Applied Mathematics in Science and Engineering

Complex and nonlinear dynamic models are characterized by intricate attributes like high dimensionality and heterogeneity, having fractional-order derivatives and constituting fractional calculus, which brings forth a thorough comprehension, control and optimization of the related dynamics and structure. This requirement, posing a daunting challenge, has grabbed attention in different fields of science where fractional derivatives and nonlinearities interact. Fractional models have, correspondingly, become relevant to deal with phenomena with memory effects in contrast with traditional models of ordinary and partial differential equations. The complicated phenomena manifested by nonlinear systems cannot be restricted to classical mechanics per se, there is rather a manifestation of mathematical behavior of solutions of differential equations to be involved. Partial differential equations are employed to describe continua such as temperature, concentration distribution, electromagnetic fields, fluids, elastic solids, quantum-mechanical probabilities, and so forth. Exploration of complex motions has become feasible through powerful computer affordances among this wide breadth of phenomena exhibited by various examples in real life. Thus, computer modeling has become a viable approach to predict the behavior of nonlinear systems which can in nature be approximately linear when small-amplitude is applicable. Despite motion being deterministic and following the laws of classical mechanics, it is also extremely sensitive to initial conditions with nonlinearities leading to chaos. This extreme sensitivity to initial conditions of the motion for nonlinear systems makes it essential to have quantitative measures at hand to characterize the degree of order and interpret complicated dynamical motion of systems accordingly. Such astounding qualities of chaos theory are due to chaos being found within trivial systems, and a system should display sensitive dependence to initial condition in the sense that neighboring orbits separate exponentially fast on average, so that a system can be called chaotic.

Chaos and fractals are integral parts of a grander subject known as dynamics which deals with change with systems evolving in time. It is dynamics that is used to analyze the behavior even if the system under consideration settles down to equilibrium, keeps repeating in cycles or displays something more complicated. Differential equations, describing the evolution of systems in continuous time, and iterated maps known as difference equations, arising in problems where time is discrete, are stated to be the main types of dynamical systems. The iterated maps are particularly useful to provide simple examples of chaos and act as tools to analyze periodic or chaotic solutions of differential equations. When compared with integer-order calculus that constitutes the mathematical basis of most control systems, fractional calculus provides better equipment to deal with the time-dependent impacts observed. Analysis and control of fractional order nonlinear systems seem to be another challenge with the observation of unknown inputs and concepts being used and analytically derived. Being a noteworthy attribute of complex systems, nonlinearity represents the various interactions between the variables in a nonlinear fashion. Emergence, feedback, adaptiveness, irreducibility, chaos, operating between order and chaos, having multilayers of structure as well as self-organization are among some of the other attributes. Multiple nonlinear complex systems demonstrate phenomena where oscillations improve periodic behaviors and synchronization of the system. Therein, the aggregate of parts is in fact not equal to the sum of their characteristics, behaviors and actions.

Fractional dynamics, along with fractional calculus, through the investigation of fractional-order integral and derivative operators with real or complex domains in science and engineering, has accordingly merged with the advances in the high-speed and applicable computing technologies; and hence, computational processing analyses, as a method of reasoning and a main pillar of majority of current research, can be of aid to tackle nonlinear dynamic problems in complex systems through novel strategies based on observations and complex data. Besides this aspect, tackling uncertainty in multiple attribute group decision-making processes is required to attain exact, precise and intelligent solutions in real life, which is characterized by the complexity in engineering, scientific, social and economic environments. To this end, to be able to provide feasible and applicable solutions within the dynamic processes of the complex nonlinear systems, methods related to analytical, numerical, simulation-related and computational analyses can be employed by taking into consideration the control-theoretic aspects thereto associated. Thus, this stance can enable to bridge the gap between mathematics and computer science besides the other wide array of sciences so that transition from integer to fractional order methods can be ensured. Along this strand, fractional derivatives and fractional differential equations are extensively employed in the mathematical modeling of diverse dynamic processes in the physical and natural world, providing ample aid for the accurate description of complex, dynamic and nonlinear behaviors of nature. All these aspects are important for the optimal prediction solutions, critical decision-making processes, optimization, quantification, multiplicity, controllability, observability and stabilization of fractal, fractional, quantum-related, neural and computational systems amongst many others.

This sophisticated and multiscale approach with computer-assisted applications and constructions has become more prominent in nonlinear dynamic systems, facilitating to achieve enabling solutions, optimization processes, numerical simulations besides technical analyses and related applications in areas like mathematics, medicine, neuroscience, engineering, physics, biology, virology, chemistry, genetics, information science, computer science, information and communication technologies, informatics, space sciences, applied sciences, public health, finance, economics and social sciences, to name some. Accordingly, Part II series of our special issue aims to provide a way towards original multi-, inter- and intradisciplinary as well as intrinsic goal-oriented research based on advanced mathematical modeling and computational foundations. Thus, we expect to get studies submitted on theoretical and applied dimensions of nonlinear, dynamic and complex systems merging mathematical analysis, methods and computing technologies to be presented in order to demonstrate the significance of novel integrative approaches in real systems and related realms.

The potential topics of our special issue include but are not limited to:

- Computational methods for dynamical systems of fractional order
- Fractional calculus of variations and optimal control with time-scale
- Fractional calculus with computational complexity
- Fractals and nonlinear dynamics
- Data-driven forecasting of high-dimensional chaotic systems
- Data-driven fractional biological modeling
- Control/optimization of dynamical systems
- Fractional hypergeometric functions and applications in science and engineering
- Synchronization of dynamic systems on time scales
- Data mining with fractional calculus methods
- Fractional order observer design for nonlinear systems
- Adaptive tracking control for multiple unknown fractional-order systems
- Nonlinear control for epidemic/biological diseases
- Mathematical epidemiological modelling for dynamic infectious/viral diseases
- Fractional differential equations with uncertainty
- Fractional dynamic processes in medicine
- Fractional calculus with artificial intelligence applications
- Image / signal analyses based on soft computing
- Fuzzy fractional calculus
- Wavelet analysis and synthesis of fractional dynamics
- Entropy of complex dynamics, processes and systems
- Nonlinear periodicity and synchronization
- Mathematical modelling of complex systems
- Quantization optimization algorithms
- Integrative machine-learning and neuroscience
- Computational medicine and fractional calculus in nonlinear systems
- Control and dynamics of multi-agent network systems
- Computational complexity
- Nonlinear integral equations within fractional calculus in nonlinear science
- Signal processing and design for scalar conservation laws
- Deterministic and stochastic fractional differential equations
- Stochastic dynamics of nonlinear dynamic systems
- Fractional calculus with uncertainties and modeling
- Fractional dynamic processes in medicine
- Fractional-calculus-based control scheme for dynamical systems with uncertainty
- Control theory, optimization and their applications in complex systems
- Computational intelligence-based methodologies and techniques
- Neural computations with fractional calculus
- Bifurcation and chaos in complex systems
- Quantum computation and optimization models
- Quantum mechanics
- Quantum information theory, information processing and / or quantum algorithms
- Special functions in fractional calculus context
- Optimal synergetic control for fractional-order systems
- Artificial Intelligence in complex systems
- Cyber-human system modeling and control with fractional order dynamics

Among the many other related points with mathematical and computational modeling aspects

Please submit through the journal submission portal and select "Advanced Fractals and Fractional Calculus - Part II" from the drop down box when prompted.

Instructions for AuthorsSubmit an Article