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E-raamat: Nonlinear Ordinary Differential Equations: Analytical Approximation and Numerical Methods

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  • Formaat: PDF+DRM
  • Ilmumisaeg: 09-May-2016
  • Kirjastus: Springer, India, Private Ltd
  • Keel: eng
  • ISBN-13: 9788132228127
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Nonlinear Ordinary Differential Equations: Analytical Approximation and Numerical MethodsSuurem pilt
  • Formaat: PDF+DRM
  • Ilmumisaeg: 09-May-2016
  • Kirjastus: Springer, India, Private Ltd
  • Keel: eng
  • ISBN-13: 9788132228127
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The book discusses the solutions to nonlinear ordinary differential equations (ODEs) using analytical and numerical approximation methods. Recently, analytical approximation methods have been largely used in solving linear and nonlinear lower-order ODEs. It also discusses using these methods to solve some strong nonlinear ODEs.  There are two chapters devoted to solving nonlinear ODEs using numerical methods, as in practice high-dimensional systems of nonlinear ODEs that cannot be solved by analytical approximate methods are common. Moreover, it studies analytical and numerical techniques for the treatment of parameter-depending ODEs.

The book explains various methods for solving nonlinear-oscillator and structural-system problems, including the energy balance method, harmonic balance method, amplitude frequency formulation, variational iteration method, homotopy perturbation method, iteration perturbation method, homotopy analysis method, simple and multiple shooting method, and the nonlinear stabilized march method. This book comprehensively investigates various new analytical and numerical approximation techniques that are used in solving nonlinear-oscillator and structural-system problems. Students often rely on the finite element method to such an extent that on graduation they have little or no knowledge of alternative methods of solving problems. To rectify this, the book introduces several new approximation techniques.

Arvustused

Nonlinear problems in science and engineering are often modeled by nonlinear ordinary differential equations (ODEs) and this book comprises a well-chosen selection of analytical and numerical methods of solving such equations. the writing style is appropriate for a textbook for graduate students. The mathematical rigorous presentation includes many example ODEs and their analytical, approximate or numerical solution, which makes an interesting read for all interested in solving nonlinear problems. (Gudula Rünger, zbMATH 1361.34001, 2017)

1 A Brief Review of Elementary Analytical Methods for Solving Nonlinear ODEs
1(32)
1.1 Introduction
1(1)
1.2 Analytical Solution of First-Order Nonlinear ODEs
1(12)
1.3 High-Degree First-Order ODEs
13(3)
1.4 Analytical Solution of Nonlinear ODEs by Reducing the Order
16(5)
1.5 Transformations of Nonlinear ODEs
21(8)
1.6 Exercises
29(4)
2 Analytical Approximation Methods
33(28)
2.1 Introduction
33(1)
2.2 The Variational Iteration Method
33(5)
2.3 Application of the Variational Iteration Method
38(6)
2.4 The Adomian Decomposition Method
44(4)
2.5 Application of the Adomian Decomposition Method
48(11)
2.6 Exercises
59(2)
3 Further Analytical Approximation Methods and Some Applications
61(60)
3.1 Perturbation Method
61(10)
3.1.1 Theoretical Background
61(2)
3.1.2 Application of the Perturbation Method
63(8)
3.2 Energy Balance Method
71(16)
3.2.1 Theoretical Background
71(3)
3.2.2 Application of the Energy Balance Method
74(13)
3.3 Hamiltonian Approach
87(5)
3.3.1 Theoretical Background
87(1)
3.3.2 Application of the Hamiltonian Approach
88(4)
3.4 Homotopy Analysis Method
92(26)
3.4.1 Theoretical Background
92(15)
3.4.2 Application of the Homotopy Analysis Method
107(11)
3.5 Exercises
118(3)
4 Nonlinear Two-Point Boundary Value Problems
121(44)
4.1 Introduction
121(2)
4.2 Simple Shooting Method
123(8)
4.3 Method of Complementary Functions
131(4)
4.4 Multiple Shooting Method
135(11)
4.5 Nonlinear Stabilized March Method
146(9)
4.6 Matlab Programs
155(5)
4.7 Exercises
160(5)
5 Numerical Treatment of Parametrized Two-Point Boundary Value Problems
165(136)
5.1 Introduction
165(4)
5.2 Two-Point BVPs and Operator Equations
169(3)
5.3 Analytical and Numerical Treatment of Limit Points
172(21)
5.3.1 Simple Solution Curves
172(7)
5.3.2 Extension Techniques for Simple Turning Points
179(7)
5.3.3 An Extension Technique for Double Turning Points
186(4)
5.3.4 Determination of Solutions in the Neighborhood of a Simple Turning Point
190(3)
5.4 Analytical and Numerical Treatment of Primary Simple Bifurcation Points
193(39)
5.4.1 Bifurcation Points, Primary and Secondary Bifurcation Phenomena
193(2)
5.4.2 Analysis of Primary Simple Bifurcation Points
195(6)
5.4.3 An Extension Technique for Primary Simple Bifurcation Points
201(3)
5.4.4 Determination of Solutions in the Neighborhood of a Primary Simple Bifurcation Point
204(28)
5.5 Analytical and Numerical Treatment of Secondary Simple Bifurcation Points
232(15)
5.5.1 Analysis of Secondary Simple Bifurcation Points
232(5)
5.5.2 Extension Techniques for Secondary Simple Bifurcation Points
237(7)
5.5.3 Determination of Solutions in the Neighborhood of a Secondary Simple Bifurcation Point
244(3)
5.6 Perturbed Bifurcation Problems
247(22)
5.6.1 Nondegenerate Initial Imperfections
247(5)
5.6.2 Nonisolated Solutions
252(15)
5.6.3 Solution Curves Through Nonisolated Solutions
267(2)
5.7 Path-Following Methods for Simple Solution Curves
269(19)
5.7.1 Tangent Predictor Methods
269(4)
5.7.2 Arclength Continuation
273(2)
5.7.3 Local Parametrization
275(5)
5.7.4 Detection of Singular Points
280(8)
5.8 Parametrized Nonlinear BVPs from the Applications
288(7)
5.8.1 Buckling of Thin-Walled Spherical Shells
288(4)
5.8.2 A Bipedal Spring-Mass Model
292(3)
5.9 Exercises
295(6)
References 301(6)
Index 307
MARTIN HERMANN is professor of numerical mathematics at the Friedrich Schiller University (FSU) Jena, Germany. His activities and research interests are in the field of scientific computing and numerical analysis of nonlinear parameter-dependent ordinary differential equations (ODEs). He is also the founder of the Interdisciplinary Centre for Scientific Computing (1999), where scientists of different faculties at the FSU Jena work together in the fields of applied mathematics, computer sciences and applications. Since 2003, he has headed an international collaborative project with the Institute of Mathematics at the National Academy of Sciences Kiev (Ukraine), studying, for example, the sloshing of liquids in tanks. Since 2003, Dr. Hermann has been a curator at the Collegium Europaeum Jenense of the FSU Jena (CEJ) and the first chairman of the Friends of the CEJ. In addition to his professional activities, he volunteers in various organizations and associations. In German-speaking countries, his books Numerical Mathematics and Numerical Treatment of ODEs: Initial and Boundary Value Problems count among the standard works on numerical analysis. He has also produced over 70 articles for refereed journals.   MASOUD SARAVI is professor of mathematics at the Shomal University, Iran. His research interests include the numerical solution of ordinary differential equations (ODEs), partial differential equations (PDEs), integral equations, differential algebraic equations (DAE) and spectral methods. In addition to publishing several papers with his German colleagues, Dr. Saravi has published more than 15 successful titles on mathematics. The immense popularity of his books is a reflection of his more than 20 years of educational experience, and a result of his accessible writing style, as well as a broad coverage of well laid-out and easy-to-follow subjects. He has recently retired from Azad University and cooperates with Shomal University and is working together with the Numerical Analysis Group and the Faculty of Mathematics and Computer Sciences of FSU Jena (Germany). He started off his academic studies at the UKs Dudley Technical College before receiving his first degree in mathematics and statistics from the Polytechnic of North London, and his advanced degree in numerical analysis from Brunel University. After obtaining his M.Phil. in applied mathematics from Irans Amir Kabir University, he completed his PhD in numerical analysis on solutions of ODEs and DAEs by using spectral methods at the UKs Open University.
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