Superconvergence analysis of multistep collocation method for delay Volterra integral equations

Transcription

1 Computational Methods for Differential Equations Vol. 4, No. 3, 216, pp Superconvergence analysis of multistep collocation method for delay Volterra integral equations P. Darania Department of Mathematics, Faculty of Science, Urmia University, P.O.Box 165, Urmia-Iran. Abstract In this paper, we will present a review of the multistep collocation method for Delay Volterra Integral Equations (DVIEs) from [4] and then, we study the superconvergence analysis of the multistep collocation method for DVIEs. Some numerical examples are given to confirm our theoretical results. Keywords. Delay integral equations, Multistep collocation method, Convergence and superconvergence analysis. 21 Mathematics Subject Classification. 65L5, 34K6, 34K Introduction Many special cases of the differential and integral delay equation, can be encountered in applications: absorption of light by interstellar matter [1], analytic number theory [8], collection of current by the pantograph of an electric locomotive [9], nonlinear dynamical system [5], probability theory on algebraic structures [11], Cherenkov radiation [9], continuum mechanics [1], and the theory of dielectric materials [7]. Moreover, delay equation is an interesting example of a functional equation with a variable delay: sufficiently complicated to provide a clue to the behavior of more general classes of such equations, but also simple enough to be tractable by relatively straightforward means. For more detail see [7] and the references therein. Consider the Volterra functional integral equation with delay function θ(t), t θ(t) g(t) + k 1 (t, s, y(s))ds + k 2 (t, s, y(s))ds, t [t, T ], y(t) = t t ϕ(t), t [θ(t ), t ), (1.1) where k 1 C(D R), D = {(t, s) : t s t T } and k 2 is assumed to be continuous in D θ R, D θ = {(t, s) : θ(t ) s θ(t)}, with I = [t, T ] and ϕ : [θ(t ), t ] R and g : [t, T ] R are at least continuous on their domains. The delay function θ satisfy in the following conditions: Received: 2 June 216 ; Accepted: 1 January

2 26 P. DARANIA (D1) θ(t) = t τ(t), with τ(t) τ > for t I; (D2) θ is strictly increasing on I; (D3) τ C d (I) for some d. We will refer to the function τ = τ(t) as the delay. Definition 1.1. The points {ξ µ : µ } generated by the recursion θ(ξ µ+1 ) = ξ µ+1 τ(ξ µ+1 ) = ξ µ, µ =, 1,..., ξ = t, are called the primary discontinuity points associated with the lag function θ(t) = t τ(t). Condition (D2) ensures that these discontinuity points have the (uniform) separation property ξ µ ξ µ 1 = τ(ξ µ ) τ >, for all µ 1. Theorem 1.2. Assume that the given functions in t θ(t) g(t) + k 1 (t, s)y(s)ds + k 2 (t, s)y(s)ds, t [t, T ], y(t) = t t ϕ(t), t [θ(t ), t ), (1.2) are continuous on their respective domains and that the delay function θ satisfies the above conditions (D1)-(D3). Then for any initial function ϕ C[θ(t ), t ] there exists a unique (bounded) y C(t, T ] solving the Volterra functional integral equation (1.1) on (t, T ] and coinciding with ϕ on [θ(t ), t ]. In general, this solution has a finite (jump) discontinuity at t = t : lim y(t) lim y(t) = ϕ(t ). t t + t t The solution is continuous at t = t if, and only if, the initial function is such that Proof. See [2]. g(t ) θ(t) t k 2 (t, s)ϕ(s)ds = ϕ(t ). It is well known that these equations typically have discontinuity in the solution or its derivatives at the initial point of integration domain. This discontinuity propagated along the integration interval giving rise to subsequent points, called singular points, which can not be determined priori and the solution derivatives in these points are smoothed out along the interval. Most of the known numerical methods for this type of equations are generally very sensitive to the singular points and therefore must have a process that detects these points and insert them into the mesh to guarantee the required accuracy. In [2], H. Brunner applied collocation type methods for numerical solution of equation (1.1) and discussed about their connection with iterated collocation methods. V. Horvat, [6] had considered the collocation methods for Volterra integral equations

3 CMDE Vol. 4, No. 3, 216, pp with delay arguments. P. Darania [4], had considered the nonlinear Volterra integral equations with constant delays θ(t) = t τ, τ >, of the form g(t) + (V y)(t) + (V τ y)(t), t I = [, T ], y(t) = (1.3) ϕ(t), t [ τ, ), where (V y)(t) = (V τ y)(t) = t k 1 (t, s, y(s))ds, (1.4) k 2 (t, s, y(s))ds, (1.5) and the given functions, ϕ : [ τ, ] R, g : I R, k 1 : D R R, D = {(t, s) : s t T } and k 2 : D τ R R, D τ = I [ τ, T τ] are at least continuous on their domains. In the present paper, we are briefly introduced the new families of multistep collocation method which has presented in [4, 3]. Also, we analyze the superconvergence analysis of the multistep collocation method when used to approximate smooth solutions of delay integral equations. Some numerical examples are given to confirm our theoretical results. 2. Preliminaries Here, we recall the multistep collocation method that have been introduced in [4, 3] Multistep collocation method. Let t n = nh, (n =,..., N, t N = T, h = τ r for som r N) define a uniform partition for I = [, T ], and let Ω N := { = t < t 1 < < t N = T }, σ := [t, t 1 ], σ n := (t n, t n+1 ] (1 n N 1). With a given mesh Ω N, we associate the set of its interior points, Z N := {t n : n = 1,..., N 1}. For a fixed N 1 and, for given integer m 1, the piecewise polynomial space S ( 1) m 1 (Z N ) is defined by S ( 1) m 1 (Z N) := {u : u σn = u h Π m 1, n N 1}, where Π m 1 denotes the set of (real) polynomials of a degree not exceeding m 1. Let u h = u σn, u S ( 1) m 1 (Z N ), for all t σ n, we have r 1 u h (t n +sh) = φ k (s)y n k + ψ j (s)u n,j, s [, 1], n = r,..., N 1, (2.1) k= where U n,j = u h (t n,j ), y n k = u h (t n k ) and φ k (s) = m i=1 r 1 s c i k c i i= i k r 1 s + i k + i, ψ j(s) = i= s + i c j + i m i=1 i j s c i c j c i. (2.2)

4 28 P. DARANIA Remark 2.1. In the following, we assume that the initial function ϕ(t) is such that τ g() k 2 (, s)ϕ(s)ds = ϕ(). The collocation solution u h will be determined by imposing the condition that u h satisfies the integral equation (1.3) on the finite set X N = {t n,j = t n + c j h} g(t) + (V u h )(t) + (V τ u h )(t), t [, T ], u h (t) = ϕ(t), t [ τ, ), (2.3) where {c j } m, c 1 < < c m 1, the set of collocation parameters. After some computations, the exact multistep collocation method is obtained by collocating both sides of (2.3) at the points t = t n,j for j = 1, 2,..., m and computing y n+1 = u h (t n+1 ): U n,j = D n,j, j = 1, 2,..., m, r 1 y n+1 = φ k (1)y n k + ψ j (1)U n,j, n = r, r + 1,..., N 1, k= (2.4) where D n,j = D(t n,j ) and D(t n,j ) = g(t n,j ) + (V u h )(t n,j ) + Φ(t n,j ), t n,j τ <, (V u h )(t n,j ) + (V τ u h )(t n,j ), t n,j τ, (2.5) Φ(t n,j ) = tn,j τ and for t n,j τ <, we have k 2 (t n,j, s, ϕ(s))ds, j = 1, 2,..., m, n =, 1,..., r 1, (2.6) [ (V τ u h )(t n,j ) = h k 2 (t n,j, t n r + sh, ϕ(t n r + sh))ds c j 1 ] 1 + k 2 (t n,j, t i + sh, ϕ(t i + sh))ds, i=n r+1 (2.7) and for t n,j τ, we get (V τ u h )(t n,j ) = h + [ n r 1 i= cj k 2 (t n,j, t i + sh, u h (t i + sh))ds ] k 2 (t n,j, t n r + sh, u n r (t n r + sh))ds, (2.8)

5 CMDE Vol. 4, No. 3, 216, pp and n 1 (V u h )(t n,j ) = h i= k 1 (t n,j, t i + sh, u h (t i + sh))ds (2.9) cj +h k 1 (t n,j, t n + sh, u h (t n + sh))ds. By using quadrature formulas with the weights w l and nodes d l, l = 1,..., µ 1, for integrating on [, 1], and the weights w j,l and nodes d j,l, l = 1,..., µ for integrating on [, c i ], with positive integers µ and µ 1, one can write Y n,j = D n,j, j = 1, 2,..., m, r 1 (2.1) y n+1 = φ k (1)y n k + ψ j (1)Y n,j, n = r, r + 1,..., N 1, where k= D(t n,j ) = g(t n,j ) + ( V u h )(t n,j ) + ( V τ u h )(t n,j ), (2.11) ( V n 1 µ 1 u h )(t n,j ) = h w l k 1 (t n,j, t i + d l h, P i (t i + d l h)) and for t n,j τ <, we have i= l=1 µ +h w j,l k 1 (t n,j, t n + d j,l h, P n (t n + d j,l h)), l=1 ( 1 µ 1 ( V τ u h )(t n,j ) = h w l k 2 (t n,j, t i + d l h, ϕ(t i + d l h)) i=n r+1 l=1 ) µ 1 + w j,l k 2 (t n,j, t n r + ξ j,l h, ϕ(t n r + ξ j,l h)), and for t n,j τ, we get l=1 ( n r 1 µ 1 ( V τ u h )(t n,j ) = h w l k 2 (t n,j, t i + d l h, P i (t i + d l h)) i= l=1 ) µ + w j,l k 2 (t n,j, t n r + d j,l h, P n r (t n r + d j,l h)), l=1 (2.12) (2.13) (2.14) and ξ j,l := c j + (1 c j )d l, w j,l := (1 c j )w l, j = 1,..., m, l = 1,..., µ 1. Also, the discretized multistep collocation polynomial, denoted by r 1 P n (t n + sh) = φ k (s)y n k + ψ j (s)y n,j, s [, 1], n = r,..., N 1. k=

6 21 P. DARANIA For more detail see [4]. (2.15) 2.2. Convergence. Let u h S ( 1) m 1 (Z N) denote the (exact) collocation solution to (1.3) defined by (2.4). In convergence analysis, we consider the linear test equation t g(t) + k y(t) = 1 (t, s)y(s)ds + k 2 (t, s)y(s)ds, t I, ϕ(t), t [ τ, ), where k 1 C(D) and k 2 C(D τ ). (2.16) Theorem 2.2. Let the given functions in (2.16) satisfy g C p (I), k 1 C p (D), k 2 C p (D τ ), ϕ C p ([ τ, ]), and for t [, τ] the integral Φ(t) := k 2 (t, s)ϕ(s)ds, (2.17) is known exactly. Also, suppose that the starting error is and y u h,[,tr ]= O(h p ), (2.18) ρ(a) < 1, (2.19) where p = m + r and ρ denotes the spectral radius and [ ] (r 1) 1 I A = r 1. (2.2) φ r 1 (1) φ r 2 (1),..., φ (1) Then for all sufficiently small h = τ r, ( r N) the constrained mesh collocation solution u h S ( 1) m 1 (Z N ) to (2.16), satisfies E Ch p, (2.21) where E(t) = y(t) u h (t) be the error of the exact collocation method (2.11) and C is positive constant not depending on h. This estimate holds for all collocation parameters {c j } with c 1 < < c m 1. Proof. See [4]. Remark 2.3. [3]. The starting values y 1, y 2,..., y r, needed in (2.4) and (2.1), may be obtained by using a suitable starting procedure, based on a classical one step method has uniform convergence order of p. Theorem 2.4. Let the assumptions of Theorem 2.2 hold, except that the integrals Φ(t) = k 2 (t, s)ϕ(s)ds, t = t n,j, n =, 1,..., r 1, are now approximated by quadrature formulas Φ(t), with corresponding quadrature errors E (t) := Φ(t) Φ(t), such that E (t) h q, (2.22)

7 CMDE Vol. 4, No. 3, 216, pp for some q >. Then the collocation solution u h S ( 1) m 1 (Z N ) satisfies, for all sufficiently small h >, E Ch p, (2.23) with p := min{m + r, q}, where C are finite constants not depending on h. Proof. See [4]. 3. Superconvergence In this section, we analyze the superconvergence of the multistep collocation method when used to approximate smooth solutions of delay integral equations. By the following theorem, we obtain local superconvergence in the interior points Z N. Theorem 3.1. Suppose that the hypothesis of Theorem 2.2 hold with p = 2m + r 1 and the collocation parameters c 1,..., c m are the solution of system c m = 1, r 1 1 i+1 β k ( k) i γ j (c j ) i (3.1) =, i = m + r,..., 2m + r 2, where β k = k= φ k (s)ds, γ j = Also, suppose that the delay integral Φ(t) = ψ j (s)ds. (3.2) k 2 (t, s, ϕ(s))ds, (3.3) can be evaluated analytically and if h = τ r is sufficiently small. Then max E(t n) Ch p. (3.4) 1 n N Proof. Without loss of generality, we assume that T = t N = Mτ for some M N. The collocation equation (2.16) may be written in the form u h (t) = δ(t) + g(t) + t k 1 (t, s)u h (s)ds + where the defect function δ vanishes on X N k 2 (t, s)u h (s)ds, t I, (3.5) δ(t n,j ) =. (3.6) Also, we have δ(t) = for t <. The collocation error E = y u h solves the integral equation [2, 6] where E(t) = δ(t) + F (t) = t k 1 (t, s)e(s)ds + F (t), t I, (3.7) k 2 (t, s)e(s)ds, t [τ, T ]. (3.8)

8 212 P. DARANIA For t [, τ], we have F (t) = and so the error equation (3.7) reduces to a classical Volterra equation, which is unique solution given by [2, 6] E(t) = δ(t) + R 1 (t, s)δ(s)ds, (3.9) where R 1 denotes the resolvent kernel associated with the given kernel k 1. As T = Mτ for some positive integer M, we may set ξ µ := µτ, µ =,..., M, and then for t [ξ µ, ξ µ+1 ], 1 µ M 1 the collocation error E(t) governed by (3.7) can be expressed in the form [6, 2] µ t iτ E(t) = δ(t) + R n,i (t, s)δ(s)ds, (3.1) i= where the function R n,i (t, s) depend on the given kernel functions k i (t, s), i = 1, 2. For t = t n [ξ µ + h, ξ µ+1 ], and h = τ r for some r N, we obtain E(t n ) = δ(t n ) + h µ i= n i r 1 ν= Now, let us consider the quadrature formula k= R n,i (t n, t ν + sh)δ(t ν + sh)ds. (3.11) r 1 f(s)ds β k f( k) + γ j f(c j ), for the computation of the integrals in (3.11), we have E(t n ) = [ r 1 µ n i r 1 δ(t n ) + h β k R n,i (t n, t ν k )δ(t ν k ) i= ν= k= µ n i r 1 + γ j R n,i (t n, t ν,j )δ(t ν,j ) + h E n,i,ν, i= ν= (3.12) where E n,i,ν the corresponding error terms. The hypothesis c m = 1 assures that t ν k are collocation points for each ν. Since the defect function vanish in the collocation points, we have δ(t ν,j ) = δ(t ν k ) =, hence E(t n ) = h µ i= n i r 1 ν= E n,i,ν, µ < n µ + 1 M, Mτ = T. (3.13) The quadrature errors in (3.13) can be bounded by E n,i,ν Ch 2m+r 1 with some finite constant C not depending on h, (see [3] and [6]). Finally, because Mτ = M rh = T = Nh, we obtain max E(t n) Ch 2m+r 1. 1 n N Note that, the variation of constants formula (3.9) is the key to the establishing of (global and local) superconvergence results for collocation solutions to such equations [2].

9 CMDE Vol. 4, No. 3, 216, pp Finally, we comment on the extension of the results in Theorem 3.1 to the nonlinear case. Instead of (3.7), the equation for the multistep collocation error E now has the form where E(t) = δ(t) + F (t) = t {k 1 (t, s, y(s)) k 1 (t, s, u h (s))}ds + F (t), t I, (3.14) {k 2 (t, s, y(s)) k 2 (t, s, u h (s))}ds, t [τ, T ]. (3.15) Under appropriate differentiability and boundness conditions for k 1 and k 2, we obtain k i (t, s, y(s)) k i (t, s, u h (s)) = k i y (t, s, y(s))e(s) k i 2 y 2 (t, s, z i(s))e 2 (s), where z i is between y and u h. The global convergence of V (u h ), V τ (u h ), V (u h ) and V τ (u h ) has already been establish [4]. So, we know that E 2 = O(h 2(m+r) ) for any {c j }. 4. Presentation of results In this section, two examples will be investigated to confirm our theoretical results. We consider the multistep collocation method with m = 2, r = 3, given by Examples 4.1 and 4.2 with the following choices of the collocation abscissas: (c 1, c 2 ) = (.7, 1) and from system (3.1), (c 1, c 2 ) = ( 21 38, 1). The method have respectively order p = 5 (convergence) and p = 6 (superconvergence). The observed orders of convergence are computed from the maximum errors at the grid points. The starting values have been obtained from the known exact solutions. All computations are performed by the Mathematica R software. Example 4.1. Consider the linear DVIEs as y(t) = g(t) + t (s + t + 1)y(s)ds + t 1 2 where g(t) such that the exact solution is y(t) = sin t. Example 4.2. Consider the nonlinear DVIEs as y(t) = g(t) + t 2 cos(t s)y 2 (s)ds + t 1 2 where g(t) such that the exact solution is y(t) = e t. (s + t 2 + 4)y(s)ds, t [, 1], 2 sin(t s)y 2 (s)ds, t [, 1], The maximum errors have been shown that for values of r = 3, m = 2 and for different values of N, at the grid points in the Table 1 and Table 2. The orders of convergence of the multistep collocation method for r = 3 and m = 2 are shown in Figure 1 and Figure 2 which they confirm the theoretical results of the Theorem 3.1

10 214 P. DARANIA Table 1. Maximum errors y u h Example 4.1. for r = 3 and m = 2 in c 1 =.7, c 2 = 1 c 1 = 21 38, c 2 = 1 N y u h y u h Figure 1. Orders of convergence of u h for r = 3 and m = 2 in Example Order of convergence Superconvergence Convergence N (the orders of convergence and superconvergence are p = m + r and p = 2m + r 1, respectively). 5. Conclusion We have shown that the multistep collocation method yields an efficient and very accurate numerical method for the approximation of solutions to DVIEs. Furthermore, the multistep collocation method have uniform order m + r for any choice of collocation parameters and local superconvergence order 2m + r 1 in mesh points

11 CMDE Vol. 4, No. 3, 216, pp Table 2. Maximum errors y u h Example 4.2. for r = 3 and m = 2 in c 1 =.7, c 2 = 1 c 1 = 21 38, c 2 = 1 N y u h y u h Figure 2. Orders of convergence of u h for r = 2, 3 and m = 2 in Example Order of convergence Superconvergence Convergence N of collocation parameters. Acknowledgment The author would like to express their gratitude for the referees of the paper for their valuable suggestions that improved the final form of the paper.

12 216 P. DARANIA References [1] V.A. Ambartsumian, On the fluctuation of the brightness of the Milky Way, Doklady Akad. Nauk USSR, 44 (1944), [2] H. Brunner,Collocation Methods for Volterra Integral and Related Functional Equations, Cambridge Monographs on Applied and Computational Mathematics, 15, Cambridge University Press, Cambridge, 24. [3] D. Conte and B. Paternoster, Multistep collocation methods for Volterra integral equations, Appl. Numer. Math., 59 (29), [4] P. Darania, Multistep collocation method for nonlinear delay integral equations, Sahand Communications in Mathematical Analysis (SCMA), 3 (216), [5] G.A. Derfel, Kato problem for functional-differential equations and difference Schrodinger operators, Operator Theory: Advances and Applications, 46 (199), [6] V. Horvat,On collocation methods for Volterra integral equations with delay arguments, Mathematical Communications, 4 (1999), [7] A. Iserles,On the generalized pantograph functional-differential equation, Euro. Jnl of Applied Mathematics, 4 (1993), [8] K. Mahler, On a special functional equation, J. London Math. Soc., 15 (194), [9] J.R. Ockendon and A.B. Tayler, The dynamics of a current collection system for an electric locomotive, Proc. Royal Soc. A., 322 (1971), [1] R. Rodeman, D.B. Longcope and L.F. Shampine, Response of a string to an accelerating mass, J. Appl. Mech., 98 (1976), [11] E.Y. Romanenko and A.N. Sharkovski, Asympototic solutions of differential-functional equations, In: Asymptotic Behaviour of Solutions of Differential Difference Equations, Inst. Math. Akad. Nauk Ukr SSR Press, (1978), 5 39.