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Separation of Variables and Exact Solutions to Nonlinear PDEs
Separation of Variables and Exact Solutions to Nonlinear PDEs is devoted to describing and applying methods of generalized and functional separation of variables used to find exact solutions of nonlinear partial differential equations (PDEs). It also presents the direct method of symmetry reductions and its more general version. In addition, the authors describe the differential constraint method, which generalizes many other exact methods. The presentation involves numerous examples of utilizing the methods to find exact solutions to specific nonlinear equations of mathematical physics. The equations of heat and mass transfer, wave theory, hydrodynamics, nonlinear optics, combustion theory, chemical technology, biology, and other disciplines are studied. Particular attention is paid to nonlinear equations of a reasonably general form that depend on one or several arbitrary functions. Such equations are the most difficult to analyze. Their exact solutions are of significant practical interest, as they are suitable to assess the accuracy of various approximate analytical and numerical methods. The book contains new material previously unpublished in monographs. It is intended for a broad audience of scientists, engineers, instructors, and students specializing in applied and computational mathematics, theoretical physics, mechanics, control theory, chemical engineering science, and other disciplines. Individual sections of the book and examples are suitable for lecture courses on partial differential equations, equations of mathematical physics, and methods of mathematical physics, for delivering special courses and for practical training.
1. Methods of Generalized Separation of Variables 2. Methods of Functional Separation of Variables 3. Direct Method of Symmetry Reductions. Weak Symmetries 4. Method of Differential Constraints
Includes bibliographical references and index.
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. Alexei I. Zhurov, Ph. D., is 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.