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This book explores the fundamental concepts of derivatives and integrals in calculus, extending their classical definitions to more advanced forms such as fractional derivatives and integrals. The derivative, which measures a function's rate of change, is paired with its counterpart, the integral, used for calculating areas and volumes. Together, they form the backbone of differential and integral equations, widely applied in science, technology, and engineering. However, discrepancies between mathematical models and experimental data led to the development of extended integral forms, such as the Riemann-Stieltjes integral and fractional integrals, which integrate functions with respect to another function or involve convolutions with kernels. These extensions also gave rise to new types of derivatives, leading to fractional derivatives and integrals with respect to another function. While there has been limited theoretical exploration in recent years, this book aims to bridge that gap. It provides a comprehensive theoretical framework covering inequalities, nonlinear ordinary differential equations, numerical approximations, and their applications. Additionally, the book delves into the existence and uniqueness of solutions for nonlinear ordinary differential equations involving these advanced derivatives, as well as the development of numerical techniques for solving them.
About the author
Abdon Atangana is a Professor of applied mathematics at the University of the Free State, Bloemfontein, South Africa.
¿lknur Koca is a Professor at Müla S¿tk¿ Koçman University, Kötekli/Müla, Türkiye
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Its strengths lie in its theoretical depth and rigor, systematic structure, and practical relevance ... . The mathematical proofs and inequality results provide a solid foundation for further research, making it a valuable resource for advanced study. ... the book is a significant contribution to fractional calculus, broadening the modeling tools available for complex systems in science and engineering. (Sameh Turki, Mathematical Reviews, December, 2025)
