scholarly journals Application of Splines of the Second Order Approximation to Volterra Integral Equations of the Second Kind. Applications in Systems Theory and Dynamical Systems

2021 ◽  
Vol 15 ◽  
pp. 63-71 ◽  
Author(s):  
I. G. Burova ◽  
G. O. Alcybeev
Keyword(s):  
Dynamical Systems ◽  
Systems Theory ◽  
Integral Equations ◽  
Numerical Experiments ◽  
Order Approximation ◽  
Second Order ◽  
Volterra Integral Equations ◽  
Order Of Approximation ◽  
Polynomial Splines ◽  
Second Order Of Approximation

This paper discusses the application of local interpolation splines of the second order of approximation for the numerical solution of Volterra integral equations of the second kind. Computational schemes based on the use of polynomial and non-polynomial splines are constructed. The advantages of the proposed method include the ability to calculate the integrals which are present in the computational methods. The application of splines to the solution of nonlinear Volterra integral equations is also discussed. The results of numerical experiments are presented

scholarly journals Splines of the Fourth Order Approximation and the Volterra Integral Equations

2021 ◽  
Vol 20 ◽  
pp. 475-488
Author(s):  
I.G. Burova ◽  
A.G. Doronina ◽  
D.E. Zhilin
Keyword(s):  
Integral Equation ◽  
Numerical Solution ◽  
Integral Equations ◽  
Fourth Order ◽  
Volterra Integral Equation ◽  
Numerical Experiments ◽  
Order Approximation ◽  
Order Of Approximation ◽  
Polynomial Splines ◽  
Local Splines

This paper is a continuation of a series of papers devoted to the numerical solution of integral equations using local interpolation splines. The main focus is given to the use of splines of the fourth order of approximation. The features of the application of the polynomial and non-polynomial splines of the fourth order of approximation to the solution of Volterra integral equation of the second kind are discussed. In addition to local splines of the Lagrangian type, integro-differential splines are also used to construct computational schemes. The comparison of the solutions obtained by different methods is carried out. The results of the numerical experiments are presented.

scholarly journals Superconvergence of interpolated collocation solutions for weakly singular Volterra integral equations of the second kind

2021 ◽  
Vol 40 (3) ◽  
Author(s):  
Qiumei Huang ◽  
Min Wang
Keyword(s):  
Integral Equations ◽  
Numerical Experiments ◽  
Volterra Integral Equations ◽  
Least Square ◽  
Convergence Order ◽  
Collocation Points ◽  
Weakly Singular ◽  
Interpolation Postprocessing ◽  
Collocation Solution

AbstractIn this paper, we discuss the superconvergence of the “interpolated” collocation solutions for weakly singular Volterra integral equations of the second kind. Based on the collocation solution $$u_h$$ u h , two different interpolation postprocessing approximations of higher accuracy: $$I_{2h}^{2m-1}u_h$$ I 2 h 2 m - 1 u h based on the collocation points and $$I_{2h}^{m}u_h$$ I 2 h m u h based on the least square scheme are constructed, whose convergence order are the same as that of the iterated collocation solution. Such interpolation postprocessing methods are much simpler in computation. We further apply this interpolation postprocessing technique to hybrid collocation solutions and similar results are obtained. Numerical experiments are shown to demonstrate the efficiency of the interpolation postprocessing methods.

MONOTONE AND CONSERVATIVE DIFFERENCE SCHEMES FOR ELLIPTIC EQUATIONS WITH MIXED DERIVATIVES

2005 ◽  
Vol 9 (2) ◽  
pp. 169-178 ◽  
Author(s):  
I.V. Rybak
Keyword(s):  
Maximum Principle ◽  
Elliptic Equations ◽  
Second Order ◽  
Difference Schemes ◽  
Stability And Convergence ◽  
Order Of Approximation ◽  
Conservative Difference Schemes ◽  
Mixed Derivatives ◽  
Conservative Difference ◽  
Second Order Of Approximation

In the paper elliptic equations with alternating‐sign coefficients at mixed derivatives are considered. For such equations new difference schemes of the second order of approximation are developed. The proposed schemes are conservative and monotone. The constructed algorithms satisfy the grid maximum principle not only for coefficients of constant signs but also for alternating‐sign coefficients at mixed derivatives. The a prioriestimates of stability and convergence in the grid norm C are obtained.

scholarly journals CONJUGATION PROBLEM ABOUT JOINTLY SEPARATE FLOW OF VISCOELASTIC AND VISCOUS FLUIDS IN THE PLANE DUCT

1999 ◽  
Vol 4 (1) ◽  
pp. 114-123
Author(s):  
V. I. Korzyuk ◽  
S. V. Lemeshevsky ◽  
P. P. Matus ◽  
V. N. Shalima
Keyword(s):  
Difference Scheme ◽  
Separate Flow ◽  
Second Order ◽  
Viscous Fluids ◽  
Spatial Variable ◽  
Order Of Approximation ◽  
Conjugation Problem ◽  
Implicit Difference Scheme ◽  
Consistency Conditions ◽  
Second Order Of Approximation

Conjugation problem about jointly separate flow of viscoelastic and viscous fluids in the plane duct is considered. The results concerning of solvability of this problem are presented. The implicit difference scheme for the conjugation problem is constructed. The consistency conditions are approximated with the second order of approximation with respect to spatial variable. The convergence of the suggested difference scheme is investigated by method of energy inequalities.

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