Bøger af Vitaly Volpert

This monograph concisely but thoroughly introduces the reader to the field of mathematical immunology. The target audience primarily comprises researchers and graduate students in the field of mathematical biology who wish to be concisely introduced into mathematical immunology.

This monograph concisely but thoroughly introduces the reader to the field of mathematical immunology. The target audience primarily comprises researchers and graduate students in the field of mathematical biology who wish to be concisely introduced into mathematical immunology.

If we had to formulate in one sentence what this book is about, it might be "e;How partial differential equations can help to understand heat explosion, tumor growth or evolution of biological species"e;. These and many other applications are described by reaction-diffusion equations. The theory of reaction-diffusion equations appeared in the first half of the last century. In the present time, it is widely used in population dynamics, chemical physics, biomedical modelling. The purpose of this book is to present the mathematical theory of reaction-diffusion equations in the context of their numerous applications. We will go from the general mathematical theory to specific equations and then to their applications. Existence, stability and bifurcations of solutions will be studied for bounded domains and in the case of travelling waves. The classical theory of reaction-diffusion equations and new topics such as nonlocal equations and multi-scale models in biology will be considered.

The theory of elliptic partial differential equations has developed during about two centuries together with electrostatics, heat and mass diffusion, hydrodynamics and many other applications. It remains one of the most rapidly developing fields of mathematics. The theory of general elliptic problems is presented in the present first volume of the book. A priori estimates, normal solvability and Fredholm property, index, operators with a parameter, nonlinear Fredholm operators are discussed. Particular attention is paid to elliptic problems in unbounded domains which are not yet sufficiently well presented in the existing literature and which require some special approaches. The second volume will be devoted to reaction-diffusion equations. Existence and bifurcations of solutions, travelling waves, spectral properties and other questions are studied in relation with numerous applications in chemical physics, biology and medicine.

The theory of elliptic partial differential equations has undergone an important development over the last two centuries. The author discusses a priori estimates, normal solvability, the Fredholm property, the index of an elliptic operator, operators with a parameter, and nonlinear Fredholm operators.