Abrar A. Khan – författare

Advanced Calculus reflects the unifying role of linear algebra in an effort to smooth readers’ transition to advanced mathematics. The book fosters the development of complete theorem-proving skills through abundant exercises while also promoting a sound approach to the study. The traditional theorems of elementary differential and integral calculus are rigorously established, presenting the foundations of calculus in a way that reorients thinking toward modern analysis. After the invention of calculus there was a tremendous flowering of new results in mathematics and of applications in science and technology. Of particular note is Fourier’s work on the propagation of heat. The need to make sense of Fourier’s wonderful ideas, and to answer an increasing number of objections and logical problems with the immensely successful methods of calculus led to an increasing interest in providing a rigorous foundation for calculus. Cauchy (1789-1857) tried to formulate the notion of the derivative in terms of the concept of limit. He was not completely successful because he lacked a rigorous notion of limit. The present book is written to meet the requirements of undergraduate and postgraduate students of several Indian Universities. The subject matter both as regards the arrangement of chapters as well the contents of each chapter has been so graded proceeding from simple to more difficult topics that a student may follow the course with great ease and interest.

Partial functional differential equation arise from many biological, chemical and physical systems which are characterized by both spatial and temporal variables and exhibit various spatiotemporal patterns. The systematic study of such equations from the dynamical systems and semi-groups point of view began in the 70s, and considerable advances have been achieved since then. A partial differential equation (PDE) is a differential equation that contains unknown multivariable functions and their partial derivatives. PDEs are used to formulate problems involving functions of several variables, and are either solved by hand, or used to create a relevant computer model. PDEs can be used to describe a wide variety of phenomena such as sound, heat, electrostatics, electrodynamics, fluid flow, or elasticity. These seemingly distinct physical phenomena can be formalised identically in terms of PDEs, which shows that they are governed by the same underlying dynamic. This book provides an introduction to the qualitative theory and applications of partial functional differential equations from the viewpoint of dynamical systems. The main emphasis of the book is on reaction-diffusion equations with delayed nonlinear reaction terms and on the joint effect of the time delay and spatial diffusion on the spatial-temporal patterns of the considered systems.