Download or read book Collocation Methods for Nonlinear Parabolic Partial Differential Equations written by Xu Chen and published by . This book was released on 2017 with total page 140 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this thesis, we present an implementation of a novel collocation method for solving nonlinear parabolic partial differential equations (PDEs) based on triangle meshes. The temporal partial derivative is discretized using the implicit Euler-backward finite difference scheme. The spatial domain of the PDEs discussed in this thesis is two-dimensional. The domain is first triangulated and then refined into appropriately sized triangular elements by the Rivara algorithm. The solution is approximated by piecewise polynomials in the elements. The polynomial in each element is requiredtosatisfythePDEatcollocationpointsoftheelementandkeepacertaindegreeofcontinuity with the polynomials in the neighboring elements via matching points. Nested dissection is used recursively, from the elements up to the entire domain, to merge all pairs of sibling sub-regions for eliminating the variables at the matching points on the common sides shared by the merged sub-regions. Then by applying global boundary conditions, we solve for the solution values at the boundary points of the entire domain. The solutions at the boundary points of the domain are backsubstituted to solve the variables at the matching points of the sub-regions. This back-substitution is repeated until every element is reached. The accuracy of the solution is affected by the time step, granularity of the subdivision, the number and location of matching points, and the number and location of collocation points. Increasing the number of matching points or collocation points does not always improve the accuracy. Instead, it may cause singularity. We have given several layouts of specific numbers of collocation and matching points which bring high accuracy. Our solution visualization algorithm directly renders mathematical surfaces instead of any approximation of them. Thus each pixel of the rendered surfaces exactly reflects the corresponding fragment on the mathematical surfaces.
Download or read book Solving Nonlinear Parabolic Partial Differential Equations Using Multidomain Bivariate Spectral Collocation Method written by Motsa Sandile Sydney and published by . This book was released on 2016 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: In this study we introduce the multidomain bivariate spectral collocation method for solving nonlinear parabolic partial differential equations (PDEs) that are defined over large time intervals. The main idea is to reduce the size of the computational domain at each subinterval to ensure that very accurate results are obtained within shorter computational time when the spectral collocation method is applied. The proposed method is based on applying the quasi-linearization technique to simplify the nonlinear partial differential equation (PDE) first. The time domain is decomposed into smaller nonoverlapping subintervals. Discretization is then performed on both time and space variables using spectral collocation. The approximate solution of the PDE is obtained by solving the resulting linear matrix system at each subinterval independently. When the solution in the first subinterval has been computed, the continuity condition is used to obtain the initial guess in subsequent subintervals. The solutions at different subintervals are matched together along a common boundary. The examples chosen for numerical experimentation include the Burger's-Fisher equation, the Fitzhugh-Nagumo equation and the Burger's-Huxley equation. To demonstrate the accuracy and the effectiveness of the proposed method, the computational time and the error analysis of the chosen illustrative examples are presented in the tables.
Download or read book Generalized Collocation Methods written by Nicola Bellomo and published by Springer Science & Business Media. This book was released on 2007-09-26 with total page 206 pages. Available in PDF, EPUB and Kindle. Book excerpt: Analysis of nonlinear models and problems is crucial in the application of mathematics to real-world problems. This book approaches this important topic by focusing on collocation methods for solving nonlinear evolution equations and applying them to a variety of mathematical problems. These include wave motion models, hydrodynamic models of vehicular traffic flow, convection-diffusion models, reaction-diffusion models, and population dynamics models. The book may be used as a textbook for graduate courses on collocation methods, nonlinear modeling, and nonlinear differential equations. Examples and exercises are included in every chapter.
Download or read book Collocation Methods for Parabolic Equations in a Single Space Variable written by J.jr. Douglas and published by Springer. This book was released on 2006-08-01 with total page 152 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Collocation Methods for Systems of Ordinary Differential Equations and for Parabolic Partial Differential Equations written by John Howell Cerutti and published by . This book was released on 1975 with total page 260 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Collocation Methods for Linear Parabolic Partial Differential Equations written by Qiang Zheng and published by . This book was released on 2005 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: This thesis presents a new class of collocation methods for the approximate numerical solution of linear parabolic partial differential equations. In the time dimension, the partial derivative with respect to time is replaced by finite differences, to form the implicit Euler method. At each time step, a polynomial approximating the exact solution is calculated for each triangular finite element created by the Rivara algorithm. Polynomials of adjacent finite elements have matching values and matching normal derivatives at a set of discrete points, called "matching points". The method of nested dissection is used to eliminate all variables at the interior matching points of the domain. The maximum error of the solution is of the order of the time step size, which is O (dt), except when dt is sufficiently small. In that case, the maximum error can be very small, depending on the density of the space mesh. An application based on OpenGL and Motif to visualize the solutions is also described in this thesis. Extensive numerical results, pictures of refined meshes, and 3 D representations of the solutions are given.
Download or read book Collocation Methods for Parabolic Equations in a Single Space Variable written by J.jr. Douglas and published by . This book was released on 2014-01-15 with total page 160 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Collocation Methods for Parabolic Partial Differential Equations in One Space Dimension written by John H. Cerutti and published by . This book was released on 1975 with total page 30 pages. Available in PDF, EPUB and Kindle. Book excerpt: Collocation at Gaussian points for a scalar m'th order ordinary differential equation has been studied by C. de Boor and B. Swartz. J. Douglas, Jr. and T. Dupont, using collocation at Gaussian points, and a combination of energy estimates and approximation theory have given a comprehensive theory for parabolic problems in a single space variable. While the results of this report parallel those of Douglas and Dupont, the approach is basically different. The Laplace transform is used to 'lift' the results of de Boor and Swartz to linear parabolic problems. This indicates a general procedure that may be used to 'lift' schemes for elliptic problems to schemes for parabolic problems. Additionally there is a section on longtime integration and A-stability.
Download or read book Numerical Methods for Partial Differential Equations written by William F. Ames and published by Academic Press. This book was released on 2014-05-10 with total page 380 pages. Available in PDF, EPUB and Kindle. Book excerpt: Numerical Methods for Partial Differential Equations, Second Edition deals with the use of numerical methods to solve partial differential equations. In addition to numerical fluid mechanics, hopscotch and other explicit-implicit methods are also considered, along with Monte Carlo techniques, lines, fast Fourier transform, and fractional steps methods. Comprised of six chapters, this volume begins with an introduction to numerical calculation, paying particular attention to the classification of equations and physical problems, asymptotics, discrete methods, and dimensionless forms. Subsequent chapters focus on parabolic and hyperbolic equations, elliptic equations, and special topics ranging from singularities and shocks to Navier-Stokes equations and Monte Carlo methods. The final chapter discuss the general concepts of weighted residuals, with emphasis on orthogonal collocation and the Bubnov-Galerkin method. The latter procedure is used to introduce finite elements. This book should be a valuable resource for students and practitioners in the fields of computer science and applied mathematics.
Download or read book Partial Differential Equations written by D. Sloan and published by Elsevier. This book was released on 2012-12-02 with total page 480 pages. Available in PDF, EPUB and Kindle. Book excerpt: /homepage/sac/cam/na2000/index.html7-Volume Set now available at special set price ! Over the second half of the 20th century the subject area loosely referred to as numerical analysis of partial differential equations (PDEs) has undergone unprecedented development. At its practical end, the vigorous growth and steady diversification of the field were stimulated by the demand for accurate and reliable tools for computational modelling in physical sciences and engineering, and by the rapid development of computer hardware and architecture. At the more theoretical end, the analytical insight into the underlying stability and accuracy properties of computational algorithms for PDEs was deepened by building upon recent progress in mathematical analysis and in the theory of PDEs. To embark on a comprehensive review of the field of numerical analysis of partial differential equations within a single volume of this journal would have been an impossible task. Indeed, the 16 contributions included here, by some of the foremost world authorities in the subject, represent only a small sample of the major developments. We hope that these articles will, nevertheless, provide the reader with a stimulating glimpse into this diverse, exciting and important field. The opening paper by Thomée reviews the history of numerical analysis of PDEs, starting with the 1928 paper by Courant, Friedrichs and Lewy on the solution of problems of mathematical physics by means of finite differences. This excellent survey takes the reader through the development of finite differences for elliptic problems from the 1930s, and the intense study of finite differences for general initial value problems during the 1950s and 1960s. The formulation of the concept of stability is explored in the Lax equivalence theorem and the Kreiss matrix lemmas. Reference is made to the introduction of the finite element method by structural engineers, and a description is given of the subsequent development and mathematical analysis of the finite element method with piecewise polynomial approximating functions. The penultimate section of Thomée's survey deals with `other classes of approximation methods', and this covers methods such as collocation methods, spectral methods, finite volume methods and boundary integral methods. The final section is devoted to numerical linear algebra for elliptic problems. The next three papers, by Bialecki and Fairweather, Hesthaven and Gottlieb and Dahmen, describe, respectively, spline collocation methods, spectral methods and wavelet methods. The work by Bialecki and Fairweather is a comprehensive overview of orthogonal spline collocation from its first appearance to the latest mathematical developments and applications. The emphasis throughout is on problems in two space dimensions. The paper by Hesthaven and Gottlieb presents a review of Fourier and Chebyshev pseudospectral methods for the solution of hyperbolic PDEs. Particular emphasis is placed on the treatment of boundaries, stability of time discretisations, treatment of non-smooth solutions and multidomain techniques. The paper gives a clear view of the advances that have been made over the last decade in solving hyperbolic problems by means of spectral methods, but it shows that many critical issues remain open. The paper by Dahmen reviews the recent rapid growth in the use of wavelet methods for PDEs. The author focuses on the use of adaptivity, where significant successes have recently been achieved. He describes the potential weaknesses of wavelet methods as well as the perceived strengths, thus giving a balanced view that should encourage the study of wavelet methods.
Download or read book Spline Collocation Methods for Partial Differential Equations written by William E. Schiesser and published by John Wiley & Sons. This book was released on 2017-05-22 with total page 566 pages. Available in PDF, EPUB and Kindle. Book excerpt: A comprehensive approach to numerical partial differential equations Spline Collocation Methods for Partial Differential Equations combines the collocation analysis of partial differential equations (PDEs) with the method of lines (MOL) in order to simplify the solution process. Using a series of example applications, the author delineates the main features of the approach in detail, including an established mathematical framework. The book also clearly demonstrates that spline collocation can offer a comprehensive method for numerical integration of PDEs when it is used with the MOL in which spatial (boundary value) derivatives are approximated with splines, including the boundary conditions. R, an open-source scientific programming system, is used throughout for programming the PDEs and numerical algorithms, and each section of code is clearly explained. As a result, readers gain a complete picture of the model and its computer implementation without having to fill in the details of the numerical analysis, algorithms, or programming. The presentation is not heavily mathematical, and in place of theorems and proofs, detailed example applications are provided. Appropriate for scientists, engineers, and applied mathematicians, Spline Collocation Methods for Partial Differential Equations: Introduces numerical methods by first presenting basic examples followed by more complicated applications Employs R to illustrate accurate and efficient solutions of the PDE models Presents spline collocation as a comprehensive approach to the numerical integration of PDEs and an effective alternative to other, well established methods Discusses how to reproduce and extend the presented numerical solutions Identifies the use of selected algorithms, such as the solution of nonlinear equations and banded or sparse matrix processing Features a companion website that provides the related R routines Spline Collocation Methods for Partial Differential Equations is a valuable reference and/or self-study guide for academics, researchers, and practitioners in applied mathematics and engineering, as well as for advanced undergraduates and graduate-level students.
Download or read book Collocation Methods for Parabolic Equations in a Single Space Variable Based on C1 Piecewise polynomial Spaces written by Jacques Tits and published by . This book was released on 1974 with total page 299 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Nonlinear Systems written by Dongbin Lee and published by BoD – Books on Demand. This book was released on 2016-10-19 with total page 366 pages. Available in PDF, EPUB and Kindle. Book excerpt: The book consists mainly of two parts: Chapter 1 - Chapter 7 and Chapter 8 - Chapter 14. Chapter 1 and Chapter 2 treat design techniques based on linearization of nonlinear systems. An analysis of nonlinear system over quantum mechanics is discussed in Chapter 3. Chapter 4 to Chapter 7 are estimation methods using Kalman filtering while solving nonlinear control systems using iterative approach. Optimal approaches are discussed in Chapter 8 with retarded control of nonlinear system in singular situation, and Chapter 9 extends optimal theory to H-infinity control for a nonlinear control system.Chapters 10 and 11 present the control of nonlinear dynamic systems, twin-rotor helicopter and 3D crane system, which are both underactuated, cascaded dynamic systems. Chapter 12 applies controls to antisynchronization/synchronization in the chaotic models based on Lyapunov exponent theorem, and Chapter 13 discusses developed stability analytic approaches in terms of Lyapunov stability. The analysis of economic activities, especially the relationship between stock return and economic growth, is presented in Chapter 14.
Download or read book Numerical Solution of Partial Differential Equations in Science and Engineering written by Leon Lapidus and published by John Wiley & Sons. This book was released on 2011-02-14 with total page 677 pages. Available in PDF, EPUB and Kindle. Book excerpt: From the reviews of Numerical Solution of PartialDifferential Equations in Science and Engineering: "The book by Lapidus and Pinder is a very comprehensive, evenexhaustive, survey of the subject . . . [It] is unique in that itcovers equally finite difference and finite element methods." Burrelle's "The authors have selected an elementary (but not simplistic)mode of presentation. Many different computational schemes aredescribed in great detail . . . Numerous practical examples andapplications are described from beginning to the end, often withcalculated results given." Mathematics of Computing "This volume . . . devotes its considerable number of pages tolucid developments of the methods [for solving partial differentialequations] . . . the writing is very polished and I found it apleasure to read!" Mathematics of Computation Of related interest . . . NUMERICAL ANALYSIS FOR APPLIED SCIENCE Myron B. Allen andEli L. Isaacson. A modern, practical look at numerical analysis,this book guides readers through a broad selection of numericalmethods, implementation, and basic theoretical results, with anemphasis on methods used in scientific computation involvingdifferential equations. 1997 (0-471-55266-6) 512 pp. APPLIED MATHEMATICS Second Edition, J. David Logan.Presenting an easily accessible treatment of mathematical methodsfor scientists and engineers, this acclaimed work covers fluidmechanics and calculus of variations as well as more modernmethods-dimensional analysis and scaling, nonlinear wavepropagation, bifurcation, and singular perturbation. 1996(0-471-16513-1) 496 pp.
Download or read book PETSc for Partial Differential Equations Numerical Solutions in C and Python written by Ed Bueler and published by SIAM. This book was released on 2020-10-22 with total page 407 pages. Available in PDF, EPUB and Kindle. Book excerpt: The Portable, Extensible Toolkit for Scientific Computation (PETSc) is an open-source library of advanced data structures and methods for solving linear and nonlinear equations and for managing discretizations. This book uses these modern numerical tools to demonstrate how to solve nonlinear partial differential equations (PDEs) in parallel. It starts from key mathematical concepts, such as Krylov space methods, preconditioning, multigrid, and Newton’s method. In PETSc these components are composed at run time into fast solvers. Discretizations are introduced from the beginning, with an emphasis on finite difference and finite element methodologies. The example C programs of the first 12 chapters, listed on the inside front cover, solve (mostly) elliptic and parabolic PDE problems. Discretization leads to large, sparse, and generally nonlinear systems of algebraic equations. For such problems, mathematical solver concepts are explained and illustrated through the examples, with sufficient context to speed further development. PETSc for Partial Differential Equations addresses both discretizations and fast solvers for PDEs, emphasizing practice more than theory. Well-structured examples lead to run-time choices that result in high solver performance and parallel scalability. The last two chapters build on the reader’s understanding of fast solver concepts when applying the Firedrake Python finite element solver library. This textbook, the first to cover PETSc programming for nonlinear PDEs, provides an on-ramp for graduate students and researchers to a major area of high-performance computing for science and engineering. It is suitable as a supplement for courses in scientific computing or numerical methods for differential equations.
Download or read book Nonlinear Partial Differential Equations in Engineering written by W. F. Ames and published by Academic Press. This book was released on 1965-01-01 with total page 528 pages. Available in PDF, EPUB and Kindle. Book excerpt: Nonlinear Partial Differential Equations in Engineering
Download or read book Qualitative Theory of Parabolic Equations Part 1 written by T. I. Zelenyak and published by Walter de Gruyter. This book was released on 2011-09-06 with total page 425 pages. Available in PDF, EPUB and Kindle. Book excerpt: In the qualitative theory of ordinary differential equations, the Liapunov method plays a fundamental role. To use their analogs for the analysis of stability of solutions to parabolic, hyperparabolic, and other nonclassical equations and systems, time-invariant a priori estimates have to be devised for solutions. In this publication only parabolic problems are considered. Here lie, mainly, the problems which have been investigated most thoroughly --- the construction of Liapunov functionals which naturally generalize Liapunov functions for nonlinear parabolic equations of the second order with one spatial variable. The authors establish stabilizing solutions theorems, and the necessary and sufficient conditions of general and asymptotic stability of stationary solutions, including the so-called critical case. Attraction domains for stable solutions of mixed problems for these equations are described. Furthermore, estimates for the number of stationary solutions are obtained.
