Delay Ordinary and Partial Differential Equations

Delay Ordinary and Partial Differential Equations is devoted to linear and nonlinear ordinary and partial differential equations with constant and variable delay. It considers qualitative features of delay differential equations and formulates typical problem statements. Exact, approximate analytical and numerical methods for solving such equations are described, including the method of steps, methods of integral transformations, method of regular expansion in a small parameter, method of matched asymptotic expansions, iteration-type methods, Adomian decomposition method, collocation method, Galerkin-type projection methods, Euler and Runge-Kutta methods, shooting method, method of lines, finite-difference methods for PDEs, methods of generalized and functional separation of variables, method of functional constraints, method of generating equations, and more.

The presentation of the theoretical material is accompanied by examples of the practical application of methods to obtain the desired solutions. Exact solutions are constructed for many nonlinear delay reaction-diffusion and wave-type PDEs that depend on one or more arbitrary functions. A review is given of the most common mathematical models with delay used in population theory, biology, medicine, economics, and other applications.

The book contains much new material previously unpublished in monographs. It is intended for a broad audience of scientists, university professors, and graduate and postgraduate students specializing in applied and computational mathematics, mathematical physics, mechanics, control theory, biology, medicine, chemical technology, ecology, economics, and other disciplines.

Individual sections of the book and examples are suitable for lecture courses on applied mathematics, mathematical physics, and differential equations for delivering special courses and for practical training.

Andrei D. Polyanin, D.Sc., Ph.D., Professor, is a well-known scientist of broad interests and is active in various areas of mathematics, mechanics, and chemical engineering sciences. Professor Polyanin graduated with honors from the Department of Mechanics and Mathematics of Moscow State University in 1974. He received his Ph.D. degree in 1981 and D.Sc. degree in 1986 at the Institute for Problems in Mechanics of the Russian (former USSR) Academy of Sciences. Since 1975, Professor Polyanin has been working at the Institute for Problems in Mechanics of the Russian Academy of Sciences; he is also Professor of Mathematics at Bauman Moscow State Technical University and at National Research Nuclear University MEPhI. He is a member of the Russian National Committee on Theoretical and Applied Mechanics and of the Mathematics and Mechanics Expert Council of the Higher Certification Committee of the Russian Federation. Professor Polyanin is an author of more than 30 books in English, Russian, German, and Bulgarian as well as over 210 research papers and three patents.

Vsevolod G. Sorokin, Ph.D., graduated from the Department of Applied Mathematics of the Bauman Moscow State Technical University in 2014. He has been working as a research scientist in the Ishlinsky Institute for Problems in Mechanics of the Russian Academy of Sciences since 2015. He defended his PhD thesis on mathematical modelling and numerical integration of reaction–diffusion systems with delay in the Bauman Moscow State Technical University in 2018. Doctor Sorokin has published about 30 research papers.

Alexei I. Zhurov, Ph.D., is an outstanding scientist in nonlinear mechanics, mathematical physics, computer algebra, biomechanics, and morphometrics. He graduated with honors from the Department of Airphysics and Space Research of the Moscow Institute of Physics and Technology in 1990. Since then has become a member of staff of the Ishlinsky Institute for Problems in Mechanics of the Russian Academy of Sciences, where he received his PhD in mechanics and fluid dynamics in 1995 and has become a senior research scientist since 1999. Since 2001, he has joined Cardiff University as a research scientist in the area of biomechanics and morphometrics. Doctor Zhurov has published over 120 research papers and three books, including Solution Methods for Nonlinear Equations of Mathematical Physics and Mechanics by A.D. Polyanin, V.F. Zaitsev, and A.I. Zhurov (Fizmatlit, 2005; in Russian).