The main target of this book is to present a new concept of Ulam-type stability, i.e., multi-stability, through the classical, well-known special functions and to obtain the best approximation error estimates by a different concept of perturbation stability including fuzzy approaches for uncertainty considerations. This stability allows us to obtain diverse approximations depending on various special functions that are initially chosen and to evaluate maximal stability and minimal error which enable us to obtain a unique optimal solution of functional equations, inequalities, and fractional equations. Stability analysis in the sense of the Ulam and its different kinds has received considerable attention from the researchers. However, how to effectively generalize the Ulam stability problems and to evaluate optimized controllability and stability are new issues. The multi-stability not only covers the previous concepts but also considers the optimization of the problem and provides a comprehensive discussion of optimizing the different types of the Ulam stabilities of mathematical models used in the natural sciences and engineering disciplines with fuzzy attitudes. Besides, this book also deals with nonlinear differential equations with various boundary conditions or initial value problems, based on the matrix Mittag-Leffler function, fixed point theory, as well as Babenko's approach to study uniqueness and existence of solutions. In general, the benefits for the readers can be concluded as follows:
1. Evaluates maximal stability with minimal error to get a unique optimal solution.
2. Discusses an optimal method of the alternative to study existence, uniqueness, and different types of Ulam stabilities under special consideration of the fuzzy approaches.
3. Delves into the new study of boundary value problems of fractional integro-differential equations with integral boundary conditions and variable coefficients.

Tofigh Allahviranloo is a full professor of applied mathematics at Istinye University in Turkey and the Islamic Azad University (IAU) Science and Research Branch in Iran. He has previously worked at several universities, including the University of Prince Edward Island in Canada, Izmir University, and Bahcesehir University in Turkey.

As a mathematician and computer scientist, Prof. Allahviranloo is passionate about multi- and interdisciplinary research. He specializes in fundamental research in fuzzy applied mathematics, particularly fuzzy differential equations, and is focused on innovative applications in applied biological sciences. Currently, he is developing a new mathematical space to address complex uncertainties and is defining a new concept of uncertainty.

Safoura Rezaei Aderyani is a Ph.D. student in Mathematics at the Iran University of Science and Technology. She has extensively published papers on Ulam's type stability, including studies on difference, differential, functional, and integral equations. Her work also explores the applications of Ulam's type stability and its connections to other mathematical areas. Additionally, Rezaei Aderyani has contributed as an editor to several papers and special volumes focused on these subjects.

Reza Saadati is a faculty member at the Department of mathematics, Iran University of Science and Technology, in Iran. Dr. Saadati has authored over 5 international books and approximately 280 papers published by renowned publishers like Elsevier, Springer, and Wiley. He is the editor of several well-known journals such as Iran J Fuzzy System and fixed point theory and algorithms for Sci. and Eng-Springer.

Chenkuan Li has been a full professor at the Department of Mathematics and Computer Science since 2006. His research activities are devoted to distribution theory, generalized integral transforms, fractional analysis, and nonlinear differential equations with fixed point theory.  He has published over 100 papers and has been supported by the Natural Sciences and Engineering Research Council of Canada for a long time.  In addition, he was awarded the Brandon University Senate Award for Excellence in Research 2015 and Teaching in 2016.