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Written by two international experts in the field, this book is the first unified survey of the advances made in the last 15 years on key non-standard and improperly posed problems for partial differential equations.This reference for mathematicians, scientists, and engineers provides an overview of the methodology typically used to study improperly posed problems. It focuses on structural stability--the continuous dependence of solutions on the initial conditions and the modeling equations--and on problems for which data are only prescribed on part of the boundary.The book addresses conti

Differential equations, Partial --- Approximation theory. --- Improperly posed problems. --- Theory of approximation --- Functional analysis --- Functions --- Polynomials --- Chebyshev systems --- Improperly posed problems in partial differential equations --- Ill-posed problems

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Ill-posed problems are encountered in countless areas of real world science and technology. A variety of processes in science and engineering is commonly modeled by algebraic, differential, integral and other equations. In a more difficult case, it can be systems of equations combined with the associated initial and boundary conditions. Frequently, the study of applied optimization problems is also reduced to solving the corresponding equations. These equations, encountered both in theoretical and applied areas, may naturally be classified as operator equations. The current textbook will focus on iterative methods for operator equations in Hilbert spaces.

Differential equations, Partial --- Iterative methods (Mathematics) --- Iteration (Mathematics) --- Numerical analysis --- Improperly posed problems in partial differential equations --- Improperly posed problems. --- Ill-posed problems --- Hilbert Space. --- Ill-posed Problem. --- Inverse Problem. --- Iterative Method. --- Operator Equation.

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Internal boundary value problems deals with the problem of determining the solution of an equation if data are given on two manifolds. One manifold is the domain boundary and the other manifold is situated inside the domain. This monograph studies three essentially ill-posed internal boundary value problems for the biharmonic equation and the Cauchy problem for the abstract biharmonic equation, both qualitatively and quantitatively. In addition, some variants of these problems and the Cauchy problem, as well as the m-dimensional case, are considered. The author introduces some new notions, such as the notion of complete solvability.

Differential equations, Partial --- Boundary value problems. --- Biharmonic equations. --- Equations, Biharmonic --- Boundary conditions (Differential equations) --- Differential equations --- Functions of complex variables --- Mathematical physics --- Initial value problems --- Improperly posed problems in partial differential equations --- Improperly posed problems. --- Ill-posed problems --- Biharmonic Equation. --- Cauchy Problem. --- Domain Boundary. --- Ill-posed. --- Internal Boundary Value Problems. --- Manifolds.

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Nonlinear inverse problems appear in many applications, and typically they lead to mathematical models that are ill-posed, i.e., they are unstable under data perturbations. Those problems require a regularization, i.e., a special numerical treatment. This book presents regularization schemes which are based on iteration methods, e.g., nonlinear Landweber iteration, level set methods, multilevel methods and Newton type methods.

519.61 --- Numerical methods of algebra --- Differential equations, Partial --- Iterative methods (Mathematics) --- Improperly posed problems. --- Iterative methods (Mathematics). --- Differential equations, Partial -- Improperly posed problems. --- Mathematics. --- Calculus --- Mathematics --- Physical Sciences & Mathematics --- Improperly posed problems --- 519.61 Numerical methods of algebra --- Iteration (Mathematics) --- Improperly posed problems in partial differential equations --- Ill-posed problems --- Inkorrekt gestelltes Problem. --- Regularisierungsverfahren. --- Iteration. --- Nichtlineares inverses Problem. --- Numerical analysis --- Iterative Regularization. --- Nonlinear Ill-Posed Problems. --- Nonlinear Inverse Problems.

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Differential equations, Partial --- Electromagnetic theory --- Integro-differential equations --- Mechanics --- 519.6 --- 681.3*G19 --- 681.3*G19 Integral equations: Fredholm equations; integro-differential equations; Volterra equations (Numerical analysis) --- Integral equations: Fredholm equations; integro-differential equations; Volterra equations (Numerical analysis) --- 519.6 Computational mathematics. Numerical analysis. Computer programming --- Computational mathematics. Numerical analysis. Computer programming --- Light, Electromagnetic theory of --- Electric fields --- Magnetic fields --- Classical mechanics --- Newtonian mechanics --- Physics --- Dynamics --- Quantum theory --- Improperly posed problems in integro-differential equations --- Improperly posed problems in partial differential equations --- Improperly posed problems --- Ill-posed problems --- Differential equations --- Mathematical physics --- Classical mechanics. Field theory --- Electromagnetism. Ferromagnetism