Spline-galerkin methods for initial-value problems with variable coefficients

1974 ◽  
pp. 189-206 ◽  
Author(s):  
Burton Wendroff
Keyword(s):  
Initial Value Problems ◽  
Variable Coefficients ◽  
Galerkin Methods ◽  
Initial Value

scholarly journals Numerical Solutions of Odd Order Linear and Nonlinear Initial Value Problems Using a Shifted Jacobi Spectral Approximations

2012 ◽  
Vol 2012 ◽  
pp. 1-25 ◽  
Author(s):  
A. H. Bhrawy ◽  
M. A. Alghamdi
Keyword(s):  
Galerkin Method ◽  
Initial Value Problems ◽  
Numerical Solutions ◽  
Initial Conditions ◽  
Variable Coefficients ◽  
Solution Technique ◽  
Initial Value ◽  
Spectral Approximations ◽  
Base Functions ◽  
And Performance

A shifted Jacobi Galerkin method is introduced to get a direct solution technique for solving the third- and fifth-order differential equations with constant coefficients subject to initial conditions. The key to the efficiency of these algorithms is to construct appropriate base functions, which lead to systems with specially structured matrices that can be efficiently inverted. A quadrature Galerkin method is introduced for the numerical solution of these problems with variable coefficients. A new shifted Jacobi collocation method based on basis functions satisfying the initial conditions is presented for solving nonlinear initial value problems. Through several numerical examples, we evaluate the accuracy and performance of the proposed algorithms. The algorithms are easy to implement and yield very accurate results.

Numerical simulation of second-order initial-value problems using a new class of variable coefficients and two-step semi-hybrid methods

SIMULATION ◽  
2021 ◽  
pp. 003754972098082
Author(s):  
Ali Shokri ◽  
Mohammad Mehdizadeh Khalsaraei ◽  
Hamid Mohammad-Sedighi ◽  
Ali Atashyar
Keyword(s):  
Numerical Simulation ◽  
Initial Value Problems ◽  
Hybrid Methods ◽  
Variable Coefficients ◽  
Second Order ◽  
Initial Value ◽  
Hybrid Schemes ◽  
New Class ◽  
New Family ◽  
Algebraic Order

In this paper, a new family of two-step semi-hybrid schemes of the 12th algebraic order is proposed for the numerical simulation of initial-value problems of second-order ordinary differential equations. The proposed methods are symmetric and belong to the family of multiderivative methods. Each method of the new family appears to be hybrid, but after implementing the hybrid terms, it will continue as a multiderivative method. Therefore, the designation semi-hybrid is used. The consistency, convergence, stability, and periodicity of the methods are investigated and analyzed. In order to show the accuracy, consistency, convergence, and stability of the proposed family, it was tested on some well-known problems, such as the undamped Duffing’s equation. The simulation results demonstrate the efficiency and advantages of the proposed method compared to the currently available methods.

Finite time blow-up for semilinear wave equations with variable coefficients

2015 ◽  
Vol 27 (4) ◽  
Author(s):  
Wei Han
Keyword(s):  
Finite Time ◽  
Initial Value Problems ◽  
Wave Equations ◽  
Blow Up ◽  
Variable Coefficients ◽  
Initial Value ◽  
Semilinear Wave Equations

AbstractThis paper is devoted to studying initial value problems for semilinear wave equations with variable coefficients with subcritical exponents for

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