Error control Gaussian collocation software for boundary value ODEs and 1D time-dependent PDEs

2019 ◽  
Vol 81 (4) ◽  
pp. 1505-1519
Author(s):  
Mark Adams ◽  
Connor Tannahill ◽  
Paul Muir
Keyword(s):  
Error Control ◽  
Boundary Value ◽  
Time Dependent

Beam Vibrations With Time-Dependent Boundary Conditions

1950 ◽  
Vol 17 (4) ◽  
pp. 377-380
Author(s):  
R. D. Mindlin ◽  
L. E. Goodman
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Separation Of Variables ◽  
Boundary Value ◽  
Time Dependent ◽  
Partial Differential ◽  
Linear Partial Differential Equations ◽  
Vibration Problems

Abstract A procedure is described for extending the method of separation of variables to the solution of beam-vibration problems with time-dependent boundary conditions. The procedure is applicable to a wide variety of time-dependent boundary-value problems in systems governed by linear partial differential equations.

On an initial boundary value problem involving Beltrami-Moses fields in electromagnetic theory

1993 ◽  
Vol 344 (1671) ◽  
pp. 235-248 ◽  
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Electromagnetic Theory ◽  
Time Dependent ◽  
Covariant Formulation ◽  
Beltrami Fields ◽  
Initial Boundary Value ◽  
Infinite Media

The so-called Harmuth ansatz consists of including autonomous magnetic sources in the time-dependent Maxwell postulates. The Beltrami fields are eigenfunctions of the curl operator, and have been used by Moses for propagation in infinite media. These developments are of relatively recent provenances in electromagnetic theory. We discuss an initial-boundary value problem (IBVP) within the framework of a manifestly covariant electromagnetic formalism by using the Harmuth ansatz. We also show how a covariant formulation of the Beltrami-Moses fields may be used for solving electromagnetic IBVPS.

Wavelet Based Solution to Time-Dependent Higher Order Non-Linear Two-Point Initial Boundary Value Problems With Non-Periodic Boundary Conditions: KdV, Boussinesq Equations

2003 ◽  
Author(s):  
H. N. Narang ◽  
Rajiv K. Nekkanti
Keyword(s):  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Periodic Boundary ◽  
Boundary Value ◽  
Periodic Boundary Conditions ◽  
Initial Boundary ◽  
Time Dependent ◽  
Initial Boundary Value Problems ◽  
Non Linear ◽  
Initial Boundary Value

The Wavelet solution for boundary-value problems is relatively new and has been mainly restricted to the solutions in data compression, image processing and recently to the solution of differential equations with periodic boundary conditions. This paper is concerned with the wavelet-based Galerkin’s solution to time dependent higher order non-linear two-point initial-boundary-value problems with non-periodic boundary conditions. The wavelet method can offer several advantages in solving the initial-boundary-value problems than the traditional methods such as Fourier series, Finite Differences and Finite Elements by reducing the computational time near singularities because of its multi-resolution character. In order to demonstrate the wavelet, we extend our prior research of solution to parabolic equations and problems with non-linear boundary conditions to non-linear problems involving KdV Equation and Boussinesq Equation. The results of the wavelet solutions are examined and they are found to compare favorably to the known solution. This paper on the whole indicates that the wavelet technique is a strong contender for solving partial differential equations with non-periodic conditions.

scholarly journals A fast nonlocal algorithm for solving Neumann-Dirichlet boundary value problems with error control

2016 ◽  
pp. 500-522
Author(s):  
Б.В. Семисалов
Keyword(s):  
Boundary Value Problems ◽  
Error Control ◽  
Dirichlet Boundary ◽  
Differential Operators ◽  
Numerical Solutions ◽  
Boundary Value ◽  
Computer Time ◽  
Elliptic Type ◽  
New Approach ◽  
High Convergence Rate

Предложен метод численного решения краевых задач Неймана-Дирихле для уравнений эллиптического типа, обеспечивающий достижение требуемой точности с низким расходом памяти и машинного времени. Метод адаптирует свойства наилучших полиномиальных приближений для построения быстросходящихся алгоритмов без насыщения на основе нелокальных чебышевских приближений. Предложен новый подход к аппроксимации дифференциальных операторов и решению полученных задач линейной алгебры. Даны оценки погрешности численного решения. Обоснован и установлен экспериментально высокий порядок сходимости предложенного метода в задачах с $C^r$-гладкими и $C^{\infty}$-гладкими решениями. Получены выражения элементов массивов, аппроксимирующих операторы производных в задачах с различными граничными условиями. Эти выражения позволят читателю быстро реализовать метод с нуля. A method for searching numerical solutions to Neumann-Dirichlet boundary value problems for differential equations of elliptic type is proposed. This method allows reaching a desired accuracy with low consumption of memory and computer time. The method adapts the properties of best polynomial approximations for construction of algorithms without saturation on the basis of nonlocal Chebyshev approximations. A new approach to the approximation of differential operators and to solving the resulting problems of linear algebra is also proposed. Estimates of numerical errors are given. A high convergence rate of the proposed method is substantiated theoretically and is shown numerically in the case of problems with $C^r$-smooth and $C^{\infty}$-smooth solutions. Expressions for arrays approximating the differential operators in problems with various types of boundary conditions are obtained. These expressions allow the reader to quickly implement the method from scratch.

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