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A NUMERICAL METHOD in TERMS of the THIRD DERIVATIVE for a DELAY INTEGRAL EQUATION from BIOMATHEMATICS Volume 6, Issue 2, Article 42, 2005

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A NUMERICAL METHOD in TERMS of the THIRD DERIVATIVE for a DELAY INTEGRAL EQUATION from BIOMATHEMATICS Volume 6, Issue 2, Article 42, 2005 Journal of Inequalities in Pure and Applied Mathematics A NUMERICAL METHOD IN TERMS OF THE THIRD DERIVATIVE FOR A DELAY INTEGRAL EQUATION FROM BIOMATHEMATICS volume 6, issue 2, article 42, 2005. Received 02 March, 2003; ALEXANDRU BICA AND CRACIUN˘ IANCU accepted 01 March, 2005. Communicated by: S.S. Dragomir Department of Mathematics University of Oradea Str. Armatei Romane 5 Oradea, 3700, Romania. EMail: [email protected] Abstract Faculty of Mathematics and Informatics Babe¸s-Bolyai University Contents Cluj-Napoca, Str. N. Kogalniceanu no.1 Cluj-Napoca, 3400, Romania. EMail: [email protected] JJ II J I Home Page Go Back Close c 2000 Victoria University ISSN (electronic): 1443-5756 Quit 024-03 Abstract This paper presents a numerical method for approximating the positive, bounded and smooth solution of a delay integral equation which occurs in the study of the spread of epidemics. We use the cubic spline interpolation and obtain an algorithm based on a perturbed trapezoidal quadrature rule. 2000 Mathematics Subject Classification: Primary 45D05; Secondary 65R32, A Numerical Method in Terms of 92C60. the Third Derivative for a Delay Key words: Delay Integral Equation, Successive approximations, Perturbed trape- Integral Equation from zoidal quadrature rule. Biomathematics Contents Alexandru Bica and Craciun˘ Iancu 1 Introduction ......................................... 3 Title Page 2 Existence and Uniqueness of the Positive Smooth Solution .. 4 3 Approximation of the Solution on the Initial Interval ....... 7 Contents 4 Main Results ........................................ 11 JJ II References J I Go Back Close Quit Page 2 of 33 J. Ineq. Pure and Appl. Math. 6(2) Art. 42, 2005 http://jipam.vu.edu.au 1. Introduction Consider the delay integral equation: Z t (1.1) x(t) = f(s, x(s))ds. t−τ This equation is a mathematical model for the spread of certain infectious diseases with a contact rate that varies seasonally. Here x(t) is the proportion of infectives in the population at time t, τ > 0, is the lenght of time in which an A Numerical Method in Terms of the Third Derivative for a Delay individual remains infectious and f(t, x(t)) is the proportion of new infectives Integral Equation from per unit time. Biomathematics There are known results about the existence of a positive bounded solution Alexandru Bica and Craciun˘ Iancu (see [3], [8]), which is periodic in certain conditions ([9]), or about the exis- tence and uniqueness of the positive periodic solution (in [10], [11]). In [8] the Title Page author obtains the existence and uniqueness of the continuous positive bounded solution, and in [6] a numerical method for the approximation of this solution Contents is provided using the trapezoidal quadrature rule. JJ II Here, we obtain the existence and uniqueness of the positive, bounded and smooth solution and use the cubic spline of interpolation from [5] to approx- J I imate this solution on the initial interval [−τ, 0]. We suppose that on [−τ, 0], Go Back the solution Φ is known only in the discrete moments t , i = 0, n, and use the i Close values Φ(ti) = yi, i = 0, n for the spline interpolation. Afterward, we outline a numerical method and an algorithm to approximate the solution on [0,T ], with Quit T > 0 fixed, using the quadrature rule from [1] and [2]. Page 3 of 33 J. Ineq. Pure and Appl. Math. 6(2) Art. 42, 2005 http://jipam.vu.edu.au 2. Existence and Uniqueness of the Positive Smooth Solution Impose the initial condition x(t) = Φ(t), t ∈ [−τ, 0] to equation (1.1) and consider T > 0, be fixed. We obtain the initial value problem: R t f(s, x(s))ds, ∀t ∈ [0,T ] t−τ (2.1) x(t) = Φ(t), ∀t ∈ [−τ, 0]. A Numerical Method in Terms of the Third Derivative for a Delay Suppose that the following conditions are fulfilled: Integral Equation from Biomathematics 1 (i) Φ ∈ C [−τ, 0] and we have Alexandru Bica and Craciun˘ Iancu 0 Z 0 b = Φ(0) = f(s, x(s))ds with Φ (0) = f(0, b) − f(−τ, Φ(−τ)); Title Page −τ Contents (ii) b > 0 and ∃a, M, β ∈ R, M > 0, 0 < a ≤ β such that a ≤ Φ(t) ≤ β, ∀t ∈ [−τ, 0]; JJ II (iii) f ∈ C([−τ, T ] × [a, β]), f(t, x) ≥ 0, f(t, y) ≤ M, ∀t ∈ [−τ, T ], ∀x ≥ J I 0, ∀y ∈ [a, β]; Go Back (iv) Mτ ≤ β and there is an integrable function g such that f(t, x) ≥ g(t), ∀t ∈ Close [−τ, T ], ∀x ≥ a and Quit Z t Page 4 of 33 g(s)ds ≥ a, ∀t ∈ [0,T ]; t−τ J. Ineq. Pure and Appl. Math. 6(2) Art. 42, 2005 http://jipam.vu.edu.au (v) ∃L > 0 such that |f(t, x) − f(t, y)| ≤ L |x − y| , ∀t ∈ [−τ, T ], ∀x, y ∈ [a, ∞). Then, we obtain the following result: Theorem 2.1. Suppose that assumptions (i)-(v) are satisfied. Then the equation (1.1) has a unique continuous solution x(t) on [−τ, T ], with a ≤ x(t) ≤ β, ∀t ∈ [−τ, T ] such that x(t) = Φ(t) for t ∈ [−τ, 0]. Also, max {|xn(t) − x(t)| : t ∈ [0,T ]} −→ 0 A Numerical Method in Terms of the Third Derivative for a Delay as n → ∞ where x (t) = Φ(t) for t ∈ [−τ, 0] , n ∈ , x (t) = b and Integral Equation from n N 0 Biomathematics t Z Alexandru Bica and Craciun˘ Iancu xn(t) = f(s, xn−1(s))ds t−τ Title Page for t ∈ [0,T ] , n ∈ N∗. Moreover, the solution x belongs to C1[−τ, T ]. Contents Proof. From [9] under the conditions (i), (ii), (iv), (v), it follows that the exis- tence of an unique positive continuous on [−τ, T ] solution for (1.1) such that JJ II x(t) ≥ a, ∀t ∈ [−τ, T ] and x(t) = Φ(t) for t ∈ [−τ, 0]. Using Theorem 2 from J I [7] we conclude that max {|xn(t) − x(t)| : t ∈ [0,T ]} −→ 0 as n → ∞ . From (iv) we see that Go Back Z t Z t Close x(t) = f(s, x(s))ds ≤ Mds = Mτ ≤ β, ∀t ∈ [0,T ]. Quit t−τ t−τ Page 5 of 33 Because a ≤ Φ(t) ≤ β for t ∈ [−τ, 0] and x(t) = Φ(t) for t ∈ [−τ, 0] we deduce that a ≤ x(t) ≤ β, ∀t ∈ [−τ, T ], and the solution is bounded. Since J. Ineq. Pure and Appl. Math. 6(2) Art. 42, 2005 http://jipam.vu.edu.au R t x is a solution for (1.1) we have x(t) = t−τ f(s, x(s))ds, for all t ∈ [0,T ], and because f ∈ C([−τ, T ] × [a, β]) we can state that x is differentiable on [0,T ] , and x0 is continuous on [0,T ]. From condition (iii) it follows that x is differentiable with x0 continuous on [−τ, 0] (including the continuity in the point t = 0). Then x ∈ C1[−τ, T ] and the proof is complete. Corollary 2.2. In the conditions of the Theorem2.1, if f ∈ C1([−τ, T ] × [a, β]), Φ ∈ C2[−τ, 0] and ∂f ∂f A Numerical Method in Terms of Φ00(0) = (0, b) + (0, b)[f(0, b) − f(−τ, , Φ(−τ))] the Third Derivative for a Delay ∂t ∂x Integral Equation from ∂f ∂f Biomathematics − (−τ, Φ(−τ)) − (−τ, Φ(−τ)) Φ0(−τ), ∂t ∂x Alexandru Bica and Craciun˘ Iancu then x ∈ C2[−τ, T ]. Title Page Proof. Follows directly from the above theorem. Contents JJ II J I Go Back Close Quit Page 6 of 33 J. Ineq. Pure and Appl. Math. 6(2) Art. 42, 2005 http://jipam.vu.edu.au 3. Approximation of the Solution on the Initial Interval Suppose that on the interval [−τ, 0] the function Φ is known only in the points, ti, i = 0, n, which form the uniform partition (3.1) ∆n : −τ = t0 < t1 < ··· < tn−1 < tn = 0, τ where we have the values Φ(ti) = yi, i = 0, n and ti+1 − ti = h = n , A Numerical Method in Terms of ∀i = 0, n − 1. the Third Derivative for a Delay Integral Equation from Let Biomathematics 1 ∆2y ∆3y ∆4y Alexandru Bica and Craciun˘ Iancu m = ∆y − 0 + 0 − 0 , and 0 h 0 2 3 4 4 1 2 3 11∆ y0 Title Page M0 = 2 ∆ y0 − ∆ y0 + , h 12 Contents where, JJ II 2 ∆y0 = y1 − y0, ∆ y0 = y2 − 2y1 + y0, J I 3 4 ∆ y0 = y3 − 3y2 + 3y1 − y0, ∆ y0 = y4 − 4y3 + 6y2 − 4y1 + y0. Go Back Close We build a cubic spline of interpolation which corresponds to the following conditions: Quit 0 00 Page 7 of 33 (3.2) Φ(ti) = yi, i = 0, n, Φ (t0) = m0, Φ (t0) = M0. J. Ineq. Pure and Appl. Math. 6(2) Art. 42, 2005 http://jipam.vu.edu.au This spline function is s ∈ C2[−τ, 0] which, according to [5], have on the each subinterval [ti−1, ti], i = 1, n, the expression: Mi − Mi−1 3 Mi−1 2 (3.3) s(t) = · (t − ti−1) + · (t − ti−1) 6hi 2 + mi−1 · (t − ti−1) + yi−1, 0 00 for t ∈ [ti−1, ti] where yi = s(ti), mi = s (ti),Mi = s (ti), i = 0, n. For these values, according to [5], there exists the recurence relations: A Numerical Method in Terms of yi − yi−1 6mi−1 Mi = 6 · − − 2Mi−1 the Third Derivative for a Delay h2 h Integral Equation from (3.4) , i = 1, n. Biomathematics yi − yi−1 1 m = 3 · − 2m − M · h Alexandru Bica and Craciun˘ Iancu i h i−1 2 i−1 Then, from [5] Lemma 2.1, there exists a unique cubic spline function of interpolation, s, which satisfy the conditions: Title Page s(t ) = y , ∀i = 0, n Contents i i 0 (3.5) s (t0) = m0 JJ II 00 s (t0) = M0, J I and on the each subinterval of ∆n is defined by the relation (3.3).
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