# Numerical methods for boundary value problems with applications to the wave equation.

• 175 Pages
• 4.93 MB
• English
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Brunel University , Uxbridge
The Physical Object ID Numbers Contributions Brunel University. Department of Mathematics and Statistics. Pagination 175p. : Open Library OL14468074M

Analytical Solution Methods for Boundary Value Problems is an extensively revised, new English language edition of the original 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. Boundary Value Problems, Sixth Edition, is the leading text on boundary value problems and Fourier series for professionals and students in engineering, science, and mathematics who work with partial differential equations.

In this updated edition, author David Powers provides a thorough overview of solving boundary value problems involving partial differential equations by the methods of separation of variables/5(18).

Boundary Value Problems is a text material on partial differential equations that teaches solutions of boundary value problems. The book also aims to build up intuition about how the solution of a problem should behave. The text consists of seven chapters.

Chapter 1 covers the important topics of Fourier Series and Edition: 2. Vibrating Strings and the One-Dimensional Wave Equation. Steady-State Temperature and Laplace’s Equation.

Eigenvalue Methods and Boundary Value Problems. Sturm–Liouville Problems and Eigenfunction Expansions. Applications of Eigenfunction Series.

Steady Periodic Solutions and Natural Frequencies.

### Details Numerical methods for boundary value problems with applications to the wave equation. FB2

Chapter 11 Boundary Value Problems and Fourier Expansions Eigenvalue Problems for y00 + λy= 0 Fourier Series I Fourier Series II Chapter 12 Fourier Solutions of Partial Differential Equations The Heat Equation The Wave Equation Laplace’s Equationin Rectangular Coordinates Numerical methods for nonlinear waves John D.

Fenton the boundary. While the linearity of this equation is crucial in the development of theoretical solutions to wave problems, it is this mutually-dependent nature, and not the nonlinearity of the boundary conditions,Cited by: Numerical methods vary in their behavior, and the many different types of differ-ential equation problems affect the performanceof numerical methods in a variety of ways.

An excellent book for “real world” examples of solving differential equations is that of Shampine, Gladwell, and Thompson .File Size: 1MB. Some methods for the numerical solution of the regularized long-wave equation, u t + u x + uu x − u xxt = 0, are described.

A solitary wave solution of the equation is used to examine the practical accuracy and efficiency of each by: The W ave Equation in 1D The wave equation for the scalar u in the one dimensional case reads ∂2u ∂t2 =c2 ∂2u ∂x2.

() The one-dimensional wave equation () can be solved exactly by d’Alembert’s method, using a Fourier transform method, or via separation of variables.

To illus-trate the idea of the d’Alembert method, let. This lecture discusses different numerical methods to solve ordinary differential equations, such as forward Euler, backward Euler, and central difference methods.

Below are simple examples of how to implement these methods in Python, based on formulas given in the lecture note (see lecture 7 on Numerical Differentiation above).

Lectures on a unified theory of and practical procedures for the numerical solution of very general classes of linear and nonlinear two point boundary-value problems. - Hide Excerpt This monograph is an account of ten lectures I presented at the Regional Research Conference on Numerical Solution of Two-Point Boundary Value Problems.

equation () is an initial value problem with respect to time and a boundary value problem with respect to space.

Numerical methods for solving initial value problems were topic of Numerical Mathematics 2. A standard approach for solving the instationary problem consists in using a so-called one-step -scheme for discretizing the temporal Cited by: 5.

Elementary Differential Equations With Boundary Value Problems. This book explains the following topics: First Order Equations, Numerical Methods, Applications of First Order Equations1em, Linear Second Order Equations, Applcations of Linear Second Order Equations, Series Solutions of Linear Second Order Equations, Laplace Transforms, Linear Higher Order.

The heat equation, the wave equation and Laplace’s equation will form a basis for study from a numerical point of view for the same reason as they did in the analytic case. In book: Advanced Numerical and Semi‐Analytical Methods for Differential Equations (pp) The purpose is to propose an improved method for inverse boundary value problems.

This method.

### Description Numerical methods for boundary value problems with applications to the wave equation. EPUB

These Proceedings of the first Chinese Conference on Numerical Methods for Partial Differential Equations covers topics such as difference methods, finite element methods, spectral methods, splitting methods, parallel algorithm etc., their theoretical foundation and applications to engineering.

Numerical methods both for boundary value problems. The focus of the book is on fundamental methods and standard fluid dynamical problems such as tracer transport, the shallow-water equations, and the Euler equations.

The emphasis is on methods appropriate for applications in atmospheric and oceanic science, but these same methods are also well suited for the simulation of wave-like flows in.

The subject of this chapter is finite-difference methods for boundary value problems. The initial focus is 1D and after discretization of space (grid generation), introduction of stencil notation, and Taylor series expansions (including detailed derivations), the simple 2nd-order central difference finite-difference equation results.5/5(1).

Explanation. Boundary value problems are similar to initial value problems.A boundary value problem has conditions specified at the extremes ("boundaries") of the independent variable in the equation whereas an initial value problem has all of the conditions specified at the same value of the independent variable (and that value is at the lower boundary of the domain.

Boundary Value Problems, Sixth Edition, is the leading text on boundary value problems and Fourier series for professionals and students in engineering, science, and mathematics who work with partial differential equations.

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In this updated edition, author David Powers provides a thorough overview of solving boundary value problems involving partial differential equations by the methods. 54 Boundary-ValueProblems for Ordinary Differential Equations: Discrete Variable Methods with g(y(a), y(b» = 0 (b) Ifthe number of differential equations in systems (a) or (a) is n, then the number of independent conditions in (b) and (b) is n.

In practice, few problems occur naturally as Size: 1MB. () Adaptive Runge-Kutta methods for nonlinear two-point boundary value problems with mild boundary layers.

Computers & Mathematics with Applications() Alternating direction adaptive grid by: A survey of numerical methods for solving linear and nonlinear problems in one and several space variables is presented, with special attention being devoted to the parabolic wave equation, the cubic Schrödinger equation, and to fourth order parabolic equations arising in vibrating beam and plate by:   The book begins with a review of direct methods for the solution of linear systems, with an emphasis on the special features of the linear systems that arise when differential equations are solved.

including ordinary and partial differential equations and initial value and boundary value problems. The techniques presented in these chapters. Program (Finite-difference method for the wave equation): to approximate the solution of utt = c2uxx over R = {(x, t): 0 ≤ x ≤ ℓ, 0 ≤ t ≤ b with u(x,0) = f(x), ut (x,0) = g(x), for 0 ≤ x ≤ ℓ, and u(0,t) = 0, u(ℓ,t) = 0, for 0 ≤ t ≤ b.

Applications of Fourier Series. Heat Conduction and Separation of Variables. Vibrating Strings and the One-Dimensional Wave Equation. Steady-State Temperature and Laplace's Equation 9 Eigenvalues and Boundary Value Problems.

Sturm-Liouville Problems and Eigenfunction Expansions. Applications of Eigenfunction Series. 4 1 Why numerical methods. This is a nontrivial issue, and the answer depends both on the problem’s mathe-matical properties as well as on the numerical algorithms used to solve the Size: 6MB.

Preview of Problems and Methods 80 Dirichlet Problems with Symmetry 81 Spherical Harmonics and the General Dirichlet Problem 83 The Helmholtz Equation with Applications to the Poisson, Heat, and Wave Equations 86 Supplement on Legendre Functions Legendre’s Diﬀerential Equation 88File Size: 1MB.

Numerical methods: boundary value problem Consider solving the quadratic equation x2 +2bx 1 = 0, where b is a parameter. The quadratic formula yields the two solutions x = b p b2 +1.

Consider the solution with b > 0 and x > 0 (the x+ solution) given by. This paper reviews some of the recent advances in developing stable and efficient numerical algorithms for solving free boundary-value problems arising from fluid dynamics and materials science.

In particular, we will consider boundary integral methods and the level-set approach for water waves, general multi-fluid interfaces, Hele–Shaw cells Cited by:.

For a boundary value problem of a second-order ordinary differential equation, the method is stated as follows.

Let ″ = (, (), ′ ()), =, = be the boundary value problem. Let y(t; a) denote the solution of the initial value problem.The multi-term time fractional derivatives are defined in the Caputo sense, whose orders belong to the intervals [0,1], [1,2), [0,2), [0,3), [2,3) and [2,4), respectively.

Some computationally effective numerical methods are proposed for simulating the multi-term time-fractional wave-diffusion by: Introductory Differential Equations with Boundary Value Problems Third Edition. Numerical Methods for First-Order Equations.

81 Euler’s Method The Wave Equation on a Circular Plate. F. Dufﬁng’s Equation File Size: KB.