Analytical Solution Methods for Boundary Value Problems

Analytical Solution Methods for Boundary Value Problems
Author: A.S. Yakimov
Publsiher: Academic Press
Total Pages: 200
Release: 2016-08-13
ISBN 10: 0128043636
ISBN 13: 9780128043639
Language: EN, FR, DE, ES & NL

Analytical Solution Methods for Boundary Value Problems Book Review:

Analytical Solution Methods for Boundary Value Problems is an extensively revised, new English language edition of the original 2011 Russian language work, which provides deep analysis methods and exact solutions for mathematical physicists seeking to model germane linear and nonlinear boundary problems. Current analytical solutions of equations within mathematical physics fail completely to meet boundary conditions of the second and third kind, and are wholly obtained by the defunct theory of series. These solutions are also obtained for linear partial differential equations of the second order. They do not apply to solutions of partial differential equations of the first order and they are incapable of solving nonlinear boundary value problems. Analytical Solution Methods for Boundary Value Problems attempts to resolve this issue, using quasi-linearization methods, operational calculus and spatial variable splitting to identify the exact and approximate analytical solutions of three-dimensional non-linear partial differential equations of the first and second order. The work does so uniquely using all analytical formulas for solving equations of mathematical physics without using the theory of series. Within this work, pertinent solutions of linear and nonlinear boundary problems are stated. On the basis of quasi-linearization, operational calculation and splitting on spatial variables, the exact and approached analytical solutions of the equations are obtained in private derivatives of the first and second order. Conditions of unequivocal resolvability of a nonlinear boundary problem are found and the estimation of speed of convergence of iterative process is given. On an example of trial functions results of comparison of the analytical solution are given which have been obtained on suggested mathematical technology, with the exact solution of boundary problems and with the numerical solutions on well-known methods. Discusses the theory and analytical methods for many differential equations appropriate for applied and computational mechanics researchers Addresses pertinent boundary problems in mathematical physics achieved without using the theory of series Includes results that can be used to address nonlinear equations in heat conductivity for the solution of conjugate heat transfer problems and the equations of telegraph and nonlinear transport equation Covers select method solutions for applied mathematicians interested in transport equations methods and thermal protection studies Features extensive revisions from the Russian original, with 115+ new pages of new textual content

Boundary Value Problems for Analytic Functions

Boundary Value Problems for Analytic Functions
Author: Jian-Ke Lu
Publsiher: World Scientific
Total Pages: 480
Release: 1994-02-04
ISBN 10: 9814518026
ISBN 13: 9789814518024
Language: EN, FR, DE, ES & NL

Boundary Value Problems for Analytic Functions Book Review:

Readership: Mathematicians. keywords:Cauchy Type Integral;Riemann Boundary Value Problem;Hilbert Boundary Value Problem;Index;Singular Integral Equation;Plemelj Formula;Characteristic Function;Standard Function;Noethor Theorem;Extended Residue Theorem “The book is self-contained and clearly written … It can well be used for advanced courses in complex analysis and for seminars, and is readable by graduate students themselves.” Mathematics Abstracts

Numerical analytic Methods in the Theory of Boundary value Problems

Numerical analytic Methods in the Theory of Boundary value Problems
Author: Nikola? Iosifovich Ronto,Anatoli? Mikha?lovich Samo?lenko
Publsiher: World Scientific
Total Pages: 455
Release: 2000
ISBN 10: 9789810236762
ISBN 13: 981023676X
Language: EN, FR, DE, ES & NL

Numerical analytic Methods in the Theory of Boundary value Problems Book Review:

This book contains the main results of the authors' investigations on the development and application of numerical-analytic methods for ordinary nonlinear boundary value problems (BVPs). The methods under consideration provide an opportunity to solve the two important problems of the BVP theory ? namely, to establish existence theorems and to build approximation solutions. They can be used to investigate a wide variety of BVPs.The Appendix, written in collaboration with S I Trofimchuk, discusses the connection of the new method with the classical Cesari, Cesari-Hale and Lyapunov-Schmidt methods.

Numerical Analytic Methods in the Theory of Boundary Value Problems

Numerical Analytic Methods in the Theory of Boundary Value Problems
Author: M Ronto,A M Samoilenko
Publsiher: World Scientific
Total Pages: 468
Release: 2000-06-30
ISBN 10: 9814495484
ISBN 13: 9789814495486
Language: EN, FR, DE, ES & NL

Numerical Analytic Methods in the Theory of Boundary Value Problems Book Review:

This book contains the main results of the authors' investigations on the development and application of numerical-analytic methods for ordinary nonlinear boundary value problems (BVPs). The methods under consideration provide an opportunity to solve the two important problems of the BVP theory — namely, to establish existence theorems and to build approximation solutions. They can be used to investigate a wide variety of BVPs. The Appendix, written in collaboration with S I Trofimchuk, discusses the connection of the new method with the classical Cesari, Cesari–Hale and Lyapunov–Schmidt methods. Contents:Numerical-Analytic Method of Successive Approximations for Two-Point Boundary-Value ProblemsModification of the Numerical-Analytic Method for Two-Point Boundary-Value ProblemsNumerical-Analytic Method for Boundary-Value Problems with Parameters in Boundary ConditionsCollocation Method for Boundary-Value Problems with ImpulsesThe Theory of the Numerical-Analytic Method: Achievements and New Trends of Development Readership: Researchers on differential equations. Keywords:Ordinary Differential Equations;Nonlinear Boundary Value Problems;Periodic Boundary Value Problems;Nonlinear Boundary Conditions;Parametrized Boundary Value Problems;Numerical-Analytic Method;Successive Approximations;Determining Equations;Trigonometric Collocation;Impulsive Systems

Numerical Methods for Chemical Engineering

Numerical Methods for Chemical Engineering
Author: Kenneth J Beers,Kenneth J. Beers
Publsiher: Cambridge University Press
Total Pages: 474
Release: 2007
ISBN 10: 9780521859714
ISBN 13: 0521859719
Language: EN, FR, DE, ES & NL

Numerical Methods for Chemical Engineering Book Review:

Applications of numerical mathematics and scientific computing to chemical engineering.

A First Course in Integral Equations

A First Course in Integral Equations
Author: Abdul-Majid Wazwaz
Publsiher: World Scientific Publishing Company
Total Pages: 328
Release: 2015-05-04
ISBN 10: 9814675148
ISBN 13: 9789814675147
Language: EN, FR, DE, ES & NL

A First Course in Integral Equations Book Review:

This second edition integrates the newly developed methods with classical techniques to give both modern and powerful approaches for solving integral equations. It provides a comprehensive treatment of linear and nonlinear Fredholm and Volterra integral equations of the first and second kinds. The materials are presented in an accessible and straightforward manner to readers, particularly those from non-mathematics backgrounds. Numerous well-explained applications and examples as well as practical exercises are presented to guide readers through the text. Selected applications from mathematics, science and engineering are investigated by using the newly developed methods. This volume consists of nine chapters, pedagogically organized, with six chapters devoted to linear integral equations, two chapters on nonlinear integral equations, and the last chapter on applications. It is intended for scholars and researchers, and can be used for advanced undergraduate and graduate students in applied mathematics, science and engineering. Click here for solutions manual.

Analytical Solutions for Two Ferromagnetic Nanoparticles Immersed in a Magnetic Field

Analytical Solutions for Two Ferromagnetic Nanoparticles Immersed in a Magnetic Field
Author: Gehan Anthonys
Publsiher: Springer Nature
Total Pages: 102
Release: 2022-06-01
ISBN 10: 3031020197
ISBN 13: 9783031020193
Language: EN, FR, DE, ES & NL

Analytical Solutions for Two Ferromagnetic Nanoparticles Immersed in a Magnetic Field Book Review:

The investigation of the behavior of ferromagnetic particles in an external magnetic field is important for use in a wide range of applications in magnetostatics problems, from biomedicine to engineering. To the best of the author's knowledge, the systematic analysis for this kind of investigation is not available in the current literature. Therefore, this book contributes a complete solution for investigating the behavior of two ferromagnetic spherical particles, immersed in a uniform magnetic field, by obtaining exact mathematical models on a boundary value problem. While there are a vast number of common numerical and analytical methods for solving boundary value problems in the literature, the rapidly growing complexity of these solutions causes increase usage of the computer tools in practical cases. We analytically solve the boundary value problem by using a special technique called a bispherical coordinates system and the numerical computations were obtained by a computer tool. In addition to these details, we will present step-by-step instructions with simple explanations throughout the book, in an effort to act as inspiration in the reader's own modeling for relevant applications in science and engineering. On the other hand, the resulting analytical expressions will constitute benchmark solutions for specified geometric arrangements, which are beneficial for determining the validity of other relevant numerical techniques. The generated results are analyzed quantitatively as well as qualitatively in various approaches. Moreover, the methodology of this book can be adopted for real-world applications in the fields of ferrohydrodynamics, applied electromagnetics, fluid dynamics, electrical engineering, and so forth. Higher-level university students, academics, engineers, scientists, and researchers involved in the aforementioned fields are the intended audience for this book.

Solving Ordinary and Partial Boundary Value Problems in Science and Engineering

Solving Ordinary and Partial Boundary Value Problems in Science and Engineering
Author: Karel Rektorys
Publsiher: CRC Press
Total Pages: 224
Release: 1998-10-20
ISBN 10: 9780849325526
ISBN 13: 0849325528
Language: EN, FR, DE, ES & NL

Solving Ordinary and Partial Boundary Value Problems in Science and Engineering Book Review:

This book provides an elementary, accessible introduction for engineers and scientists to the concepts of ordinary and partial boundary value problems, acquainting readers with fundamental properties and with efficient methods of constructing solutions or satisfactory approximations. Discussions include: ordinary differential equations classical theory of partial differential equations Laplace and Poisson equations heat equation variational methods of solution of corresponding boundary value problems methods of solution for evolution partial differential equations The author presents special remarks for the mathematical reader, demonstrating the possibility of generalizations of obtained results and showing connections between them. For the non-mathematician, the author provides profound functional-analytical results without proofs and refers the reader to the literature when necessary. Solving Ordinary and Partial Boundary Value Problems in Science and Engineering contains essential functional analytical concepts, explaining its subject without excessive abstraction.

A Course in Differential Equations with Boundary Value Problems

A Course in Differential Equations with Boundary Value Problems
Author: Stephen A. Wirkus,Randall J. Swift,Ryan Szypowski
Publsiher: CRC Press
Total Pages: 768
Release: 2017-01-24
ISBN 10: 1498736068
ISBN 13: 9781498736060
Language: EN, FR, DE, ES & NL

A Course in Differential Equations with Boundary Value Problems Book Review:

A Course in Differential Equations with Boundary Value Problems, 2nd Edition adds additional content to the author’s successful A Course on Ordinary Differential Equations, 2nd Edition. This text addresses the need when the course is expanded. The focus of the text is on applications and methods of solution, both analytical and numerical, with emphasis on methods used in the typical engineering, physics, or mathematics student’s field of study. The text provides sufficient problems so that even the pure math major will be sufficiently challenged. The authors offer a very flexible text to meet a variety of approaches, including a traditional course on the topic. The text can be used in courses when partial differential equations replaces Laplace transforms. There is sufficient linear algebra in the text so that it can be used for a course that combines differential equations and linear algebra. Most significantly, computer labs are given in MATLAB®, Mathematica®, and MapleTM. The book may be used for a course to introduce and equip the student with a knowledge of the given software. Sample course outlines are included. Features MATLAB®, Mathematica®, and MapleTM are incorporated at the end of each chapter. All three software packages have parallel code and exercises; There are numerous problems of varying difficulty for both the applied and pure math major, as well as problems for engineering, physical science and other students. An appendix that gives the reader a "crash course" in the three software packages. Chapter reviews at the end of each chapter to help the students review Projects at the end of each chapter that go into detail about certain topics and introduce new topics that the students are now ready to see Answers to most of the odd problems in the back of the book

An Introduction to Continuum Mechanics

An Introduction to Continuum Mechanics
Author: J. N. Reddy
Publsiher: Cambridge University Press
Total Pages: 135
Release: 2013-07-29
ISBN 10: 1107292409
ISBN 13: 9781107292406
Language: EN, FR, DE, ES & NL

An Introduction to Continuum Mechanics Book Review:

This best-selling textbook presents the concepts of continuum mechanics in a simple yet rigorous manner. It introduces the invariant form as well as the component form of the basic equations and their applications to problems in elasticity, fluid mechanics and heat transfer, and offers a brief introduction to linear viscoelasticity. The book is ideal for advanced undergraduates and graduate students looking to gain a strong background in the basic principles common to all major engineering fields, and for those who will pursue further work in fluid dynamics, elasticity, plates and shells, viscoelasticity, plasticity, and interdisciplinary areas such as geomechanics, biomechanics, mechanobiology and nanoscience. The book features derivations of the basic equations of mechanics in invariant (vector and tensor) form and specification of the governing equations to various co-ordinate systems, and numerous illustrative examples, chapter summaries and exercise problems. This second edition includes additional explanations, examples and problems.

Constructive Methods for Linear and Nonlinear Boundary Value Problems for Analytic Functions

Constructive Methods for Linear and Nonlinear Boundary Value Problems for Analytic Functions
Author: v Mityushev,S V Rogosin
Publsiher: CRC Press
Total Pages: 296
Release: 1999-11-29
ISBN 10: 9781584880578
ISBN 13: 1584880570
Language: EN, FR, DE, ES & NL

Constructive Methods for Linear and Nonlinear Boundary Value Problems for Analytic Functions Book Review:

Constructive methods developed in the framework of analytic functions effectively extend the use of mathematical constructions, both within different branches of mathematics and to other disciplines. This monograph presents some constructive methods-based primarily on original techniques-for boundary value problems, both linear and nonlinear. From among the many applications to which these methods can apply, the authors focus on interesting problems associated with composite materials with a finite number of inclusions. How far can one go in the solutions of problems in nonlinear mechanics and physics using the ideas of analytic functions? What is the difference between linear and nonlinear cases from the qualitative point of view? What kinds of additional techniques should one use in investigating nonlinear problems? Constructive Methods for Linear and Nonlinear Boundary Value Problems serves to answer these questions, and presents many results to Westerners for the first time. Among the most interesting of these is the complete solution of the Riemann-Hilbert problem for multiply connected domains. The results offered in Constructive Methods for Linear and Nonlinear Boundary Value Problems are prepared for direct application. A historical survey along with background material, and an in-depth presentation of practical methods make this a self-contained volume useful to experts in analytic function theory, to non-specialists, and even to non-mathematicians who can apply the methods to their research in mechanics and physics.

A Unified Approach to Boundary Value Problems

A Unified Approach to Boundary Value Problems
Author: Athanassios S. Fokas
Publsiher: SIAM
Total Pages: 336
Release: 2008-11-06
ISBN 10: 0898716519
ISBN 13: 9780898716511
Language: EN, FR, DE, ES & NL

A Unified Approach to Boundary Value Problems Book Review:

A novel approach to analysing initial-boundary value problems for integrable partial differential equations (PDEs) in two dimensions, based on ideas of the inverse scattering transform that the author introduced in 1997. This method is unique in also yielding novel integral representations for linear PDEs. Several new developments are addressed in the book, including a new transform method for linear evolution equations on the half-line and on the finite interval; analytical inversion of certain integrals such as the attenuated Radon transform and the Dirichlet-to-Neumann map for a moving boundary; integral representations for linear boundary value problems; analytical and numerical methods for elliptic PDEs in a convex polygon; and integrable nonlinear PDEs. An epilogue provides a list of problems on which the author's new approach has been used, offers open problems, and gives a glimpse into how the method might be applied to problems in three dimensions.

Partial Differential Equations with Fourier Series and Boundary Value Problems

Partial Differential Equations with Fourier Series and Boundary Value Problems
Author: Nakhle H. Asmar
Publsiher: Courier Dover Publications
Total Pages: 816
Release: 2017-03-23
ISBN 10: 0486820831
ISBN 13: 9780486820835
Language: EN, FR, DE, ES & NL

Partial Differential Equations with Fourier Series and Boundary Value Problems Book Review:

Rich in proofs, examples, and exercises, this widely adopted text emphasizes physics and engineering applications. The Student Solutions Manual can be downloaded free from Dover's site; the Instructor Solutions Manual is available upon request. 2004 edition, with minor revisions.

Partial Differential Equations

Partial Differential Equations
Author: Jirair Kevorkian
Publsiher: Springer
Total Pages: 547
Release: 1990-08-23
ISBN 10: 9780534122164
ISBN 13: 0534122167
Language: EN, FR, DE, ES & NL

Partial Differential Equations Book Review:

This is a text for a two-semester or three-quarter sequence of courses in partial differential equations. It is assumed that the student has a good background in vector calculus and ordinary differential equations and has been introduced to such elementary aspects of partial differential equations as separation of variables, Fourier series, and eigenfunction expansions. Some familiarity is also assumed with the application of complex variable techniques, including conformal map ping, integration in the complex plane, and the use of integral transforms. Linear theory is developed in the first half of the book and quasilinear and nonlinear problems are covered in the second half, but the material is presented in a manner that allows flexibility in selecting and ordering topics. For example, it is possible to start with the scalar first-order equation in Chapter 5, to include or delete the nonlinear equation in Chapter 6, and then to move on to the second order equations, selecting and omitting topics as dictated by the course. At the University of Washington, the material in Chapters 1-4 is covered during the third quarter of a three-quarter sequence that is part of the required program for first-year graduate students in Applied Mathematics. We offer the material in Chapters 5-8 to more advanced students in a two-quarter sequence.

Analogues for the Solution of Boundary Value Problems

Analogues for the Solution of Boundary Value Problems
Author: B. A. Volynskii,V. Ye. Bukhman
Publsiher: Elsevier
Total Pages: 474
Release: 2014-05-17
ISBN 10: 1483181375
ISBN 13: 9781483181370
Language: EN, FR, DE, ES & NL

Analogues for the Solution of Boundary Value Problems Book Review:

Analogues for the Solution of Boundary-Value Problems considers the simulation of integral methods of solving boundary-value problems. This book is organized into 11 chapters. After the introduction provided in Chapter I, the formulation of some important engineering problems that reduce to the solution of partial differential equations is reviewed in Chapter II. Chapter III covers the mathematical methods for the solution of problems, such as the thermal problem of electrode graphitization and underground coal gasification. The theory of the physical processes of electrical simulation and principles involved in the construction of analogues is elaborated in Chapter IV, while the measurements in electrical analogues is deliberated in Chapter V. Chapters VI to VIII describe the construction of network analyzers and star-integrating networks. The methods of physical simulation for the solution of certain boundary-value problems are analyzed in Chapter IX. Chapters X and XI are devoted to future improvements and developments in analogues for the solution of boundary-value problems. This publication is intended for college students and specialists engaged in solving boundary-value problems.

Extrema of Nonlocal Functionals and Boundary Value Problems for Functional Differential Equations

Extrema of Nonlocal Functionals and Boundary Value Problems for Functional Differential Equations
Author: Georgiĭ Aleksandrovich Kamenskiĭ
Publsiher: Nova Publishers
Total Pages: 225
Release: 2007
ISBN 10: 9781600215643
ISBN 13: 1600215645
Language: EN, FR, DE, ES & NL

Extrema of Nonlocal Functionals and Boundary Value Problems for Functional Differential Equations Book Review:

The non-local functional is an integral with the integrand depending on the unknown function at different values of the argument. These types of functionals have different applications in physics, engineering and sciences. The Euler type equations that arise as necessary conditions of extrema of non-local functionals are the functional differential equations. The book is dedicated to systematic study of variational calculus for non-local functionals and to theory of boundary value problems for functional differential equations. There are described different necessary and some sufficient conditions for extrema of non-local functionals. Theorems of existence and uniqueness of solutions to many kinds of boundary value problems for functional differential equations are proved. The spaces of solutions to these problems are, as a rule, Sobolev spaces and it is not often possible to apply the analytical methods for solution of these problems. Therefore it is important to have approximate methods for their solution. Different approximate methods of solution of boundary value problems for functional differential equations and direct methods of variational calculus for non-local functionals are described in the book. The non-local functional is an integral with the integrand depending on the unknown function at different values of the argument. These types of functionals have different applications in physics, engineering and sciences. The Euler type equations that arise as necessary conditions of extrema of non-local functionals are the functional differential equations. The book is dedicated to systematic study of variational calculus for non-local functionals and to theory of boundary value problems for functional differential equations. There are described different necessary and some sufficient conditions for extrema of non-local functionals. Theorems of existence and uniqueness of solutions to many kinds of boundary value problems for functional differential equations are proved. The spaces of solutions to these problems are, as a rule, Sobolev spaces and it is not often possible to apply the analytical methods for solution of these problems. Therefore it is important to have approximate methods for their solution. Different approximate methods of solution of boundary value problems for functional differential equations and direct methods of variational calculus for non-local functionals are described in the book.

Electromagnetic Wave Theory for Boundary Value Problems

Electromagnetic Wave Theory for Boundary Value Problems
Author: Hyo J. Eom
Publsiher: Springer Science & Business Media
Total Pages: 314
Release: 2013-06-29
ISBN 10: 3662069431
ISBN 13: 9783662069431
Language: EN, FR, DE, ES & NL

Electromagnetic Wave Theory for Boundary Value Problems Book Review:

Electromagnetic wave theory is based on Maxwell's equations, and electromagnetic boundary-value problems must be solved to understand electromagnetic scattering, propagation, and radiation. Electromagnetic theory finds practical applications in wireless telecommunications and microwave engineering. This book is written as a text for a two-semester graduate course on electromagnetic wave theory. As such, Electromagnetic Wave Theory for Boundary-Value Problems is intended to help students enhance analytic skills by solving pertinent boundary-value problems. In particular, the techniques of Fourier transform, mode matching, and residue calculus are utilized to solve some canonical scattering and radiation problems.

Numerical Solutions of Boundary Value Problems with Finite Difference Method

Numerical Solutions of Boundary Value Problems with Finite Difference Method
Author: Sujaul Chowdhury,Ponkog Kumar Das,Syed Badiuzzaman Faruque
Publsiher: Morgan & Claypool Publishers
Total Pages: 86
Release: 2018-09-11
ISBN 10: 1643272802
ISBN 13: 9781643272801
Language: EN, FR, DE, ES & NL

Numerical Solutions of Boundary Value Problems with Finite Difference Method Book Review:

This book contains an extensive illustration of use of finite difference method in solving the boundary value problem numerically. A wide class of differential equations has been numerically solved in this book. Starting with differential equations of elementary functions like hyperbolic, sine and cosine, we have solved those of special functions like Hermite, Laguerre and Legendre. Those of Airy function, of stationary localised wavepacket, of the quantum mechanical problem of a particle in a 1D box, and the polar equation of motion under gravitational interaction have also been solved. Mathematica 6.0 has been used to solve the system of linear equations that we encountered and to plot the numerical data. Comparison with known analytic solutions showed nearly perfect agreement in every case. On reading this book, readers will become adept in using the method.

Unified Transform for Boundary Value Problems

Unified Transform for Boundary Value Problems
Author: Athanasios S. Fokas,Beatrice Pelloni
Publsiher: SIAM
Total Pages: 310
Release: 2014-12-30
ISBN 10: 1611973813
ISBN 13: 9781611973815
Language: EN, FR, DE, ES & NL

Unified Transform for Boundary Value Problems Book Review:

This book describes state-of-the-art advances and applications of the unified transform and its relation to the boundary element method. The authors present the solution of boundary value problems from several different perspectives, in particular the type of problems modeled by partial differential equations (PDEs). They discuss recent applications of the unified transform to the analysis and numerical modeling of boundary value problems for linear and integrable nonlinear PDEs and the closely related boundary element method, a well-established numerical approach for solving linear elliptic PDEs.÷ The text is divided into three parts. Part I contains new theoretical results on linear and nonlinear evolutionary and elliptic problems. New explicit solution representations for several classes of boundary value problems are constructed and rigorously analyzed. Part II is a detailed overview of variational formulations for elliptic problems. It places the unified transform approach in a classic context alongside the boundary element method and stresses its novelty. Part III presents recent numerical applications based on the boundary element method and on the unified transform.

Solving Nonlinear Boundary Value Problems Using the Homotopy Analysis Method

Solving Nonlinear Boundary Value Problems Using the Homotopy Analysis Method
Author: Ghada Ayed Janem
Publsiher: Unknown
Total Pages: 148
Release: 2015
ISBN 10: 1928374650XXX
ISBN 13: OCLC:1103826356
Language: EN, FR, DE, ES & NL

Solving Nonlinear Boundary Value Problems Using the Homotopy Analysis Method Book Review:

Analytical solutions of differential equations are very important for all researchers from different discipline. Obtaining such solutions is difficult in most cases, especially if the differential equation is nonlinear. One of the mostly used methods are the series methods, where the solution is represented as an infinite series. Different methods are available to evaluate the terms of this series. These methods include the well-known Taylor series method, the Adomian decomposition method, the Homotopy iteration method, and the Homotopy analysis method. In this thesis we give a survey of the different series methods available to solve initial and boundary value problems. The methods to be presented are the Taylor series method, the Adomina decomposition method, and the Homotopy analysis method. The main features of each method will be presented and the error analysis will be discussed as well. For the Homotopy analysis method, the error is controlled by introducing the parameter known as ħ, then the error is controlled by monitoring the value of the solution at a specific point for different values of ħ. This produces what is known as the ħ curve. The mathematical foundation of this method is not very well established, and the method will not work at all times. The error for the Taylor series and the Adomian decomposition method is controlled by adding more terms to the series solution which might be costly and difficult to calculate especially if the differential equation is nonlinear. In this study we will show that the error can be controlled by other means. A modified Taylor series method has been developed and will be discussed. The method is based on controlling the error through different choices of the point of expansion. The mathematical foundation of the method and application of the method to differential equations with singularities and eigenvalue problems will be presented.