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AUFGABENSAMMLUNG ZU DEN GEWÖHNLICHEN UND PARTIELLEN DIFFENRE
The main theme is the integration of the theory of linear PDE and the theory of finite difference and finite element methods. For each type of PDE, elliptic, parabolic, and hyperbolic, the text contains one chapter on the mathematical theory of the differential equation, followed by one chapter on finite difference methods and one on finite element methods. The chapters on elliptic equations are preceded by a chapter on the two-point boundary value problem for ordinary differential equations. Similarly, the chapters on time-dependent problems are preceded by a chapter on the initial-value problem for ordinary differential equations. There is also one chapter on the elliptic eigenvalue problem and eigenfunction expansion. The presentation does not presume a deep knowledge of mathematical and functional analysis. The required background on linear functional analysis and Sobolev spaces is reviewed in an appendix. The book is suitable for advanced undergraduate and beginning graduate students of applied mathematics and engineering.
In 1964 the author's mono graph "Differential- und Integral-Un gleichungen," with the subtitle "und ihre Anwendung bei Abschätzungs und Eindeutigkeitsproblemen" was published. The present volume grew out of the response to the demand for an English translation of this book. In the meantime the literature on differential and integral in equalities increased greatly. We have tried to incorporate new results as far as possible. As a matter of fact, the Bibliography has been almost doubled in size. The most substantial additions are in the field of existence theory. In Chapter I we have included the basic theorems on Volterra integral equations in Banach space (covering the case of ordinary differential equations in Banach space). Corresponding theorems on differential inequalities have been added in Chapter II. This was done with a view to the new sections; dealing with the line method, in the chapter on parabolic differential equations. Section 35 contains an exposition of this method in connection with estimation and convergence. An existence theory for the general nonlinear parabolic equation in one space variable based on the line method is given in Section 36. This theory is considered by the author as one of the most significant recent applications of in equality methods. We should mention that an exposition of Krzyzanski's method for solving the Cauchy problem has also been added. The numerous requests that the new edition include a chapter on elliptic differential equations have been satisfied to some extent.
This treatment presents most of the methods for solving ordinary differential equations and systematic arrangements of more than 2,000 equations and their solutions. The material is organized so that standard equations can be easily found. Plus, the substantial number and variety of equations promises an exact equation or a sufficiently similar one. 1960 edition.
The Faculty of Mathematics and Geoinformation of the TU Wien has existed as such since the division of the early, very large Faculty of Technical Sciences in 2004. It provides its own study programmes in both subjects, as well as ensuring the mathematical and geometrical basic education of the students of all seven other faculties. The faculty also conducts research in broad and highly crucial focal areas. The current volume is part of a comprehensive commemorative series published in 2015 for the bicentennial memorial of the TU Wien providing information on the research activities, teaching tasks, and history of the Faculty of Mathematics and Geoinformation, in particular over the last 50 years. Special attention has been paid to the exceptional scientific achievements of faculty members.
Maximum Principles are central to the theory and applications of second-order partial differential equations and systems. This self-contained text establishes the fundamental principles and provides a variety of applications.
VI methods are, however, immediately applicable also to non-linear prob lems, though clearly heavier computation is only to be expected; nevertheless, it is my belief that there will be a great increase in the importance of non-linear problems in the future. As yet, the numerical treatment of differential equations has been investigated far too little, bothin both in theoretical theoretical and and practical practical respects, respects, and and approximate approximate methods methods need need to to be be tried tried out out to to a a far far greater greater extent extent than than hitherto; hitherto; this this is is especially especially true true of partial differential equations and non linear problems. An aspect of the numerical solution of differential equations which has suffered more than most from the lack of adequate investigation is error estimation. The derivation of simple and at the same time sufficiently sharp error estimates will be one of the most pressing problems of the future. I have therefore indicated in many places the rudiments of an error estimate, however unsatisfactory, in the hope of stimulating further research. Indeed, in this respect the book can only be regarded as an introduction. Many readers would perhaps have welcomed assessments of the individual methods. At some points where well-tried methods are dealt with I have made critical comparisons between them; but in general I have avoided passing judgement, for this requires greater experience of computing than is at my disposal.
The quantitative understanding of changes in cell types, referred to as cell type transitions, is fundamental to advance fields such as stem cell research, immunology, and cancer therapies. This thesis provides a mathematical modeling framework to simulate and analyze cell type transitions. The novel methodological approaches and models presented here address diverse levels which are essential in this context: Gene regulatory network models represent the cell type-determining gene expression dynamics. Here, a novel construction method for gene regulatory network models is introduced, which allows to transfer results from generic low-dimensional to realistic high-dimensional gene regulatory network models. For populations of cells, a generalized model class is proposed that accounts for multiple cell types, division numbers, and the full label distribution. Analysis and solution methods are presented for this new model class, which cover common cell population experiments and allow to exploit the full information from data. The modeling and analysis methods presented here connect formerly isolated approaches, and thereby contribute to a holistic framework for the quantitative understanding of cell type transitions.