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This textbook introduces both to the theory and numerics of partial differential equations (PDEs) which is rather unique for German textbooks.For all basic types of PDEs and boundary conditions, existence and uniqueness results are provided and numerical schemes are presented.

Differential equations, Partial --- Numerical solutions --- Boundary Value Problem. --- Finite Difference Scheme. --- Finite Element Method. --- Numerical Mathematics. --- Partial Differential Equation.

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The text is intended for students who wish a concise and rapid introduction to some main topics in PDEs, necessary for understanding current research, especially in nonlinear PDEs. Organized on three parts, the book guides the reader from fundamental classical results, to some aspects of the modern theory and furthermore, to some techniques of nonlinear analysis. Compared to other introductory books in PDEs, this work clearly explains the transition from classical to generalized solutions and the natural way in which Sobolev spaces appear as completions of spaces of continuously differentiable functions with respect to energetic norms. Also, special attention is paid to the investigation of the solution operators associated to elliptic, parabolic and hyperbolic non-homogeneous equations anticipating the operator approach of nonlinear boundary value problems. Thus the reader is made to understand the role of linear theory for the analysis of nonlinear problems.

Differential equations, Linear --- Differential equations, Partial --- Linear differential equations --- Linear systems --- Partial differential equations --- Elliptic Equation. --- Heat Equation. --- Partial Differential Equation. --- Sobolev Space. --- Wave Equation.

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This book deals with the reliable verification of the accuracy of approximate solutions which is one of the central problems in modern applied analysis.After giving an overview of the methods developed for models based on partial differential equations, the author derives computable a posteriori error estimates by using methods of the theory of partial differential equations and functional analysis. These estimates are applicable to approximate solutions computed by various methods.

Differential equations, Partial. --- Error analysis (Mathematics) --- Errors, Theory of --- Instrumental variables (Statistics) --- Mathematical statistics --- Numerical analysis --- Statistics --- Partial differential equations --- A-posteriori-Abschätzung. --- Error analysis (Mathematics). --- Partielle Differentialgleichung. --- Partial Differential Equation, Approximate Solution, A Posteriori Error Estimates.

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Green's functions represent one of the classical and widely used issues in the area of differential equations. This monograph is looking at applied elliptic and parabolic type partial differential equations in two variables. The elliptic type includes the Laplace, static Klein-Gordon and biharmonic equation. The parabolic type is represented by the classical heat equation and the Black-Scholes equation which has emerged as a mathematical model in financial mathematics. The book is attractive for practical needs: It contains many easily computable or computer friendly representations of Green's functions, includes all the standard Green's functions and many novel ones, and provides innovative and new approaches that might lead to Green's functions. The book is a useful source for everyone who is studying or working in the fields of science, finance, or engineering that involve practical solution of partial differential equations.

Green's functions. --- Functions, Green's --- Functions, Induction --- Functions, Source --- Green functions --- Induction functions --- Source functions --- Differential equations --- Potential theory (Mathematics) --- Elliptic. --- Green's Function. --- Parabolic. --- Partial Differential Equation.

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This book is the third volume of three volume series recording the "Radon Special Semester 2011 on Multiscale Simulation & Analysis in Energy and the Environment" taking place in Linz, Austria, October 3-7, 2011. This book surveys recent developments in the analysis of wave propagation problems. The topics covered include aspects of the forward problem and problems in inverse problems, as well as applications in the earth sciences. Wave propagation problems are ubiquitous in environmental applications such as seismic analysis, acoustic and electromagnetic scattering. The design of efficient numerical methods for the forward problem, in which the scattered field is computed from known geometric configurations is very challenging due to the multiscale nature of the problems. Even more challenging are inverse problems where material parameters and configurations have to be determined from measurements in conjunction with the forward problem. This book contains review articles covering several state-of-the-art numerical methods for both forward and inverse problems. This collection of survey articles focusses on the efficient computation of wave propagation and scattering is a core problem in numerical mathematics, which is currently of great research interest and is central to many applications in energy and the environment. Two generic applications which resonate strongly with the central aims of the Radon Special Semester 2011 are forward wave propagation in heterogeneous media and seismic inversion for subsurface imaging. As an example of the first application, modelling of absorption and scattering of radiation by clouds, aerosol and precipitation is used as a tool for interpretation of (e.g.) solar, infrared and radar measurements, and as a component in larger weather/climate prediction models in numerical weather forecasting. As an example of the second application, inverse problems in wave propagation in heterogeneous media arise in the problem of imaging the subsurface below land or marine deposits. The book records the achievements of Workshop 3 "Wave Propagation and Scattering, Inverse Problems and Applications in Energy and the Environment". It brings together key numerical mathematicians whose interest is in the analysis and computation of wave propagation and scattering problems, and in inverse problems, together with practitioners from engineering and industry whose interest is in the applications of these core problems.

Radio wave propagation. --- Radio waves --- Diffraction --- Propagation of radio waves --- Wave-motion, Theory of --- Diffraction. --- Scattering. --- Propagation --- 537.8 --- 537.8 Electromagnetism. Electromagnetic field. Electrodynamics. Maxwell theory --- Electromagnetism. Electromagnetic field. Electrodynamics. Maxwell theory --- Cloaking. --- Finite Element Method. --- Helmholtz Equation. --- Inverse Problem. --- Partial Differential Equation. --- Wave Propagation.