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This book presents a concise introduction to a unified Hilbert space approach to the mathematical modelling of physical phenomena which has been developed over recent years by Picard and his co-workers. The main focus is on time-dependent partial differential equations with a particular structure in the Hilbert space setting that ensures well-posedness and causality, two essential properties of any reasonable model in mathematical physics or engineering. As a unique feature, this powerful tool for tackling time-dependent partial differential equations is subsequently applied to many equations. By means of illustrative examples, from the straightforward to the more complex, the authors show that many of the classical models in mathematical physics as well as more recent models of novel materials and interactions are covered, or can be restructured to be covered, by this unified Hilbert space approach. The reader should require only a basic foundation in the theory of Hilbert spaces and operators therein. For convenience, however, some of the more technical background requirements are covered in detail in the appendix. The theory is kept as elementary as possible, making the material suitable for a senior undergraduate or master’s level course. In addition, researchers in a variety of fields whose work involves partial differential equations and applied operator theory will also greatly benefit from this approach to structuring their mathematical models in order that the general theory can be applied to ensure the essential properties of well-posedness and causality.

Differential equations, Partial. --- Hilbert space. --- Mathematical physics. --- Physical mathematics --- Physics --- Banach spaces --- Hyperspace --- Inner product spaces --- Partial differential equations --- Mathematics --- Partial differential equations. --- Partial Differential Equations.

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In this book the authors use a technique based on recurrence relations to study the convergence of the Newton method under mild differentiability conditions on the first derivative of the operator involved. The authors’ technique relies on the construction of a scalar sequence, not majorizing, that satisfies a system of recurrence relations, and guarantees the convergence of the method. The application is user-friendly and has certain advantages over Kantorovich’s majorant principle. First, it allows generalizations to be made of the results obtained under conditions of Newton-Kantorovich type and, second, it improves the results obtained through majorizing sequences. In addition, the authors extend the application of Newton’s method in Banach spaces from the modification of the domain of starting points. As a result, the scope of Kantorovich’s theory for Newton’s method is substantially broadened. Moreover, this technique can be applied to any iterative method. This book is chiefly intended for researchers and (postgraduate) students working on nonlinear equations, as well as scientists in general with an interest in numerical analysis.

Operator theory. --- Numerical analysis. --- Integral equations. --- Differential equations. --- Partial differential equations. --- Operator Theory. --- Numerical Analysis. --- Integral Equations. --- Ordinary Differential Equations. --- Partial Differential Equations. --- Partial differential equations --- 517.91 Differential equations --- Differential equations --- Equations, Integral --- Functional equations --- Functional analysis --- Mathematical analysis --- Banach spaces. --- Differential algebra. --- Algebra, Differential --- Differential fields --- Algebraic fields --- Functions of complex variables --- Generalized spaces --- Topology