In today’s world, everything is changing rapidly such as technology, lifestyle, which affects us very much.
Due to change in technology, now days come in our life when we need to change each and everything human-friendly as well as a technology base. In this direction, we need to change our thought process and come out from the purview of our life to the open sky. As I am a mathematician, I think we must think about the change in Mathematical thinking and mathematical problems. From the past century, we studied mathematics for integers only, but from the last decade, mathematics involves fractional values also. Many mathematicians are working on the new field of mathematics as fractional calculus. They are introducing the new concept of mathematical calculation by the use of fractional calculus.
The subject of fractional calculus has applications in diverse and widespread fields of engineering and science such as electromagnetic, viscoelasticity, fluid mechanics, electrochemistry, biological population models, optics, and signals processing. It has been widely used to model physical and engineering processes that are found to be best described by fractional differential equations.
The fractional derivative models are used for accurate modeling of those systems that require accurate modeling of damping. In these fields, various analytical and numerical methods to new problems have been proposed in recent years including their applications. This special issue on “Fractional Calculus and its Applications in Applied Mathematics and Other Sciences” is devoted to studying the recent works in the above fields of fractional calculus done by the leading researchers and eminent Mathematician.
Mathematical modeling of real-life problems usually results in fractional differential equations and various other problems involving special functions of mathematical physics as well as their extensions and generalizations in one or more variables. In addition, most physical phenomena of ecological systems, fluid dynamics, electricity, quantum mechanics, and many other models are controlled by fractional order PDEs within their domain of validity.
Therefore, it becomes increasingly important to be familiar with all recently developed methods and traditional methods for solving fractional order PDEs and the implementations of these methods.
Fractional calculus is also used in Oil industries.
Fractional calculus can provide a concise model that occurs in biological tissues for the description of the dynamic events. Such a description is important for gaining an understanding of the underlying multi-scale processes that occur when, for example, tissues are mechanically stressed or electrically stimulated. The fractional calculus has been applied successfully in chemistry, physics, and materials science to describe dielectrics, electrodes and viscoelastic materials over extended ranges of time and frequency.
In heat and mass transfer, for example, the half-order fractional integral is the natural mathematical connection between material gradients (thermal) and the diffusion of heat or ions. Since the material properties of tissue arise from the nanoscale and microscale architecture of subcellular, cellular, and extracellular networks, the challenge for the bioengineer is to develop new dynamic models that predict macroscale behavior from microscale observations and measurements.
The complexity of all living systems is expressed in the structure and function of each tissue and cell. Thus, the biological functions of cardiac muscle, articular cartilage, and the spinal cord, for example, are embedded in the three-dimensional structure of each tissue’s cells, extracellular matrix, and overall anatomical organization. In the heart, tight electrical contacts between cardiac cells ensure that the pacemaker signals are distributed sequentially to the atria and ventricles; in the knee, the multiple layers within hyaline cartilage distribute transient loads by the rapid movement of ions and water; while in the axons of the spinal cord, sensory input and reflexes are expressed via electrical signals – action potentials – that are directed through complex neural networks.
The physiologist seeks to understand such complex behavior by gently probing the cell and tissue environment and by developing mathematical models that describe the resulting perturbations (e.g., ECG changes, gait variation, evoked potential latency). These mathematical models are typically constructed using linear differential equations (LDE) and provide a means for predicting the time variation of the experimentally measured fields, forces, and flows that regulate biomechanical, neural and hormonal processes.
Fractional calculus models provide a relatively simple way to describe the physical and electrical properties of heterogeneous, complex, and composite biomaterials. There is a multiscale generalization inherent in the definition of the fractional derivative that accurately represents interactions occurring over a wide range of time or space. Thus, we can avoid excessive segmentation or compartmentalization of tissues into subsystems or subunits — a system reduction that often creates more computational and compositional complexity than can be experimentally evaluated.
Finally, fractional calculus models suggest new experiments and measurements that can shed light on the meaning of biological system structure and dynamics. Thus, by applying fractional calculus to model the behavior of cells and tissues, we can begin to unravel the inherent complexity of individual molecules and membranes in a way that leads to an improved understanding of the overall biological function and behavior of living systems.
From this study, we come to know that, mathematics is changed rapidly and soon we would have a new field of mathematics to study.
