surprising that he has maintained long and productive collaborations with his former students, postdoctoral fellows, and researchers from various disciplines.

6. Current Issue

The first article of this Special Issue, authored by Wang and Zhai [14], introduces fast explicit numerical schemes for solving the extended Fisher–Kolmogorov (EFK) equation, which is an important model for phase transitions and bistable phenomena. Their approach combines the integrating factor Runge–Kutta method and the Fourier spectral method to obtain a high-precision numerical method for the EFK model. Further, three numerical experiments were conducted to investigate the performance and applicability of their reported method, focusing on convergence analysis and the behavior of diminishing energy. Cui et al. [15] presented efficient finite difference schemes for solving one- and two-dimensional wave equations with fractional damping. The authors proved the unconditional stability and convergence of the proposed difference schemes by testing them on several numerical examples. Yao and Weng [16] introduced a numerical method based on an operator splitting collocation scheme for solving the nonlinear Schrödinger equation. The equation is split into linear and nonlinear parts utilizing a second-order operator splitting method combined with the barycentric Lagrange interpolation collocation method for spatial discretization and the Crank–Nicolson scheme for temporal discretization. The presented numerical experiments validated the accuracy, mass, and energy conservation of the reported method. In another study featured in this Special Issue, Ivanov and Shelyag [17] developed stable periodic solutions for a simple form delay differential equation with nonlinear negative periodic feedback. The theoretical derivations and exact analytical calculations of the considered equation are further verified using extensive numerical simulations. De la Sen [18] investigates the loss of the Sturm–Liouville property of time-varying second-order differential equations in the presence of delayed dynamics. The time-delay systems naturally arise in several real-world scenarios, including biological processes such as epidemic modelling, ecology and medical systems, and the sunflower equation. Furthermore, they play an important role in diffusion phenomena, war and peace problems, and missile/anti-missile strategies. Such delay models are also very relevant in control theory and applications. Wang et al. [19] introduced a conservative and compact finite difference scheme aimed at preserving both the mass change rate and energy when solving the sixth-order Boussinesq equation with surface tension. Through the theoretical analysis, the authors demonstrate that the proposed scheme achieves fourth-order accuracy in spatial discretization and second-order accuracy in temporal discretization. The solvability, convergence, and stability of the scheme are rigorously proven using the discrete energy method. Furthermore, a series of numerical experiments are conducted to highlight the effectiveness and reliability of the scheme for long-time simulations. Torokhti and Soto-Quiros [20] proposed a method for the best constructive approximation of a nonlinear operator within probability spaces. This approach is highly adaptable, and can be extended to several applications, such as the development of efficient algorithms for signal reconstruction and denoising. Moreover, the proposed approach proves valuable in modeling and estimating dynamic systems with stochastic inputs, enhancing predictive accuracy and enabling more effective adaptive control strategies.

Barua and Povitsky [21] developed a comprehensive model to simulate the reactor-scale flow dynamics in chemical vapor deposition, a common industrial process that incorporates a complex combination of fluid flow, chemical reactions, and surface deposition. The model couples the Finite-volume and the Direct Simulation Monte Carlo methods to simulate the complex interactions around the micro- and nano-scale fibers. In another study, Mahoney and Povitsky [22] introduced the Method of Fundamental Solutions (MFS) to model the chemical vapor infiltration with a surface chemical reaction. Their approach computed the dynamics of pore size and location to evaluate the quality of the material obtained by chemical vapor infiltration. The deposition rates are calculated from the fibers’ surface concentrations obtained by MFS, and the geometry is updated at each time step for modelling the pore filling over time. The developed model fidelity and integrity are evaluated through comparison with known analytical results for simple geometry as well as published experimental data. In [23], Povitsky introduced a meshless method of fundamental solutions to model particle sedimentation near corrugated (rough) surfaces. The proposed method is able to accurately predict the particle velocity and trajectory within the Stokes flow regime as they move along the corrugated surface under the action of gravity. The developed method holds significant potential for applications in biological systems and microfluidic devices.

Dutta and Layton [24] developed a computational model for simulating the epithelial transport of electrolytes and water along the nephrons in a male rat kidney. The developed model is able to predict the renal transport of Ca²⁺ and Mg²⁺ as well as other electrolytes and water under different physiological conditions. The developed models can serve as a valuable tool for in silico studies on kidney adaptation to alterations in Ca²⁺ and Mg²⁺ homeostasis in different conditions, such as pregnancy, diabetes, and chronic kidney disease. Bianchi et al. [25] reported a network-based study of the dynamics of Aβ and τ proteins in Alzheimer’s disease. Four network-based models are presented to capture the spatial and temporal dynamics of Aβ and τ proteins in the brain, which were then directly compared with the clinical data to enhance understanding of disease pathology. In another study, Drapaca [26] introduced a mathematical model to investigate the impact of Alzheimer’s drug donepezil hydrochloride on neuronal viscoelasticity and action potentials. The model couples the controlled drug release, transport to the brain, and its effect on action potential. Findings suggest that combining various drug release modalities and dosages could enhance the therapeutic efficacy of donepezil. Jabeen and Ilie [27] introduced a finite-difference strategy for approximating second-order parametric sensitivities for stochastic discrete models of biochemically reacting systems. The proposed strategy employs adaptive tau-leaping schemes and couples perturbed and nominal processes to enhance sensitivity estimation efficiency. The advantages of the proposed technique are demonstrated through applications to various biochemically significant system models. In another study featured in this Special Issue, Sacco et al. [28] reported a reduced-order model for cell volume homeostasis to simulate aqueous humor production in the human eye. The model computes the cell volume dynamics and fluid velocity by coupling intracellular reactions, transmembrane solute exchange, and Starling’s Law. The extension of proposed computational framework could serve as a virtual laboratory to further test in vivo experiments and machine learning-based data analysis, assisting in the prevention and treatment of ocular diseases, such as glaucoma.

Hojati et al. [29] proposed a spatio-temporal indexing data structure for distributed networks, DSTree, that operates in the data layer using a distributed InterPlanetary File System (IPFS) network. This proposed framework enables peer-to-peer (P2P) data sharing and queuing without the need for any third-party central entities. The study highlighted that, for time-series data, such as storing sensor data, DSTree improves spatio-temporal query performance by approximately 40% for small and medium datasets. However, challenges remain, such as 20% slower insertion speed and limited semantic query capabilities. The study concluded by highlighting the need for future research efforts from GIScience and related fields to enhance decentralized applications and standardize geographic data sharing on IPFS networks. In another study, Omar et al. [30] analyzed and optimized a propagation model for malware in multi-cloud environments (MCEs) incorporating the effects of Brownian motion. The study explores the MCEs’ network dynamics using six-state virtual machines (VMs), deterministic and stochastic dynamics models, and four popular multi-cloud environments: media, healthcare, finance, and educational servers. The analysis covers solution existence, uniqueness, equilibrium, and stability for both modeling approaches. After that, the two developed modeling approaches’ solution existence, uniqueness, equilibrium, and stability are carefully investigated. Using an optimal control strategy, the models are tested to maintain VM security levels and minimize malware propagation. The reported results showed the effectiveness of the control methods in managing malware infections.

In conclusion, this Special Issue highlights novel findings of the state-of-the-art computational models that integrate multiscale, multiphysics, and data-driven methodologies, along with artificial intelligence and machine learning techniques. We believe that this Special Issue will provide the readers with ample knowledge and lead to the generation of novel ideas for future advancement, pushing the boundaries of knowledge in mathematical and computational sciences and their diverse applications including natural, social, and engineering systems.

Finally, we extend our sincere gratitude to all the authors who contributed high-quality articles [14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30] to this Special Issue, as well as to the referees whose insights and expertise were invaluable.