Choose an application

• Reference Manager
• EndNote
• RefWorks (Direct export to RefWorks)

Die vierte, durchgesehene und ergänzte Auflage dieses Standardlehrbuchs folgt weiterhin konsequent der Linie, den Leser auf solider theoretischer Basis direkt zu praktisch bewährten Methoden zu führen - von der Herleitung über die Analyse bis hin zu Fragen der Implementierung. Dies macht das Buch sowohl für Mathematiker als auch für Naturwissenschaftler und Ingenieure attraktiv. Das Lehrbuch eignet sich als Vorlesungsbegleitung für Studierende ebenso wie zum Selbststudium für im Beruf stehende Naturwissenschaftler. Es setzt lediglich Grundkenntnisse der Analysis (entsprechend Vorlesung Höhere Mathematik bei Physikern und Ingenieuren) sowie der Numerischen Mathematik (Einführungsvorlesung) voraus.

Differential equations --- 517.91 Differential equations --- Numerical solutions. --- Boundary Value Problem. --- Differential Equation. --- Initial Value Problem. --- Numerical Method.

Choose an application

• Reference Manager
• EndNote
• RefWorks (Direct export to RefWorks)

Since the end of the 19th century when the prominent Norwegian mathematician Sophus Lie created the theory of Lie algebras and Lie groups and developed the method of their applications for solving differential equations, his theory and method have continuously been the research focus of many well-known mathematicians and physicists. This book is devoted to recent development in Lie theory and its applications for solving physically and biologically motivated equations and models. The book contains the articles published in two Special Issue of the journal Symmetry, which are devoted to analysis and classification of Lie algebras, which are invariance algebras of real-word models; Lie and conditional symmetry classification problems of nonlinear PDEs; the application of symmetry-based methods for finding new exact solutions of nonlinear PDEs (especially reaction-diffusion equations) arising in applications; the application of the Lie method for solving nonlinear initial and boundary-value problems (especially those for modelling processes with diffusion, heat transfer, and chemotaxis).

Lie algebra/group --- invariance algebra of nonlinear PDE --- Lie symmetry --- nonlinear boundary-value problem --- (generalized) conditional symmetry --- symmetry of (initial) boundary-value problem --- invariant solution --- exact solution --- non-Lie solution --- Q-conditional symmetry --- representation of Lie algebra --- nonclassical symmetry --- invariance algebra of PDE

Choose an application

• Reference Manager
• EndNote
• RefWorks (Direct export to RefWorks)

The aim of this monograph is to give a thorough and self-contained account of functions of (generalized) bounded variation, the methods connected with their study, their relations to other important function classes, and their applications to various problems arising in Fourier analysis and nonlinear analysis. In the first part the basic facts about spaces of functions of bounded variation and related spaces are collected, the main ideas which are useful in studying their properties are presented, and a comparison of their importance and suitability for applications is provided, with a particular emphasis on illustrative examples and counterexamples. The second part is concerned with (sometimes quite surprising) properties of nonlinear composition and superposition operators in such spaces. Moreover, relations with Riemann-Stieltjes integrals, convergence tests for Fourier series, and applications to nonlinear integral equations are discussed. The only prerequisite for understanding this book is a modest background in real analysis, functional analysis, and operator theory. It is addressed to non-specialists who want to get an idea of the development of the theory and its applications in the last decades, as well as a glimpse of the diversity of the directions in which current research is moving. Since the authors try to take into account recent results and state several open problems, this book might also be a fruitful source of inspiration for further research.

Functions of bounded variation. --- Bounded variables, Functions of --- Bounded variation, Functions of --- BV functions --- Functions of bounded variables --- Functions of real variables --- Boundary Value Problem. --- Bounded Variation. --- Continuity Properties. --- Fourier Analysis. --- Monotonicity Properties. --- Nonlinear Composition Operators. --- Nonlinear Integral Equation.

Choose an application

• Reference Manager
• EndNote
• RefWorks (Direct export to RefWorks)

Differential equations with impulses arise as models of many evolving processes that are subject to abrupt changes, such as shocks, harvesting, and natural disasters. These phenomena involve short-term perturbations from continuous and smooth dynamics, whose duration is negligible in comparison with the duration of an entire evolution. In models involving such perturbations, it is natural to assume these perturbations act instantaneously or in the form of impulses. As a consequence, impulsive differential equations have been developed in modeling impulsive problems in physics, population dynamics, ecology, biotechnology, industrial robotics, pharmacokinetics, optimal control, and so forth. There are also many different studies in biology and medicine for which impulsive differential equations provide good models. During the last 10 years, the authors have been responsible for extensive contributions to the literature on impulsive differential inclusions via fixed point methods. This book is motivated by that research as the authors endeavor to bring under one cover much of those results along with results by other researchers either affecting or affected by the authors' work. The questions of existence and stability of solutions for different classes of initial value problems for impulsive differential equations and inclusions with fixed and variable moments are considered in detail. Attention is also given to boundary value problems. In addition, since differential equations can be viewed as special cases of differential inclusions, significant attention is also given to relative questions concerning differential equations. This monograph addresses a variety of side issues that arise from its simpler beginnings as well.

Boundary value problems. --- Differential equations. --- Prediction theory. --- Stochastic processes. --- Random processes --- Probabilities --- Forecasting theory --- Stochastic processes --- 517.91 Differential equations --- Differential equations --- Boundary conditions (Differential equations) --- Functions of complex variables --- Mathematical physics --- Initial value problems --- Boundary Value Problem. --- Condensing. --- Contraction. --- Controllability. --- Differential Inclusion. --- Filippov's Theorem. --- Hyperbolic Differential Inclusion. --- Impulsive Functional Differential Equation. --- Infinite Delay. --- Normal Cone. --- Relaxation. --- Seeping Process. --- Stability. --- Stochastic Differential Equation. --- Variable Times. --- Viable Solution.

Choose an application

• Reference Manager
• EndNote
• RefWorks (Direct export to RefWorks)

The charm of Mathematical Physics resides in the conceptual difficulty of understanding why the language of Mathematics is so appropriate to formulate the laws of Physics and to make precise predictions. Citing Eugene Wigner, this “unreasonable appropriateness of Mathematics in the Natural Sciences” emerged soon at the beginning of the scientific thought and was splendidly depicted by the words of Galileo: “The grand book, the Universe, is written in the language of Mathematics.” In this marriage, what Bertrand Russell called the supreme beauty, cold and austere, of Mathematics complements the supreme beauty, warm and engaging, of Physics. This book, which consists of nine articles, gives a flavor of these beauties and covers an ample range of mathematical subjects that play a relevant role in the study of physics and engineering. This range includes the study of free probability measures associated with p-adic number fields, non-commutative measures of quantum discord, non-linear Schrödinger equation analysis, spectral operators related to holomorphic extensions of series expansions, Gibbs phenomenon, deformed wave equation analysis, and optimization methods in the numerical study of material properties.

Research & information: general --- Mathematics & science --- prolongation structure --- mNLS equation --- Riemann-Hilbert problem --- initial-boundary value problem --- free probability --- primes --- p-adic number fields --- Banach *-probability spaces --- weighted-semicircular elements --- semicircular elements --- truncated linear functionals --- FCM fuel --- thermal–mechanical performance --- failure probability --- silicon carbide --- quantum discord --- non-commutativity measure --- dynamic models --- Gibbs phenomenon --- quasi-affine --- shift-invariant system --- dual tight framelets --- oblique extension principle --- B-splines --- crack growth behavior --- particle model --- intersecting flaws --- uniaxial compression --- reinforced concrete --- retaining wall --- optimization --- bearing capacity --- particle swarm optimization --- PSO --- generalized Fourier transform --- deformed wave equation --- Huygens’ principle --- representation of ??(2,ℝ) --- holomorphic extension --- spherical Laplace transform --- non-Euclidean Fourier transform --- Fourier–Legendre expansion