DEVIATION OF THE ERROR ESTIMATION FOR VOLTERRA INTEGRO-DIFFERENTIAL EQUATIONS

Abstract

References 11

Metrics

doi:10.1016/S0252-9602(18)30817-8

Abstract:

In this paper, we study an efficient asymptotically correction of a-posteriori error estimator for the numerical approximation of Volterra integro-differential equations by piecewise polynomial collocation method. The deviation of the error for Volterra integrodifferential equations by using the defect correction principle is defined. Also, it is shown that for m degree piecewise polynomial collocation method, our method provides O(h^m+1) as the order of the deviation of the error. The theoretical behavior is tested on examples and it is shown that the numerical results confirm the theoretical part.

Key words: Volterra integro-differential, defect correction principle, piecewise polynomial, collocation, finite difference, error analysis

Mohammad ZAREBNIA, Reza PARVAZ, Amir SABOOR BAGHERZADEH. DEVIATION OF THE ERROR ESTIMATION FOR VOLTERRA INTEGRO-DIFFERENTIAL EQUATIONS[J].Acta mathematica scientia,Series B, 2018, 38(4): 1322-1344.

Trendmd

[1]	Zhuang W. Existence and uniqueness of solutions of nonlinear integro-differential equations of Volterra type in a Banach space. Applicable Analysis, 1986, 22(2):157-166
[2]	Brunner H. On the numerical solution of nonlinear Volterra integro-differential equations. BIT Numerical Mathematics, 1973, 13(4):381-390
[3]	Zhao J, Corless R M. Compact finite difference method for integro-differential equations. Applied Mathematics and Computation, 2006, 177(1):271-288
[4]	Turkyilmazoglu M. An effective approach for numerical solutions of high-order Fredholm integro-differential equations. Applied Mathematics and Computation, 2014, 227:384-398
[5]	Turkyilmazoglu M. High-order nonlinear Volterra Fredholm-Hammerstein integro-differential equations and their effective computation. Applied Mathematics and Computation, 2014, 247:410-416
[6]	El-Gendi S E. Chebyshev solution of differential, integral and integro-differential equations. Computer Journal, 1969, 12(3):282-287
[7]	Stetter H J. The defect correction principle and discretization methods. Numerische Mathematik, 1978, 29(4):425-443
[8]	Böhmer K, Hemker P W, Stetter H J. The defect correction approach//Defect Correction Methods, Theory and Applications. Springer, 1984:1-32
[9]	Saboor Bagherzadeh A. Defect-based Error Estimation for Higher Order Differential Equations[D]. Vienna:Vienna University of Technology, 2011
[10]	Auzinger W, Koch O, Bagherzadeh A S. Error estimation based on locally weighted defect for boundary value problems in second order ordinary differential equations. BIT Numerical Mathematics, 2014, 54(4):873-900
[11]	Stoer J, Bulirsch R. Introduction to Numerical Analysis. Springer Science & Business Media, 2013

Just accepted

Online first

Just accepted

Online first

Viewed

Full text

	From	local

	Times	6
	Rate	100%

Abstract

107

Just accepted	Online first	Issue

0	0	107

	From	Others

	Times	108
	Rate	100%

Cited

Web of Science	Crossref	ScienceDirect	Search for Citations in Google Scholar >>


This page requires you have already subscribed to WoS.

Shared

Discussed

[1]	Yunxia WEI, Yanping CHEN. LEGENDRE SPECTRAL COLLOCATION METHOD FOR VOLTERRA-HAMMERSTEIN INTEGRAL EQUATION OF THE SECOND KIND [J]. Acta mathematica scientia,Series B, 2017, 37(4): 1105-1114.
[2]	Nermina MUJAKOVIC, Nelida CRNJARIC-ZIC. GLOBAL SOLUTION TO 1D MODEL OF A COMPRESSIBLE VISCOUS MICROPOLAR HEAT-CONDUCTING FLUID WITH A FREE BOUNDARY [J]. Acta mathematica scientia,Series B, 2016, 36(6): 1541-1576.
[3]	WEN Juan, HE Yin-Nian, WANG Xue-Min, HE Mi-Hui. TWO-LEVEL MULTISCALE FINITE ELEMENT METHODS FOR THE STEADY NAVIER-STOKES PROBLEM [J]. Acta mathematica scientia,Series B, 2014, 34(3): 960-972.
[4]	YANG Yin, CHEN Yan-Ping, HUANG Yun-Qing. CONVERGENCE ANALYSIS OF THE JACOBI SPECTRAL-COLLOCATION METHOD FOR FRACTIONAL INTEGRO-DIFFERENTIAL EQUATIONS [J]. Acta mathematica scientia,Series B, 2014, 34(3): 673-690.
[5]	ZHANG Fa-Yong, GUO Bai-Ling. ATTRACTORS FOR FULLY DISCRETE FINITE DIFFERENCE SCHEME OF DISSIPATIVE ZAKHAROV EQUATIONS [J]. Acta mathematica scientia,Series B, 2012, 32(6): 2431-2452.
[6]	FANG Neng-Sheng, YING Long-An. ANALYSIS OF FDTD TO UPML FOR MAXWELL EQUATIONS IN POLAR COORDINATES [J]. Acta mathematica scientia,Series B, 2011, 31(5): 2007-2032.
[7]	YUAN Yi-Rang. THE UPWIND FINITE DIFFERENCE METHOD FOR MOVING BOUNDARY VALUE PROBLEM OF COUPLED SYSTEM [J]. Acta mathematica scientia,Series B, 2011, 31(3): 857-881.
[8]	H. Kim, M. Laforest, D. Yoon. AN ADAPTIVE VERSION OF GLIMM'S SCHEME [J]. Acta mathematica scientia,Series B, 2010, 30(2): 428-446.
[9]	Ying Longan. THE ASYMPTOTIC LIMIT FOR A COMBUSTION MODEL IN REGARD TO INFINITE REACTION RATE [J]. Acta mathematica scientia,Series B, 2008, 28(4): 924-932.
[10]	Ma Ning. A FINITE ELEMENT COLLOCATION METHOD FOR TWO-PHASE INCOMPRESSIBLE#br# IMMISCIBLE PROBLEMS [J]. Acta mathematica scientia,Series B, 2007, 27(4): 875-885.
[11]	Wang Xiaofeng; Cao Guangfu. GALERKIN-PETROV METHODS OF TOEPLITZ OPERATORS ON DIRICHLET SPACE [J]. Acta mathematica scientia,Series B, 2007, 27(2): 308-316.
[12]	Yuan-Yi-Rang. FINITE DIFFERENCE FRACTIONAL STEP METHODS FOR THE TRANSIENT BEHAVIOR OF A SEMICONDUCTOR DEVICE [J]. Acta mathematica scientia,Series B, 2005, 25(3): 427-438.
[13]	YUAN Yi-Rang. THE UPWIND OPERATOR SPLITTING FINITE DIFFERENCE METHOD FOR COMPRESSIBLE TWO-PHASE DISPLACEMENT PROBLEM AND ANALYSIS [J]. Acta mathematica scientia,Series B, 2002, 22(4): 489-499.
[14]	JIANG Da-Qing, WEI Jun-Jie. EXISTENCE OF POSITIVE PERIODIC SOLUTIONS FOR VOLTERRA INTERGO-DIFFERENTIAL EQUATIONS [J]. Acta mathematica scientia,Series B, 2001, 21(4): 553-560.
[15]	DU Jin-Yuan. THE COLLOCATION METHODS FOR SINGULAR INTEGRAL EQUATIONS WITH CAUCHY KERNELS [J]. Acta mathematica scientia,Series B, 2000, 20(3): 289-302.