Calculus is a unique branch of mathematics discovered by Sir Isaac Newton. A fun fact is that Isaac Newton apparently invented calculus when an outbreak of plague forced him to spend a year in idleness in a village. Truly, a genius.
Calculus is the study of continuous change and is also referred to as infinitesimal calculus, because it usually deals with infinitesimal quantities, or small quantities. It is used in almost every branch of science, especially in physics. Besides that it is also used in classical mechanics, electromagnetism, economics, and even in medicine.
Calculus is differentiated into differential calculus, and integral calculus. Differential calculus is the study of the rate of change and slopes and curves. Integral calculus deals with the accumulation of quantities and area under curves.
Ordinary calculus deals only with integers, or whole numbers (such as 1, 2, 3…). Fractional calculus deals with non-integer quantities such as fractions and complex numbers (e.g. 1/3, 13/9…).
Fractional calculus has attracted the interest of a large number of mathematicians, scientists, and engineers over the last three decades. Fractional differential equations (FDEs) have use in various fields such as physics, chemistry, economics, biology, etc. These equations are applied in control theory, viscoelasticity, porous media, electrochemistry, and electromagnets.
Abstract nonlinear systems have been well studied in ordinary differential equations, and are now being applied to fractional differential systems. Fractional differential systems are also known as nonlinear evolution FDEs (EFDEs).
Although a lot of research has been done in this field, such as, transforming partial FDEs into non-autonomous EFDEs (NEFDEs), and exploring the local and global existence of mild solutions to a family of NEFDEs, there has not been any application of this to existing mathematical models to show the efficacy of their proposed model.
In a previous study, the existence of mild solutions has been extended to autonomous semilinear fractional integrodifferential equation (FIDEs) along with nonlocal conditions. FIDEs are equations that involve both fractional derivatives and fractional integrals. They are used in physics, biology, control systems, and finance. However, these results too were not applied to mathematical models to prove their efficacy.
In this study, the authors Mr. Kamla Kant Mishra and Prof. Shruti Dubey from the Department of Mathematics, Indian Institute of Technology (IIT) Madras, Chennai, India, have extended previous works done in this field to a family of non-autonomous FIDEs under nonlocal conditions.
This work further explored the existence of local and global mild solutions of more general FIDEs in the context of time-variant dynamics and nonlocal conditions.
FIDEs under nonlocal conditions offer a more comprehensive framework for modelling complex systems. They also provide a more accurate description of phenomena exhibiting long-range dependencies such as anomalous diffusion or viscoelastic materials.
The aim of the authors was to investigate the nonlinear delay FIDEs more deeply and establish outcomes in addition to what has already been achieved.
Future research in this area could use a weaker condition for proving the existence and uniqueness of mild solutions. This could open up a new direction for studying the behaviour of FDEs in a broader class of functions.
Dr. Muslim Malik, Associate Professor at the School of Mathematical & Statistical Sciences, Indian Institute of Technology (IIT) Mandi, Kamand, India, acknowledged the importance of this study with the following comments: “The article makes a substantial contribution to the field of fractional differential equations and stands out due to its integration of various advanced mathematical tools, including fractional calculus, resolvent operator theory, Banach contraction mapping, and fixed-point techniques. The use of these methodologies exemplifies the innovative application of mathematical theories. By combining fractional calculus with resolvent operator theory, the article introduces a robust framework for addressing complex equations, presenting these approaches in a cohesive and comprehensive manner. Additionally, the examples provided at the conclusion of the article underscore the practical relevance of the findings, effectively illustrating how the theoretical results can be applied to real-world scenarios.”
Article by Akshay Anantharaman
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