A Study of Error Estimation for Second Order Fredholm Integro-Differential Equations
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DOI: 10.1007/s13226-020-0459-8
A STUDY OF ERROR ESTIMATION FOR SECOND ORDER FREDHOLM INTEGRO-DIFFERENTIAL EQUATIONS R. Parvaz∗ , M. Zarebnia∗ and A. Saboor Bagherzadeh∗∗ ∗ Department
of Mathematics, University of Mohaghegh Ardabili, 56199-11367 Ardabil, Iran
∗∗ Department
of Applied Mathematics, Faculty of Mathematics,
Ferdowsi University of Mashhad, Mashhad, Iran e-mails: [email protected]; [email protected]; [email protected] (Received 23 August 2016; after final revision 12 May 2019; accepted 26 June 2019) In this work, we study efficient asymptotically correct a posteriori error estimates for the numerical approximation of second order Fredholm integro-differential equations. We use the defect correction principle to find the deviation of the error estimation and show that collocation method by using m degree piecewise polynomial provides order O(hm+2 ) for the deviation of the error. Also, the theoretical behavior is tested on examples and it is shown that the numerical results confirm theoretical analysis. Key words : Deviation of the error; collocation; finite difference; exact finite difference; integrodifferential. 2010 Mathematics Subject Classification : 41A25; 45J05; 65N35.
1. I NTRODUCTION The second order Fredholm integro-differential (SFID) equations is defined in the following form ¡ ¢ y 00 (t) =F t, y(t), y 0 (t), z[y](t) , t ∈ I := [a, b], y(a) = r1 , with
y(b) = r2 , Z
z[y](t) := a
b
¡ ¢ K t, s, y(s) ds,
(1.1) (1.2)
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R. PARVAZ, M. ZAREBNIA AND A. S. BAGHERZADEH
where a, b, r1 , r2 ∈ (−∞, ∞) and b > a. We define W and S as follows W := {(t, y, y 0 , z) ; t ∈ I&y, y 0 , z ∈ (−∞, ∞)}, S := {(t, s, u) ; t, s ∈ I&u ∈ (−∞, ∞)}. In this paper we shall assume that F and K are uniformly continuous in W and S, respectively. We say that z[y](t) is linear if we can write z[y](t) as Z b z[y](t) = Λ(t, s)y(s)ds, a
where Λ(t, s) is sufficiently smooth in J := {(t, s) ; t, s ∈ I}. Also, we say that F is semilinear if ¡ ¢ we can write F t, y(t), y 0 (t), z[y](t) as ¡ ¢ F t, y(t), y 0 (t), z[y](t) = a1 (t)y 0 (t) + a2 (t)y(t) + a3 (t) + z[y](t). In the nonlinear case, we assume that F (t, y, y 0 , z), Ft (t, y, y 0 , z), Fy (t, y, y 0 , z), Fy0 (t, y, y 0 , z) and Fz (t, y, y 0 , z) are Lipschitz-continuous. Also when z[y](t) is nonlinear we assume that K(t, s, u) and Ku (t, s, u) are Lipschitz-continuous. We say SFID equation with boundary condition (1.2) is linear if we can write (1.1) as follows y 00 (t) = a1 (t)y 0 (t) + a2 (t)y(t) + a3 (t) + z[y](t), t ∈ [a, b],
(1.3)
where z[y](t) is linear. Also, in the linear case we assume that a1 (t), a2 (t) and a3 (t) are sufficiently smooth in I. The piecewise polynomial collocation method for integro-differential equations problem is studied in [2, 4, 9]. Also other methods for the integro-differential equations are studied in [7, 8, 10]. The defect correction principle is introduced in [3, 6]. The deviation of the error estimation for linear and nonlinear second order boundary value problem can be found in [1, 11]. The error estimati
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