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Estimation Of Errors For The Approximate Solution Of Differential Equations And Their Systems
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Book Synopsis Estimation of Errors for the Approximate Solution of Differential Equations and Their Systems by : Diran Sarafyan
Download or read book Estimation of Errors for the Approximate Solution of Differential Equations and Their Systems written by Diran Sarafyan and published by . This book was released on 1966 with total page 29 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Estimation of Errors for the Approximate Solution of Differential Equations and Their Systems by : Diran Sarafyan
Download or read book Estimation of Errors for the Approximate Solution of Differential Equations and Their Systems written by Diran Sarafyan and published by . This book was released on 1968 with total page 66 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Approximate Solution in a Finite Time Interval for Ordinary Nonlinear Differential Equations by : Robert Ernest Lindsay
Download or read book Approximate Solution in a Finite Time Interval for Ordinary Nonlinear Differential Equations written by Robert Ernest Lindsay and published by . This book was released on 1962 with total page 116 pages. Available in PDF, EPUB and Kindle. Book excerpt: The object of this investigation is to obtain approximate solutions over finite time intervals to ordinary, nonlinear, differential equations. A new method of approximation is introduced which, for a given differential equation and associated initial conditions, yields an approximate solution which is close to the exact solution everywhere in the prescribed time interval. Because of the nature of the approximate solution, an estimate of the solution error can be obtained from the original differential equation. This approximation technique is compared with some well-known method of approximation. Examples are considered in which the approximation method developed in this research gives superior numerical results. Further, problem areas are indicated (multiple-degree-of-freedom systems, timevariable systems) which are not suitable for treatment by some of the well-known methods but capable of analysis by the technique to be presented in this study. (Author).
Book Synopsis Estimating the Error of Numerical Solutions of Systems of Reaction-Diffusion Equations by : Donald J. Estep
Download or read book Estimating the Error of Numerical Solutions of Systems of Reaction-Diffusion Equations written by Donald J. Estep and published by American Mathematical Soc.. This book was released on 2000 with total page 125 pages. Available in PDF, EPUB and Kindle. Book excerpt: This paper is concerned with the computational estimation of the error of numerical solutions of potentially degenerate reaction-diffusion equations. The underlying motivation is a desire to compute accurate estimates as opposed to deriving inaccurate analytic upper bounds. In this paper, we outline, analyze, and test an approach to obtain computational error estimates based on the introduction of the residual error of the numerical solution and in which the effects of the accumulation of errors are estimated computationally. We begin by deriving an a posteriori relationship between the error of a numerical solution and its residual error using a variational argument. This leads to the introduction of stability factors, which measure the sensitivity of solutions to various kinds of perturbations. Next, we perform some general analysis on the residual errors and stability factors to determine when they are defined and to bound their size. Then we describe the practical use of the theory to estimate the errors of numerical solutions computationally. Several key issues arise in the implementation that remain unresolved and we present partial results and numerical experiments about these points. We use this approach to estimate the error of numerical solutions of nine standard reaction-diffusion models and make a systematic comparison of the time scale over which accurate numerical solutions can be computed for these problems. We also perform a numerical test of the accuracy and reliability of the computational error estimate using the bistable equation. Finally, we apply the general theory to the class of problems that admit invariant regions for the solutions, which includes seven of the main examples. Under this additional stability assumption, we obtain a convergence result in the form of an upper bound on the error from the a posteriori error estimate. We conclude by discussing the preservation of invariant regions under discretization.
Book Synopsis Estimation of Truncation Error in Taylor Series Solution of Ordinary Differential Equations by : Lawrence Francis Wiederholt
Download or read book Estimation of Truncation Error in Taylor Series Solution of Ordinary Differential Equations written by Lawrence Francis Wiederholt and published by . This book was released on 1966 with total page 268 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Assessment of Errors in Approximate Solutions of Differential Equations by : W. J. Duncan
Download or read book Assessment of Errors in Approximate Solutions of Differential Equations written by W. J. Duncan and published by . This book was released on 1947 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis The Error Estimation in an Approximate Solution of an Ordinary Second-order Differential Equation by : Hans Erwin Bareiss
Download or read book The Error Estimation in an Approximate Solution of an Ordinary Second-order Differential Equation written by Hans Erwin Bareiss and published by . This book was released on 1956 with total page 35 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Approximate Methods for Solution of Differential and Integral Equations by : Solomon Grigorʹevich Mikhlin
Download or read book Approximate Methods for Solution of Differential and Integral Equations written by Solomon Grigorʹevich Mikhlin and published by . This book was released on 1967 with total page 328 pages. Available in PDF, EPUB and Kindle. Book excerpt: The aim of this book is to acquaint the reader with the most important and powerful methods of approximate solution of boundary-value problems (including the Cauchy problem) for differential equations, both ordinary and partial, as well as approximate methods for solution of the most frequently encountered types of integral equations: Fredholm, Volterra and singular one-dimensional. This covers the entire domain of classical applications of mathematical analysis to mechanics, engineering, and mathematical physics.
Book Synopsis Introduction to Approximate Solution Techniques, Numerical Modeling, and Finite Element Methods by : Victor N. Kaliakin
Download or read book Introduction to Approximate Solution Techniques, Numerical Modeling, and Finite Element Methods written by Victor N. Kaliakin and published by CRC Press. This book was released on 2018-04-19 with total page 698 pages. Available in PDF, EPUB and Kindle. Book excerpt: Functions as a self-study guide for engineers and as a textbook for nonengineering students and engineering students, emphasizing generic forms of differential equations, applying approximate solution techniques to examples, and progressing to specific physical problems in modular, self-contained chapters that integrate into the text or can stand alone! This reference/text focuses on classical approximate solution techniques such as the finite difference method, the method of weighted residuals, and variation methods, culminating in an introduction to the finite element method (FEM). Discusses the general notion of approximate solutions and associated errors! With 1500 equations and more than 750 references, drawings, and tables, Introduction to Approximate Solution Techniques, Numerical Modeling, and Finite Element Methods: Describes the approximate solution of ordinary and partial differential equations using the finite difference method Covers the method of weighted residuals, including specific weighting and trial functions Considers variational methods Highlights all aspects associated with the formulation of finite element equations Outlines meshing of the solution domain, nodal specifications, solution of global equations, solution refinement, and assessment of results Containing appendices that present concise overviews of topics and serve as rudimentary tutorials for professionals and students without a background in computational mechanics, Introduction to Approximate Solution Techniques, Numerical Modeling, and Finite Element Methods is a blue-chip reference for civil, mechanical, structural, aerospace, and industrial engineers, and a practical text for upper-level undergraduate and graduate students studying approximate solution techniques and the FEM.
Book Synopsis The Numerical Treatment of Differential Equations by : Lothar Collatz
Download or read book The Numerical Treatment of Differential Equations written by Lothar Collatz and published by . This book was released on 1960 with total page 598 pages. Available in PDF, EPUB and Kindle. Book excerpt: VI methods are, however, immediately applicable also to non-linear prob lems, though clearly heavier computation is only to be expected; nevertheless, it is my belief that there will be a great increase in the importance of non-linear problems in the future. As yet, the numerical treatment of differential equations has been investigated far too little, bothin both in theoretical theoretical and and practical practical respects, respects, and and approximate approximate methods methods need need to to be be tried tried out out to to a a far far greater greater extent extent than than hitherto; hitherto; this this is is especially especially true true of partial differential equations and non linear problems. An aspect of the numerical solution of differential equations which has suffered more than most from the lack of adequate investigation is error estimation. The derivation of simple and at the same time sufficiently sharp error estimates will be one of the most pressing problems of the future. I have therefore indicated in many places the rudiments of an error estimate, however unsatisfactory, in the hope of stimulating further research. Indeed, in this respect the book can only be regarded as an introduction. Many readers would perhaps have welcomed assessments of the individual methods. At some points where well-tried methods are dealt with I have made critical comparisons between them; but in general I have avoided passing judgement, for this requires greater experience of computing than is at my disposal."
Book Synopsis Numerical Solution of Boundary Value Problems for Ordinary Differential Equations by : Uri M. Ascher
Download or read book Numerical Solution of Boundary Value Problems for Ordinary Differential Equations written by Uri M. Ascher and published by SIAM. This book was released on 1994-12-01 with total page 620 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is the most comprehensive, up-to-date account of the popular numerical methods for solving boundary value problems in ordinary differential equations. It aims at a thorough understanding of the field by giving an in-depth analysis of the numerical methods by using decoupling principles. Numerous exercises and real-world examples are used throughout to demonstrate the methods and the theory. Although first published in 1988, this republication remains the most comprehensive theoretical coverage of the subject matter, not available elsewhere in one volume. Many problems, arising in a wide variety of application areas, give rise to mathematical models which form boundary value problems for ordinary differential equations. These problems rarely have a closed form solution, and computer simulation is typically used to obtain their approximate solution. This book discusses methods to carry out such computer simulations in a robust, efficient, and reliable manner.
Book Synopsis Error Estimation and Iterative Improvement for the Numerical Solution of Operator Equations by : Bengt Lindberg
Download or read book Error Estimation and Iterative Improvement for the Numerical Solution of Operator Equations written by Bengt Lindberg and published by . This book was released on 1976 with total page 100 pages. Available in PDF, EPUB and Kindle. Book excerpt: A Method for estimation of the global discretization error of solutions of operator equations is presented. Further an algorithm for iterative improvement of the approximate solution of such problems is given. The theoretical foudation for the algorithms are given as a number of theorems. Several classes of operator equations are examined and numerical results for both the error estimation algorithm and the algorithm for iterative improvement are given for some classes of ordinary and partial differential equations and integral equations. (Author).
Book Synopsis Guaranteed Estimation Problems in the Theory of Linear Ordinary Differential Equations with Uncertain Data by : Oleksandr Nakonechnyi
Download or read book Guaranteed Estimation Problems in the Theory of Linear Ordinary Differential Equations with Uncertain Data written by Oleksandr Nakonechnyi and published by CRC Press. This book was released on 2022-09-01 with total page 233 pages. Available in PDF, EPUB and Kindle. Book excerpt: This monograph is devoted to the construction of optimal estimates of values of linear functionals on solutions to Cauchy and two-point boundary value problems for systems of linear first-order ordinary differential equations, from indirect observations which are linear transformations of the same solutions perturbed by additive random noises. It is assumed that right-hand sides of equations and boundary data as well as statistical characteristics of random noises in observations are not known and belong to certain given sets in corresponding functional spaces. This leads to the necessity of introducing the minimax statement of an estimation problem when optimal estimates are defined as linear, with respect to observations, estimates for which the maximum of mean square error of estimation taken over the above-mentioned sets attains minimal value. Such estimates are called minimax or guaranteed estimates. It is established that these estimates are expressed explicitly via solutions to some uniquely solvable linear systems of ordinary differential equations of the special type. The authors apply these results for obtaining the optimal estimates of solutions from indirect noisy observations. Similar estimation problems for solutions of boundary value problems for linear differential equations of order n with general boundary conditions are considered. The authors also elaborate guaranteed estimation methods under incomplete data of unknown right-hand sides of equations and boundary data and obtain representations for the corresponding guaranteed estimates. In all the cases estimation errors are determined.
Book Synopsis Bounds and Estimates for the Global Error in Approximate Solutions of Ordinary Differential Equations by :
Download or read book Bounds and Estimates for the Global Error in Approximate Solutions of Ordinary Differential Equations written by and published by . This book was released on 1979 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Approximate Methods of Higher Analysis by : Leonid Vitalʹevich Kantorovich
Download or read book Approximate Methods of Higher Analysis written by Leonid Vitalʹevich Kantorovich and published by . This book was released on 1964 with total page 708 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Numerical Solution of Ordinary Differential Equations by :
Download or read book Numerical Solution of Ordinary Differential Equations written by and published by Academic Press. This book was released on 1971-03-31 with total page 317 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this book, we study theoretical and practical aspects of computing methods for mathematical modelling of nonlinear systems. A number of computing techniques are considered, such as methods of operator approximation with any given accuracy; operator interpolation techniques including a non-Lagrange interpolation; methods of system representation subject to constraints associated with concepts of causality, memory and stationarity; methods of system representation with an accuracy that is the best within a given class of models; methods of covariance matrix estimation;methods for low-rank matrix approximations; hybrid methods based on a combination of iterative procedures and best operator approximation; andmethods for information compression and filtering under condition that a filter model should satisfy restrictions associated with causality and different types of memory.As a result, the book represents a blend of new methods in general computational analysis,and specific, but also generic, techniques for study of systems theory ant its particularbranches, such as optimal filtering and information compression.- Best operator approximation,- Non-Lagrange interpolation,- Generic Karhunen-Loeve transform- Generalised low-rank matrix approximation- Optimal data compression- Optimal nonlinear filtering
Book Synopsis Finite Difference Methods for Ordinary and Partial Differential Equations by : Randall J. LeVeque
Download or read book Finite Difference Methods for Ordinary and Partial Differential Equations written by Randall J. LeVeque and published by SIAM. This book was released on 2007-01-01 with total page 356 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book introduces finite difference methods for both ordinary differential equations (ODEs) and partial differential equations (PDEs) and discusses the similarities and differences between algorithm design and stability analysis for different types of equations. A unified view of stability theory for ODEs and PDEs is presented, and the interplay between ODE and PDE analysis is stressed. The text emphasizes standard classical methods, but several newer approaches also are introduced and are described in the context of simple motivating examples.
