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This book deals with methods for solving nonstiff ordinary differential equations. The first chapter describes the historical development of the classical theory, and the second chapter includes a modern treatment of Runge-Kutta and extrapolation methods. Chapter three begins with the classical theory of multistep methods, and concludes with the theory of general linear methods. The reader will benefit from many illustrations, a historical and didactic approach, and computer programs which help him/her learn to solve all kinds of ordinary differential equations. This new edition has been rewritten and new material has been included.

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Numerical methods that preserve properties of Hamiltonian systems, reversible systems, differential equations on manifolds and problems with highly oscillatory solutions are the subject of this book. A complete self-contained theory of symplectic and symmetric methods, which include Runge-Kutta, composition, splitting, multistep and various specially designed integrators, is presented and their construction and practical merits are discussed. The long-time behaviour of the numerical solutions is studied using a backward error analysis (modified equations) combined with KAM theory. The book is illustrated by many figures, it treats applications from physics and astronomy and contains many numerical experiments and comparisons of different approaches. The second edition is substantially revised and enlarged, with many improvements in the presentation and additions concerning in particular non-canonical Hamiltonian systems, highly oscillatory mechanical systems, and the dynamics of multistep methods.

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The monograph presents a new approach to modeling and studying dynamic processes in the simplest two-link systems, where the input and output connections are somehow connected or interact with each other. It is shown that, neglecting nonlinearities, these systems can be described by analogous differential equations. The new method explicitly takes into account fluctuations in the speed of movement and tension. Using the Laplace transformations allowed us to derive equations describing the dynamics in a more general form. In this case, the elasticity of the system was considered as a function depending, in addition to the line parameters, on the oscillation frequency, which under certain conditions may lead to loss of stability. The theory of stability of open-loop systems with distributed parameters is obtained. For the cases of using internal feedbacks on the intermediate coordinate (pressure, voltage), a new frequency method for estimating stability was developed. The book contains new numerical methods based on the Runge-Kutta method, which made it possible to accurately simulate dynamic processes in nonlinear systems with lumped and distributed parameters.

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This book comprises of six chapters and discusses the heat and mass transfer effects on a laminar, two dimensional, steady/unsteady, free/mixed convection flow of a viscous incompressible electrically conducting and radiating fluid past a semi-infinite vertical porous moving plate / stretching sheet / stretching cylinder/ horizontal circular cylinder bounded by a porous (or non-porous) medium, in the presence of various effects such as MHD, chemical reaction, viscous dissipation, thermophoresis, Brownian motion, thermally stratified medium, variable thermal conductivity and partial slip. The book is divided into six chapters. The first chapter deals with unsteady case, whereas the later five chapters deal with steady case. The governing equations of the flow under consideration were solved with the associated boundary conditions by using a traditional perturbation method or Runge - Kutta fourth order scheme along with shooting technique or Crank-Nicolson finite difference scheme.

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This research presents numerical solution of Whitham-Broer-Kaup (WBK) shallow water model using some finite difference methods such as the explicit, implicit, Crank-Nicolson, exponential, Richardson, and DuFort-Frankel methods. The local truncation errors and consistency are also studied for these methods. Furthermore, the radial basis function-pseudospectral method is successfully applied for solving WBK model numerically. In this method, the model is discretized in the space x which leads to a system of ordinary differential equations with independent variable time t. Then the forward Euler's and fourth-order Runge-Kutta methods used to solve this system of ordinary differential equations. Finally, two different problems are considered to illustrate the efficiency of the methods. A comparative study is made between the approximate solution obtained by the presented numerical methods and the exact solution. The MATLAB codes have been written for all methods as can be found in the appendix.

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The three body problem is one important object of space dynamics, because this problem has many applications on space dynamics objects. Also this problem attract many scientist during last three centuries. The motion of space craft around binary asteroid can be consider as restricted three body problem so that, the treatment this object was done. Where equations of motioncan be solved by using numerical integration methods (Runge Kutta fourth order).

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Acoustic fluid solid interaction (AFSI) has been a subject of high interest for the last decade and the literature is now extensive. Noise transmission into and radiation from a rectangular cavity through a flexible structure in a noisy/thermal environment at supersonic flow is studied. In the present book, the pre/post -flutter regions are considered in a nonlinear study. The governing equations of flexible panel are constructed using Von Karman theory, first-order piston theory and a modal cavity formulation, also a stationary homogeneous turbulent-boundary-layer (TBL) is modeled based on experimental data through stochastic modeling. The thermal environment and external incident random noise are also considered and the far-field sound is predicted with a boundary integral method. The Rayleigh-Ritz approach is adopted to discretize the partial differential equations of the AFSI system, and the resulting ordinary differential equations (ODEs) are solved numerically by the fourth-order Runge-Kutta with Matlab. additionally, Matlab codes are presented in the Appendix section.

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