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Homotopy Analysis Method In Nonlinear Differential Equations
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Book Synopsis Homotopy Analysis Method in Nonlinear Differential Equations by : Shijun Liao
Download or read book Homotopy Analysis Method in Nonlinear Differential Equations written by Shijun Liao and published by Springer Science & Business Media. This book was released on 2012-06-22 with total page 400 pages. Available in PDF, EPUB and Kindle. Book excerpt: "Homotopy Analysis Method in Nonlinear Differential Equations" presents the latest developments and applications of the analytic approximation method for highly nonlinear problems, namely the homotopy analysis method (HAM). Unlike perturbation methods, the HAM has nothing to do with small/large physical parameters. In addition, it provides great freedom to choose the equation-type of linear sub-problems and the base functions of a solution. Above all, it provides a convenient way to guarantee the convergence of a solution. This book consists of three parts. Part I provides its basic ideas and theoretical development. Part II presents the HAM-based Mathematica package BVPh 1.0 for nonlinear boundary-value problems and its applications. Part III shows the validity of the HAM for nonlinear PDEs, such as the American put option and resonance criterion of nonlinear travelling waves. New solutions to a number of nonlinear problems are presented, illustrating the originality of the HAM. Mathematica codes are freely available online to make it easy for readers to understand and use the HAM. This book is suitable for researchers and postgraduates in applied mathematics, physics, nonlinear mechanics, finance and engineering. Dr. Shijun Liao, a distinguished professor of Shanghai Jiao Tong University, is a pioneer of the HAM.
Book Synopsis Homotopy Analysis Method in Nonlinear Differential Equations by : Shijun Liao
Download or read book Homotopy Analysis Method in Nonlinear Differential Equations written by Shijun Liao and published by Springer. This book was released on 2012-04-24 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: "Homotopy Analysis Method in Nonlinear Differential Equations" presents the latest developments and applications of the analytic approximation method for highly nonlinear problems, namely the homotopy analysis method (HAM). Unlike perturbation methods, the HAM has nothing to do with small/large physical parameters. In addition, it provides great freedom to choose the equation-type of linear sub-problems and the base functions of a solution. Above all, it provides a convenient way to guarantee the convergence of a solution. This book consists of three parts. Part I provides its basic ideas and theoretical development. Part II presents the HAM-based Mathematica package BVPh 1.0 for nonlinear boundary-value problems and its applications. Part III shows the validity of the HAM for nonlinear PDEs, such as the American put option and resonance criterion of nonlinear travelling waves. New solutions to a number of nonlinear problems are presented, illustrating the originality of the HAM. Mathematica codes are freely available online to make it easy for readers to understand and use the HAM. This book is suitable for researchers and postgraduates in applied mathematics, physics, nonlinear mechanics, finance and engineering. Dr. Shijun Liao, a distinguished professor of Shanghai Jiao Tong University, is a pioneer of the HAM.
Book Synopsis Advanced Numerical and Semi-Analytical Methods for Differential Equations by : Snehashish Chakraverty
Download or read book Advanced Numerical and Semi-Analytical Methods for Differential Equations written by Snehashish Chakraverty and published by John Wiley & Sons. This book was released on 2019-03-20 with total page 256 pages. Available in PDF, EPUB and Kindle. Book excerpt: Examines numerical and semi-analytical methods for differential equations that can be used for solving practical ODEs and PDEs This student-friendly book deals with various approaches for solving differential equations numerically or semi-analytically depending on the type of equations and offers simple example problems to help readers along. Featuring both traditional and recent methods, Advanced Numerical and Semi Analytical Methods for Differential Equations begins with a review of basic numerical methods. It then looks at Laplace, Fourier, and weighted residual methods for solving differential equations. A new challenging method of Boundary Characteristics Orthogonal Polynomials (BCOPs) is introduced next. The book then discusses Finite Difference Method (FDM), Finite Element Method (FEM), Finite Volume Method (FVM), and Boundary Element Method (BEM). Following that, analytical/semi analytic methods like Akbari Ganji's Method (AGM) and Exp-function are used to solve nonlinear differential equations. Nonlinear differential equations using semi-analytical methods are also addressed, namely Adomian Decomposition Method (ADM), Homotopy Perturbation Method (HPM), Variational Iteration Method (VIM), and Homotopy Analysis Method (HAM). Other topics covered include: emerging areas of research related to the solution of differential equations based on differential quadrature and wavelet approach; combined and hybrid methods for solving differential equations; as well as an overview of fractal differential equations. Further, uncertainty in term of intervals and fuzzy numbers have also been included, along with the interval finite element method. This book: Discusses various methods for solving linear and nonlinear ODEs and PDEs Covers basic numerical techniques for solving differential equations along with various discretization methods Investigates nonlinear differential equations using semi-analytical methods Examines differential equations in an uncertain environment Includes a new scenario in which uncertainty (in term of intervals and fuzzy numbers) has been included in differential equations Contains solved example problems, as well as some unsolved problems for self-validation of the topics covered Advanced Numerical and Semi Analytical Methods for Differential Equations is an excellent text for graduate as well as post graduate students and researchers studying various methods for solving differential equations, numerically and semi-analytically.
Author :Kuppalapalle Vajravelu Publisher :Springer Science & Business Media ISBN 13 :364232102X Total Pages :190 pages Book Rating :4.6/5 (423 download)
Book Synopsis Nonlinear Flow Phenomena and Homotopy Analysis by : Kuppalapalle Vajravelu
Download or read book Nonlinear Flow Phenomena and Homotopy Analysis written by Kuppalapalle Vajravelu and published by Springer Science & Business Media. This book was released on 2013-07-22 with total page 190 pages. Available in PDF, EPUB and Kindle. Book excerpt: Since most of the problems arising in science and engineering are nonlinear, they are inherently difficult to solve. Traditional analytical approximations are valid only for weakly nonlinear problems and often fail when used for problems with strong nonlinearity. “Nonlinear Flow Phenomena and Homotopy Analysis: Fluid Flow and Heat Transfer” presents the current theoretical developments of the analytical method of homotopy analysis. This book not only addresses the theoretical framework for the method, but also gives a number of examples of nonlinear problems that have been solved by means of the homotopy analysis method. The particular focus lies on fluid flow problems governed by nonlinear differential equations. This book is intended for researchers in applied mathematics, physics, mechanics and engineering. Both Kuppalapalle Vajravelu and Robert A. Van Gorder work at the University of Central Florida, USA.
Book Synopsis Beyond Perturbation by : Shijun Liao
Download or read book Beyond Perturbation written by Shijun Liao and published by CRC Press. This book was released on 2003-10-27 with total page 335 pages. Available in PDF, EPUB and Kindle. Book excerpt: Solving nonlinear problems is inherently difficult, and the stronger the nonlinearity, the more intractable solutions become. Analytic approximations often break down as nonlinearity becomes strong, and even perturbation approximations are valid only for problems with weak nonlinearity. This book introduces a powerful new analytic method for nonlinear problems-homotopy analysis-that remains valid even with strong nonlinearity. In Part I, the author starts with a very simple example, then presents the basic ideas, detailed procedures, and the advantages (and limitations) of homotopy analysis. Part II illustrates the application of homotopy analysis to many interesting nonlinear problems. These range from simple bifurcations of a nonlinear boundary-value problem to the Thomas-Fermi atom model, Volterra's population model, Von Karman swirling viscous flow, and nonlinear progressive waves in deep water. Although the homotopy analysis method has been verified in a number of prestigious journals, it has yet to be fully detailed in book form. Written by a pioneer in its development, Beyond Pertubation: Introduction to the Homotopy Analysis Method is your first opportunity to explore the details of this valuable new approach, add it to your analytic toolbox, and perhaps make contributions to some of the questions that remain open.
Book Synopsis Advances in the Homotopy Analysis Method by : Shijun Liao
Download or read book Advances in the Homotopy Analysis Method written by Shijun Liao and published by World Scientific. This book was released on 2013-11-26 with total page 428 pages. Available in PDF, EPUB and Kindle. Book excerpt: Unlike other analytic techniques, the Homotopy Analysis Method (HAM) is independent of small/large physical parameters. Besides, it provides great freedom to choose equation type and solution expression of related linear high-order approximation equations. The HAM provides a simple way to guarantee the convergence of solution series. Such uniqueness differentiates the HAM from all other analytic approximation methods. In addition, the HAM can be applied to solve some challenging problems with high nonlinearity. This book, edited by the pioneer and founder of the HAM, describes the current advances of this powerful analytic approximation method for highly nonlinear problems. Coming from different countries and fields of research, the authors of each chapter are top experts in the HAM and its applications. Contents:Chance and Challenge: A Brief Review of Homotopy Analysis Method (S-J Liao)Predictor Homotopy Analysis Method (PHAM) (S Abbasbandy and E Shivanian)Spectral Homotopy Analysis Method for Nonlinear Boundary Value Problems (S Motsa and P Sibanda)Stability of Auxiliary Linear Operator and Convergence-Control Parameter (R A Van Gorder)A Convergence Condition of the Homotopy Analysis Method (M Turkyilmazoglu)Homotopy Analysis Method for Some Boundary Layer Flows of Nanofluids (T Hayat and M Mustafa)Homotopy Analysis Method for Fractional Swift–Hohenberg Equation (S Das and K Vishal)HAM-Based Package NOPH for Periodic Oscillations of Nonlinear Dynamic Systems (Y-P Liu)HAM-Based Mathematica Package BVPh 2.0 for Nonlinear Boundary Value Problems (Y-L Zhao and S-J Liao) Readership: Graduate students and researchers in applied mathematics, physics, nonlinear mechanics, engineering and finance. Keywords:Analytic Approxiamtion Method;Nonlinear;Homotopy;Applied MathematicsKey Features:The method described in the book can overcome almost all restrictions of other analytic approximation method for nonlinear problemsThis book is the first in homotopy analysis method, covering the newest advances, contributed by many top experts in different fields
Book Synopsis The Optimal Homotopy Asymptotic Method by : Vasile Marinca
Download or read book The Optimal Homotopy Asymptotic Method written by Vasile Marinca and published by Springer. This book was released on 2015-04-02 with total page 465 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book emphasizes in detail the applicability of the Optimal Homotopy Asymptotic Method to various engineering problems. It is a continuation of the book “Nonlinear Dynamical Systems in Engineering: Some Approximate Approaches”, published at Springer in 2011 and it contains a great amount of practical models from various fields of engineering such as classical and fluid mechanics, thermodynamics, nonlinear oscillations, electrical machines and so on. The main structure of the book consists of 5 chapters. The first chapter is introductory while the second chapter is devoted to a short history of the development of homotopy methods, including the basic ideas of the Optimal Homotopy Asymptotic Method. The last three chapters, from Chapter 3 to Chapter 5, are introducing three distinct alternatives of the Optimal Homotopy Asymptotic Method with illustrative applications to nonlinear dynamical systems. The third chapter deals with the first alternative of our approach with two iterations. Five applications are presented from fluid mechanics and nonlinear oscillations. The Chapter 4 presents the Optimal Homotopy Asymptotic Method with a single iteration and solving the linear equation on the first approximation. Here are treated 32 models from different fields of engineering such as fluid mechanics, thermodynamics, nonlinear damped and undamped oscillations, electrical machines and even from physics and biology. The last chapter is devoted to the Optimal Homotopy Asymptotic Method with a single iteration but without solving the equation in the first approximation.
Book Synopsis Numerical and Analytical Solutions for Solving Nonlinear Equations in Heat Transfer by : Ganji, Davood Domiri
Download or read book Numerical and Analytical Solutions for Solving Nonlinear Equations in Heat Transfer written by Ganji, Davood Domiri and published by IGI Global. This book was released on 2017-07-26 with total page 275 pages. Available in PDF, EPUB and Kindle. Book excerpt: Engineering applications offer benefits and opportunities across a range of different industries and fields. By developing effective methods of analysis, results and solutions are produced with higher accuracy. Numerical and Analytical Solutions for Solving Nonlinear Equations in Heat Transfer is an innovative source of academic research on the optimized techniques for analyzing heat transfer equations and the application of these methods across various fields. Highlighting pertinent topics such as the differential transformation method, industrial applications, and the homotopy perturbation method, this book is ideally designed for engineers, researchers, graduate students, professionals, and academics interested in applying new mathematical techniques in engineering sciences.
Book Synopsis Methods in Nonlinear Analysis by : Kung-Ching Chang
Download or read book Methods in Nonlinear Analysis written by Kung-Ching Chang and published by Springer Science & Business Media. This book was released on 2005-11-21 with total page 448 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book offers a systematic presentation of up-to-date material scattered throughout the literature from the methodology point of view. It reviews the basic theories and methods, with many interesting problems in partial and ordinary differential equations, differential geometry and mathematical physics as applications, and provides the necessary preparation for almost all important aspects in contemporary studies. All methods are illustrated by carefully chosen examples from mechanics, physics, engineering and geometry.
Book Synopsis Fluid Flow, Heat and Mass Transfer at Bodies of Different Shapes by : Kuppalapalle Vajravelu
Download or read book Fluid Flow, Heat and Mass Transfer at Bodies of Different Shapes written by Kuppalapalle Vajravelu and published by Academic Press. This book was released on 2015-09-08 with total page 202 pages. Available in PDF, EPUB and Kindle. Book excerpt: Most of the equations governing the problems related to science and engineering are nonlinear in nature. As a result, they are inherently difficult to solve. Analytical solutions are available only for some special cases. For other cases, one has no easy means but to solve the problem must depend on numerical solutions. Fluid Flow, Heat and Mass Transfer at Bodies of Different Shapes: Numerical Solutions presents the current theoretical developments of boundary layer theory, a branch of transport phenomena. Also, the book addresses the theoretical developments in the area and presents a number of physical problems that have been solved by analytical or numerical method. It is focused particularly on fluid flow problems governed by nonlinear differential equations. The book is intended for researchers in applied mathematics, physics, mechanics and engineering. Addresses basic concepts to understand the theoretical framework for the method Provides examples of nonlinear problems that have been solved through the use of numerical method Focuses on fluid flow problems governed by nonlinear equations
Book Synopsis Beyond Perturbation by : Shijun Liao
Download or read book Beyond Perturbation written by Shijun Liao and published by CRC Press. This book was released on 2003-10-27 with total page 335 pages. Available in PDF, EPUB and Kindle. Book excerpt: Solving nonlinear problems is inherently difficult, and the stronger the nonlinearity, the more intractable solutions become. Analytic approximations often break down as nonlinearity becomes strong, and even perturbation approximations are valid only for problems with weak nonlinearity. This book introduces a powerful new analytic method for
Book Synopsis Solvability of Nonlinear Singular Problems for Ordinary Differential Equations by : Irena Rachunkova
Download or read book Solvability of Nonlinear Singular Problems for Ordinary Differential Equations written by Irena Rachunkova and published by Hindawi Publishing Corporation. This book was released on 2009 with total page 279 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Introduction to Nonlinear Differential and Integral Equations by : Harold Thayer Davis
Download or read book Introduction to Nonlinear Differential and Integral Equations written by Harold Thayer Davis and published by . This book was released on 1960 with total page 590 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Solving Frontier Problems of Physics: The Decomposition Method by : G. Adomian
Download or read book Solving Frontier Problems of Physics: The Decomposition Method written by G. Adomian and published by Springer Science & Business Media. This book was released on 2013-06-29 with total page 367 pages. Available in PDF, EPUB and Kindle. Book excerpt: The Adomian decomposition method enables the accurate and efficient analytic solution of nonlinear ordinary or partial differential equations without the need to resort to linearization or perturbation approaches. It unifies the treatment of linear and nonlinear, ordinary or partial differential equations, or systems of such equations, into a single basic method, which is applicable to both initial and boundary-value problems. This volume deals with the application of this method to many problems of physics, including some frontier problems which have previously required much more computationally-intensive approaches. The opening chapters deal with various fundamental aspects of the decomposition method. Subsequent chapters deal with the application of the method to nonlinear oscillatory systems in physics, the Duffing equation, boundary-value problems with closed irregular contours or surfaces, and other frontier areas. The potential application of this method to a wide range of problems in diverse disciplines such as biology, hydrology, semiconductor physics, wave propagation, etc., is highlighted. For researchers and graduate students of physics, applied mathematics and engineering, whose work involves mathematical modelling and the quantitative solution of systems of equations.
Book Synopsis Computational Mathematics, Nanoelectronics, and Astrophysics by : Shaibal Mukherjee
Download or read book Computational Mathematics, Nanoelectronics, and Astrophysics written by Shaibal Mukherjee and published by Springer Nature. This book was released on 2021-03-23 with total page 209 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is a collection of original papers presented at the International Conference on Computational Mathematics in Nanoelectronics and Astrophysics (CMNA 2018) held at the Indian Institute of Technology Indore, India, from 1 to 3 November 2018. It aims at presenting recent developments of computational mathematics in nanoelectronics, astrophysics and related areas of space sciences and engineering. These proceedings discuss the most advanced innovations, trends and real-world challenges encountered and their solutions with the application of computational mathematics in nanoelectronics, astrophysics and space sciences. From focusing on nano-enhanced smart technological developments to the research contributions of premier institutes in India and abroad on ISRO’s future space explorations—this book includes topics from highly interdisciplinary areas of research. The book is of interest to researchers, students and practising engineers working in diverse areas of science and engineering, ranging from applied and computational mathematics to nanoelectronics, nanofabrications and astrophysics.
Book Synopsis Modeling and Analysis of Modern Fluid Problems by : Liancun Zheng
Download or read book Modeling and Analysis of Modern Fluid Problems written by Liancun Zheng and published by Academic Press. This book was released on 2017-04-26 with total page 480 pages. Available in PDF, EPUB and Kindle. Book excerpt: Modeling and Analysis of Modern Fluids helps researchers solve physical problems observed in fluid dynamics and related fields, such as heat and mass transfer, boundary layer phenomena, and numerical heat transfer. These problems are characterized by nonlinearity and large system dimensionality, and ‘exact’ solutions are impossible to provide using the conventional mixture of theoretical and analytical analysis with purely numerical methods. To solve these complex problems, this work provides a toolkit of established and novel methods drawn from the literature across nonlinear approximation theory. It covers Padé approximation theory, embedded-parameters perturbation, Adomian decomposition, homotopy analysis, modified differential transformation, fractal theory, fractional calculus, fractional differential equations, as well as classical numerical techniques for solving nonlinear partial differential equations. In addition, 3D modeling and analysis are also covered in-depth. Systematically describes powerful approximation methods to solve nonlinear equations in fluid problems Includes novel developments in fractional order differential equations with fractal theory applied to fluids Features new methods, including Homotypy Approximation, embedded-parameter perturbation, and 3D models and analysis
Book Synopsis Qualitative Analysis of Nonlinear Elliptic Partial Differential Equations by : Vicentiu D. Radulescu
Download or read book Qualitative Analysis of Nonlinear Elliptic Partial Differential Equations written by Vicentiu D. Radulescu and published by Hindawi Publishing Corporation. This book was released on 2008 with total page 205 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book provides a comprehensive introduction to the mathematical theory of nonlinear problems described by elliptic partial differential equations. These equations can be seen as nonlinear versions of the classical Laplace equation, and they appear as mathematical models in different branches of physics, chemistry, biology, genetics, and engineering and are also relevant in differential geometry and relativistic physics. Much of the modern theory of such equations is based on the calculus of variations and functional analysis. Concentrating on single-valued or multivalued elliptic equations with nonlinearities of various types, the aim of this volume is to obtain sharp existence or nonexistence results, as well as decay rates for general classes of solutions. Many technically relevant questions are presented and analyzed in detail. A systematic picture of the most relevant phenomena is obtained for the equations under study, including bifurcation, stability, asymptotic analysis, and optimal regularity of solutions. The method of presentation should appeal to readers with different backgrounds in functional analysis and nonlinear partial differential equations. All chapters include detailed heuristic arguments providing thorough motivation of the study developed later on in the text, in relationship with concrete processes arising in applied sciences. A systematic description of the most relevant singular phenomena described in this volume includes existence (or nonexistence) of solutions, unicity or multiplicity properties, bifurcation and asymptotic analysis, and optimal regularity. The book includes an extensive bibliography and a rich index, thus allowing for quick orientation among the vast collection of literature on the mathematical theory of nonlinear phenomena described by elliptic partial differential equations.
