Formal Algorithmic Elimination For Pdes

Investigating the correspondence between systems of partial differential equations and their analytic solutions using a formal approach, this monograph presents algorithms to determine the set of analytic solutions of such a system and conversely to find differential equations whose set of solutions coincides with a given parametrized set of analytic functions. After giving a detailed introduction to Janet bases and Thomas decomposition, the problem of finding an implicit description of certain sets of analytic functions in terms of differential equations is addressed. Effective methods of varying generality are developed to solve the differential elimination problems that arise in this context. In particular, it is demonstrated how the symbolic solution of partial differential equations profits from the study of the implicitization problem. For instance, certain families of exact solutions of the Navier-Stokes equations can be computed.

This book aims at reviewing recent progress in the direction of algebraic and symbolic computation methods for functional systems, e.g. ODE systems, differential time-delay equations, difference equations and integro-differential equations. In the nineties, modern algebraic theories were introduced in mathematical systems theory and in control theory. Combined with real algebraic geometry, which was previously introduced in control theory, the past years have seen a flourishing development of algebraic methods in control theory. One of the strengths of algebraic methods lies in their close connections to computations. The use of the above-mentioned algebraic theories in control theory has been an important source of motivation to develop effective versions of these theories (when possible). With the development of computer algebra and computer algebra systems, symbolic methods for control theory have been developed over the past years. The goal of this book is to propose a partial state of the art in this direction. To make recent results more easily accessible to a large audience, the chapters include materials which survey the main mathematical methods and results and which are illustrated with explicit examples.

This book constitutes the refereed proceedings of the 21st International Workshop on Computer Algebra in Scientific Computing, CASC 2019, held in Moscow, Russia, in August 2019. The 28 full papers presented together with 2 invited talks were carefully reviewed and selected from 44 submissions. They deal with cutting-edge research in all major disciplines of computer algebra. The papers cover topics such as polynomial algebra, symbolic and symbolic-numerical computation, applications of symbolic computation for investigating and solving ordinary differential equations, applications of CASs in the investigation and solution of celestial mechanics problems, and in mechanics, physics, and robotics.

This book constitutes the refereed proceedings of the 22nd International Workshop on Computer Algebra in Scientific Computing, CASC 2020, held in Linz, Austria, in September 2020. The conference was held virtually due to the COVID-19 pandemic. The 34 full papers presented together with 2 invited talks were carefully reviewed and selected from 41 submissions. They deal with cutting-edge research in all major disciplines of computer algebra. The papers cover topics such as polynomial algebra, symbolic and symbolic-numerical computation, applications of symbolic computation for investigating and solving ordinary differential equations, applications of CAS in the investigation and solution of celestial mechanics problems, and in mechanics, physics, and robotics.

This book constitutes the proceedings of the 17th International Workshop on Computer Algebra in Scientific Computing, CASC 2015, held in Aachen, Germany, in September 2015. The 35 full papers presented in this volume were carefully reviewed and selected from 42 submissions. They deal with the ongoing progress both in theoretical computer algebra and its expanding applications. New and closer interactions are fostered by combining the area of computer algebra methods and systems and the application of the tools of computer algebra for the solution of problems in scientific computing.

This book constitutes the proceedings of the 17th International Conference on Quantitative Evaluation Systems, QEST 2020, held in Vienna, Austria, in August/September 2020. The 12 full papers presented together with 7 short papers were carefully reviewed and selected from 42 submissions. The papers cover topics such as classic measures involving performance and reliability, quantification of properties that are classically qualitative, such as safety, correctness, and security as well as analytic studies, diversity in the model formalisms and methodologies employed, and development of new formalisms and methodologies.

This edited volume presents a fascinating collection of lecture notes focusing on differential equations from two viewpoints: formal calculus (through the theory of Gröbner bases) and geometry (via quiver theory). Gröbner bases serve as effective models for computation in algebras of various types. Although the theory of Gröbner bases was developed in the second half of the 20th century, many works on computational methods in algebra were published well before the introduction of the modern algebraic language. Since then, new algorithms have been developed and the theory itself has greatly expanded. In comparison, diagrammatic methods in representation theory are relatively new, with the quiver varieties only being introduced – with big impact – in the 1990s. Divided into two parts, the book first discusses the theory of Gröbner bases in their commutative and noncommutative contexts, with a focus on algorithmic aspects and applications of Gröbner bases to analysis on systems of partial differential equations, effective analysis on rings of differential operators, and homological algebra. It then introduces representations of quivers, quiver varieties and their applications to the moduli spaces of meromorphic connections on the complex projective line. While no particular reader background is assumed, the book is intended for graduate students in mathematics, engineering and related fields, as well as researchers and scholars.

This book provides a systematic and uniform presentation of elimination methods and the underlying theories, along the central line of decomposing arbitrary systems of polynomials into triangular systems of various kinds. Highlighting methods based on triangular sets, the book also covers the theory and techniques of resultants and Gröbner bases. The methods and their efficiency are illustrated by fully worked out examples and their applications to selected problems such as from polynomial ideal theory, automated theorem proving in geometry and the qualitative study of differential equations. The reader will find the formally described algorithms ready for immediate implementation and applicable to many other problems. Suitable as a graduate text, this book offers an indispensable reference for everyone interested in mathematical computation, computer algebra (software), and systems of algebraic equations.

This book deals with the general topic “Numerical solution of partial differential equations (PDEs)” with a focus on adaptivity of discretizations in space and time. By and large, introductory textbooks like “Numerical Analysis in Modern Scientific Computing” by Deuflhard and Hohmann should suffice as a prerequisite. The emphasis lies on elliptic and parabolic systems. Hyperbolic conservation laws are treated only on an elementary level excluding turbulence. Numerical Analysis is clearly understood as part of Scientific Computing. The focus is on the efficiency of algorithms, i.e. speed, reliability, and robustness, which directly leads to the concept of adaptivity in algorithms. The theoretical derivation and analysis is kept as elementary as possible. Nevertheless required somewhat more sophisticated mathematical theory is summarized in comprehensive form in an appendix. Complex relations are explained by numerous figures and illustrating examples. Non-trivial problems from regenerative energy, nanotechnology, surgery, and physiology are inserted. The text will appeal to graduate students and researchers on the job in mathematics, science, and technology. Conceptually, it has been written as a textbook including exercises and a software list, but at the same time it should be well-suited for self-study.