scholarly journals The stability of collocation methods for VIDEs of second order

2005 ◽  
Vol 2005 (7) ◽  
pp. 1049-1066 ◽  
Author(s):  
Edris Rawashdeh ◽  
Dave McDowell ◽  
Leela Rakesh
Keyword(s):  
Differential Equation ◽  
Jordan Block ◽  
Solution Procedure ◽  
Polynomial Spline ◽  
Second Order ◽  
Collocation Methods ◽  
Spline Functions ◽  
Stability Criteria ◽  
Spline Collocation ◽  
The Stability

Simplest results presented here are the stability criteria of collocation methods for the second-order Volterra integro differential equation (VIDE) by polynomial spline functions. The polynomial spline collocation method is stable if all eigenvalues of a matrix are in the unit disk and all eigenvalues with|λ|=1belong to a1×1Jordan block. Also many other conditions are derived depending upon the choice of collocation parameters used in the solution procedure.

scholarly journals The stability of collocation methods for higher-order Volterra integro-differential equations

2005 ◽  
Vol 2005 (19) ◽  
pp. 3075-3089 ◽  
Author(s):  
Edris Rawashdeh ◽  
Dave McDowell ◽  
Leela Rakesh
Keyword(s):  
Differential Equations ◽  
Collocation Method ◽  
Numerical Stability ◽  
Polynomial Spline ◽  
Higher Order ◽  
Collocation Methods ◽  
New Method ◽  
Spline Collocation ◽  
Spline Collocation Method ◽  
The Stability

The numerical stability of the polynomial spline collocation method for general Volterra integro-differential equation is being considered. The convergence and stability of the new method are given and the efficiency of the new method is illustrated by examples. We also proved the conjecture suggested by Danciu in 1997 on the stability of the polynomial spline collocation method for the higher-order integro-differential equations.

scholarly journals Polynomial spline collocation methods for second-order Volterra integrodifferential equations

2004 ◽  
Vol 2004 (56) ◽  
pp. 3011-3022 ◽  
Author(s):  
Edris Rawashdeh ◽  
David Mcdowell ◽  
Leela Rakesh
Keyword(s):  
Integrodifferential Equation ◽  
Polynomial Spline ◽  
Second Order ◽  
Integrodifferential Equations ◽  
Collocation Methods ◽  
Spline Collocation ◽  
Initial Value ◽  
Volterra Integrodifferential Equation

We have presented a method for the construction of an approximation to the initial-value second-order Volterra integrodifferential equation (VIDE). The polynomial spline collocation methods described here give a superconvergence to the solution of the equation.

scholarly journals Some Properties of Approximate Solutions of Linear Differential Equations

Mathematics ◽  
2019 ◽  
Vol 7 (9) ◽  
pp. 806 ◽  
Author(s):  
Ginkyu Choi Soon-Mo Choi ◽  
Jaiok Jung ◽  
Roh
Keyword(s):  
Differential Equation ◽  
Linear Differential Equation ◽  
Approximate Solutions ◽  
Small Error ◽  
Second Order ◽  
Differentiable Functions ◽  
Constant Coefficients ◽  
Classical Integral ◽  
Linear Differential ◽  
The Stability

In this paper, we will consider the Hyers-Ulam stability for the second order inhomogeneous linear differential equation, u ′ ′ ( x ) + α u ′ ( x ) + β u ( x ) = r ( x ) , with constant coefficients. More precisely, we study the properties of the approximate solutions of the above differential equation in the class of twice continuously differentiable functions with suitable conditions and compare them with the solutions of the homogeneous differential equation u ′ ′ ( x ) + α u ′ ( x ) + β u ( x ) = 0 . Several mathematicians have studied the approximate solutions of such differential equation and they obtained good results. In this paper, we use the classical integral method, via the Wronskian, to establish the stability of the second order inhomogeneous linear differential equation with constant coefficients and we will compare our result with previous ones. Specially, for any desired point c ∈ R we can have a good approximate solution near c with very small error estimation.

G-INFLATION: INFLATION WITH THE MOST GENERAL SECOND-ORDER FIELD EQUATIONS

2012 ◽  
Vol 10 ◽  
pp. 77-84
Author(s):  
TSUTOMU KOBAYASHI ◽  
MASAHIDE YAMAGUCHI ◽  
JUN'ICHI YOKOYAMA
Keyword(s):  
Power Spectra ◽  
Second Order ◽  
Stability Criteria ◽  
Cosmological Perturbations ◽  
Field Equations ◽  
Special Cases ◽  
Single Field ◽  
The Stability

In this talk, we have discussed generalized Galileons as a framework to develop the most general single-field inflation models ever, (Generalized) G-inflation, containing previous examples such as k-inflation, extended inflation, and new Higgs inflation as special cases. We have also investigated the background and perturbation evolution in this model, calculating the most general quadratic actions for tensor and scalar cosmological perturbations to give the stability criteria and the power spectra of primordial fluctuations.

scholarly journals Solving the Modified Regularized Long Wave Equations via Higher Degree B-Spline Algorithm

2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Pshtiwan Othman Mohammed ◽  
Manar A. Alqudah ◽  
Y. S. Hamed ◽  
Artion Kashuri ◽  
Khadijah M. Abualnaja
Keyword(s):  
Solitary Waves ◽  
Wave Motion ◽  
Wave Equations ◽  
The Other ◽  
Collocation Methods ◽  
Spline Collocation ◽  
B Spline ◽  
Long Wave ◽  
Current Article ◽  
The Stability

The current article considers the sextic B-spline collocation methods (SBCM1 and SBCM2) to approximate the solution of the modified regularized long wave ( MRLW ) equation. In view of this, we will study the solitary wave motion and interaction of higher (two and three) solitary waves. Also, the modified Maxwellian initial condition into solitary waves is studied. Moreover, the stability analysis of the methods has been discussed, and these will be unconditionally stable. Moreover, we have calculated the numerical conserved laws and error norms L 2 and L ∞ to demonstrate the efficiency and accuracy of the method. The numerical examples are presented to illustrate the applications of the methods and to compare the computed results with the other methods. The results show that our proposed methods are more accurate than the other methods.

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