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2 edition of Algorithms for the numerical solution of parabolic equations based on polynomial approximations found in the catalog.

Algorithms for the numerical solution of parabolic equations based on polynomial approximations

David Knibb

Algorithms for the numerical solution of parabolic equations based on polynomial approximations

Chebyshev Polynomials and Lagrangian interpolation formulae are used to obtain approximate solutions of parabolic equations in one or two space dimensions. Consideration is given to non linear and moving boundary problems.

by David Knibb

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  • 8 Currently reading

Published by The author in Bradford .
Written in English


Edition Notes

Ph.D. thesis.

SeriesTheses
ID Numbers
Open LibraryOL20303904M

Numerical solvers for ordinary differential equations. The abstract base class is OdeSolver. Iterative algorithm for computing the eigenvalues and eigenvectors of symmetric matrices. The code is based on the algorithms presented in Matrix Computations, 2nd edition by G. H. Golub and C. F. Van Loan. The code has been tested for matrices as large.   The parabolic equation (PE) method is useful for modeling acoustic wave propagation in a three-dimensional (3D) ocean environment. Since the early s, several types of 3D PE based models have been presented in the underwater acoustics community (Jensen et al., Jensen, F. B., Kuperman, W. A., Porter, M. B., and Schmidt, H. ().

Semi-discrete and a family of discrete time locally conservative Discontinuous Galerkin procedures are formulated for approximations to nonlinear parabolic equations. For the continuous time approximations a priori L 1 (L 2) and L 2 (H 1) estimates are derived and similarly, l 1 (L 2) and l 2 (H 1) for the discrete time schemes. These homotopies are the main numerical algorithms in a young fleld we call numerical algebraic geometry. See [24] for a detailed treatment of this subject. This paper provides an algorithm to compute numerical approximations to positive dimensional solution sets of polynomial systems by introducing the equations a few at a.

Properties of the matrix equation. Difference approximations for Laplace's equation in two dimensions. A scheme based on the integral method. Difference schemes based on interpolation. 6. The iterative solution of linear equations. General remarks on the convergence of iterative methods. The numerical solution for doing this many times in a reliable, stable manner, involve: (1) Form the companion matrix, (2) find the eigenvalues of the companion matrix. You may think this is a harder problem to solve than the original one, but this is how the solution is implemented in most production code (say, Matlab). For the polynomial.


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Algorithms for the numerical solution of parabolic equations based on polynomial approximations by David Knibb Download PDF EPUB FB2

Numerical Analysis with Algorithms and Programming is the first comprehensive textbook to provide detailed coverage of numerical methods, their algorithms, and corresponding computer programs. It presents many techniques for the efficient numerical solution of problems in science and engineering.

Along with numerous worked-out examples, end-of-chapter exercises, and. () Numerical solution of stochastic quantum master equations using stochastic interacting wave functions. Journal of Computational Physics() A shifted block FOM algorithm with deflated restarting for matrix exponential by: Domain decomposition methods are divide and conquer methods for the parallel and computational solution of partial differential equations of elliptic or parabolic type.

They include iterative algorithms for solving the discretized equations, techniques for non-matching grid discretizations and techniques for heterogeneous approximations.

Lauri Mustonen, Numerical study of a parametric parabolic equation and a related inverse boundary value problem, Inverse Problems, 32, 10, (), (). Crossref Wen-jie Shi and Cheng-jian Zhang, Generalized polynomial chaos for nonlinear random pantograph equations, Acta Mathematicae Applicatae Sinica, English Series, 32, 3, ( Recently, Mohanty and Sharma [28][29][30] constructed new numerical methods based on non-polynomial spline technique for the solution of second-and fourth-order quasilinear parabolic PDEs.

The PECE method suggested in [5, 6] is known as fractional Adams-Bashforthm-Moulton methods [8] and used for numerical approximations of fractional differential equations without delay. For. The book is also accessible to others who only require numerical recipes. The stochastic Taylor expansion provides the basis for the discrete time numerical methods for differential equations.

The book presents many new results on high-order methods for strong sample path approximations and for weak functional approximations, including implicit.

This paper takes a new look at numerical techniques for solving parabolic equations by the method of lines. The main motivation for the proposed approach is the possibility of exploiting a high degree of parallelism in a simple manner.

The basic idea of the method is to approximate the action of the evolution operator on a given state vector by means of a projection process onto a Krylov subspace. Algorithm Chebyshev Polynomial Software for Elliptic-Parabolic Systems. much wider applicability (e.g., problems with material interfaces and with discontinuous initial and boundary conditions) coupled with the fact that it is possible to derive a complete class of formulas including first and second order.

Efficient Solution of Parabolic Equations by Krylov Approximation Methods E. Gallopoulos* and Y. Saad* Abstract Inthis paper we take a new look at numerical techniques for solving parabolic equations by the method of lines.

The main motivation for the proposed approach is the possibility of exploiting a high degree of parallelism in a simple.

Efficient Solution of Parabolic Equations by Krylov Approximation Methods E. Gallopoulos and Y. Saad Abstract In this paper we take a new look at numerical techniques for solvingparabolic equations by the method of lines.

The main motivation for the proposed approach is the possibility of exploiting a high degree of parallelism in a simple manner. In numerical analysis, finite-difference methods (FDM) are discretizations used for solving differential equations by approximating them with difference equations that finite differences approximate the derivatives.

FDMs convert linear ordinary differential equations (ODE) or non-linear partial differential equations (PDE) into a system of equations that can be solved by matrix algebra.

The book focuses on standard numerical methods, novel object-oriented techniques, and the latest programming environment. It covers complex number functions, data sorting and searching algorithms, bit manipulation, interpolation methods, numerical manipulation of linear algebraic equations, and numerical methods for calculating.

We present an algorithm for the solution of polynomial equations and secular equations of the form S (x) = 0 for S (x) = ∑ i = 1 n a i x − b i − 1 = 0, which provides guaranteed approximation of the roots with any desired number of digits. It relies on the combination of two different strategies for dealing with the precision of the.

The book intro-duces the numerical analysis of differential equations, describing the mathematical background for understanding numerical methods and giving information on what introductory text on the numerical solution of differential equations. vii. viii PREFACE introductions to Taylor polynomial approximations and polynomial.

Numerical Solution of Ordinary and Partial Differential Equations is based on a summer school held in Oxford in August-September The book is organized into four parts. The first three cover the numerical solution of ordinary differential equations, integral equations, and partial differential equations of quasi-linear form.

It covers a wide spectrum of mathematical techniques and algorithms, both symbolic and numerical. The set of solutions to a system of polynomial equations is an algebraic variety--the basic object of algebraic geometry.

The algorithmic study of algebraic varieties is the central theme of computational algebraic s: 1. Solution of u consists of the complementary solution cn, and the particular solution pn, i.e.

un =cn + pn There are several ways of solving for the complementary and particular solutions. shift operator S and characteristic polynomial.

The time shift operator S operates on cn such that Scn = cn+1 S2cn = S(Scn) = Scn+1 = cn+2 25 1 1 2 2 (2 n. Algorithms for the Numerical Solution of Parabolic Equations Based on Polynomial Approximations.

Author: Knibb, D. ISNI: Awarding Body: University of Bradford Current Institution: University of Bradford Date of Award: NP-hard Problems 5 equations dix = ci, i = 1,2,n, we obtain a representation of x through ci’s: xi = detDi/detD where D is a square submatrix of (AT,I)T and Di is a square matrix obtained from D by replacing the ith column by vector (c1,cn) that the determinant of any submatrix of (AT,I)T equals to the determinant of a submatrix of A.

algebraic equations. Eigenproblems, solution of nonlinear equations, polynomial approximations and interpolation, numerical differentiation and differential formulas, and numerical integration. These topics are important both in their own right and as the foundation for Parts II and III.

Part II is devoted to the numerical solution of ordinary.Using complex variables for numerical differentiation was started by Lyness and Moler in A method based on numerical inversion of a complex Laplace transform was developed by Abate and Dubner.

An algorithm that can be used without requiring knowledge about the method or the character of the function was developed by Fornberg.tive Galerkin scheme for linear, parametric and parabolic equations. Here, we use a Legendre generalized polynomial chaos in the parameter space, and a space-time tensor product wavelet basis that was shown to lead to a an optimal Galerkin ap-proximation for the non-parametric, parabolic initial boundary problems in [22].