An Introductory Monograph on Solution Behaviour of Dynamical Equations

Title: An Introductory Monograph on Solution Behaviour of Dynamical Equations

Author: Dr. Nimai Sarkar

Publisher: Cogniverse Press, Jorhat, Assam, India

First Edition: July 2026

DOI: 10.5281/zenodo.21241948

ISBN: 978-81-688771-4-6 (Print Edition)

e-ISBN: 978-81-688771-5-3 (Digital Edition)

Cover Designing: Cogniverse Press Digital Team

Copyright: © Author

• Dr. Nimai Sarkar – Assistant Professor, Department of Mathematics, School of Advanced Sciences, VIT-AP University, Amaravati, Andhra Pradesh, India

Cogniverse Press

Nakari Gaon, Borigaon Siding, Jorhat – 1, Assam, India

Dynamical equations form a cornerstone of modern mathematical analysis and applied sciences, providing powerful frameworks for describing systems influenced by time, memory, delays, accumulated effects, nonlocal interactions, and external perturbations. Since explicit analytical solutions are often unavailable for realistic models, understanding the qualitative behaviour of solutions—including existence, uniqueness, stability, boundedness, positivity, continuous dependence, attractivity, and long-term dynamics—becomes essential.

This monograph presents a concise and systematic introduction to the qualitative behaviour of various classes of dynamical equations. Beginning with classical ordinary differential equations, the discussion gradually progresses to delay and neutral differential equations, followed by fractional and integro-differential equations that incorporate memory, hereditary effects, and nonlocal interactions. The objective is to provide readers with a clear mathematical understanding of how different structural characteristics influence solution behaviour.

Written for postgraduate students, research scholars, early-career researchers, educators, and professionals in applied mathematics, the monograph also serves readers from engineering, physics, biology, control theory, and related disciplines who employ dynamical models in their research. Emphasis is placed on qualitative analysis rather than lengthy technical derivations, allowing the work to function both as an introductory reference and a foundation for advanced study.

The monograph identifies several emerging research directions, including fractional delay systems, neutral fractional equations, Volterra–Fredholm integro-differential models, Ulam-type stability, positivity-preserving numerical methods, and mathematical models involving memory and nonlocal effects. It is intended to encourage further research by providing a coherent overview of the subject and highlighting important open problems.

The author expresses sincere gratitude to teachers, colleagues, collaborators, friends, and well-wishers for their encouragement and support, while welcoming constructive feedback from readers for future editions.

Dr. Nimai Sarkar

The author gratefully acknowledges the blessings of Almighty God and extends heartfelt thanks to teachers, colleagues, collaborators, family members, friends, students, readers, and well-wishers whose encouragement and support made this work possible. Appreciation is also expressed to the publishing house for its guidance and assistance in bringing the monograph to publication. Constructive suggestions from readers are welcomed and will contribute to improving future editions. Any remaining errors or shortcomings remain solely the responsibility of the author.

• Dynamical Equations
• Ordinary Differential Equations
• Delay Differential Equations
• Neutral Differential Equations
• Fractional Calculus
• Fractional Dynamical Systems
• Integro-Differential Equations
• Volterra–Fredholm Models
• Solution Behaviour Analysis
• Stability and Boundedness
• Ulam Stability
• Nonlinear Analysis
• Mathematical Modelling
• Memory and Nonlocal Effects
• Applied Mathematics