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.
ALEXANDRU BICA AND CRĂCIUN IANCU
Department of Mathematics
University of Oradea
Str. Armatei Romane 5
Oradea, 3700, Romania.
EMail: smbica@yahoo.com
Faculty of Mathematics and Informatics
Babeş-Bolyai University
Cluj-Napoca, Str. N. Kogalniceanu no.1
Cluj-Napoca, 3400, Romania.
EMail: ciancu@math.ubbcluj.ro
Received 02 March, 2003;
accepted 01 March, 2005.
Communicated by: S.S. Dragomir
Abstract
Contents
JJ
J
II
I
Home Page
Go Back
Close
c
2000
Victoria University
ISSN (electronic): 1443-5756
024-03
Quit
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,
92C60.
Key words: Delay Integral Equation, Successive approximations, Perturbed trapezoidal quadrature rule.
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2
Existence and Uniqueness of the Positive Smooth Solution . . 4
3
Approximation of the Solution on the Initial Interval . . . . . . . 7
4
Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
References
A Numerical Method in Terms of
the Third Derivative for a Delay
Integral Equation from
Biomathematics
Alexandru Bica and Crăciun Iancu
Title Page
Contents
JJ
J
II
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
individual remains infectious and f (t, x(t)) is the proportion of new infectives
per unit time.
There are known results about the existence of a positive bounded solution
(see [3], [8]), which is periodic in certain conditions ([9]), or about the existence and uniqueness of the positive periodic solution (in [10], [11]). In [8] the
author obtains the existence and uniqueness of the continuous positive bounded
solution, and in [6] a numerical method for the approximation of this solution
is provided using the trapezoidal quadrature rule.
Here, we obtain the existence and uniqueness of the positive, bounded and
smooth solution and use the cubic spline of interpolation from [5] to approximate this solution on the initial interval [−τ, 0]. We suppose that on [−τ, 0],
the solution Φ is known only in the discrete moments ti , i = 0, n, and use the
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
T > 0 fixed, using the quadrature rule from [1] and [2].
A Numerical Method in Terms of
the Third Derivative for a Delay
Integral Equation from
Biomathematics
Alexandru Bica and Crăciun Iancu
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
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:
 Rt
 t−τ f (s, x(s))ds, ∀t ∈ [0, T ]
(2.1)
x(t) =

Φ(t),
∀t ∈ [−τ, 0].
Suppose that the following conditions are fulfilled:
(i) Φ ∈ C 1 [−τ, 0] and we have
Z 0
0
b = Φ(0) =
f (s, x(s))ds with Φ (0) = f (0, b) − f (−τ, Φ(−τ ));
A Numerical Method in Terms of
the Third Derivative for a Delay
Integral Equation from
Biomathematics
Alexandru Bica and Crăciun Iancu
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];
(iii) f ∈ C([−τ, T ] × [a, β]), f (t, x) ≥ 0, f (t, y) ≤ M, ∀t ∈ [−τ, T ], ∀x ≥
0, ∀y ∈ [a, β];
(iv) M τ ≤ β and there is an integrable function g such that f (t, x) ≥ g(t), ∀t ∈
[−τ, T ], ∀x ≥ a and
Z t
g(s)ds ≥ a, ∀t ∈ [0, T ];
JJ
J
II
I
Go Back
Close
Quit
Page 4 of 33
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
as n → ∞ where xn (t) = Φ(t) for t ∈ [−τ, 0] , n ∈ N , x0 (t) = b and
Z t
xn (t) =
f (s, xn−1 (s))ds
A Numerical Method in Terms of
the Third Derivative for a Delay
Integral Equation from
Biomathematics
Alexandru Bica and Crăciun Iancu
t−τ
for t ∈ [0, T ] , n ∈ N∗ . Moreover, the solution x belongs to C 1 [−τ, T ].
Proof. From [9] under the conditions (i), (ii), (iv), (v), it follows that the existence of an unique positive continuous on [−τ, T ] solution for (1.1) such that
x(t) ≥ a, ∀t ∈ [−τ, T ] and x(t) = Φ(t) for t ∈ [−τ, 0]. Using Theorem 2 from
[7] we conclude that max {|xn (t) − x(t)| : t ∈ [0, T ]} −→ 0 as n → ∞ . From
(iv) we see that
Z t
Z t
x(t) =
f (s, x(s))ds ≤
M ds = M τ ≤ β, ∀t ∈ [0, T ].
t−τ
t−τ
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
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 5 of 33
J. Ineq. Pure and Appl. Math. 6(2) Art. 42, 2005
http://jipam.vu.edu.au
Rt
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 ∈ C 1 [−τ, T ] and the proof is complete.
Corollary 2.2. In the conditions of the Theorem2.1, if f ∈ C 1 ([−τ, T ] ×
[a, β]), Φ ∈ C 2 [−τ, 0] and
∂f
∂f
Φ (0) =
(0, b) +
(0, b) [f (0, b) − f (−τ, , Φ(−τ ))]
∂t
∂x
∂f
∂f
−
(−τ, Φ(−τ )) −
(−τ, Φ(−τ )) Φ0 (−τ ),
∂t
∂x
00
then x ∈ C 2 [−τ, T ].
Proof. Follows directly from the above theorem.
A Numerical Method in Terms of
the Third Derivative for a Delay
Integral Equation from
Biomathematics
Alexandru Bica and Crăciun Iancu
Title Page
Contents
JJ
J
II
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,
where we have the values Φ(ti ) = yi , i = 0, n and ti+1 − ti = h =
∀i = 0, n − 1.
Let
∆2 y0 ∆3 y0 ∆4 y0
1
∆y0 −
+
−
, and
m0 =
h
2
3
4
11∆4 y0
1
2
3
,
M0 = 2 ∆ y0 − ∆ y 0 +
h
12
τ
,
n
where,
∆y0 = y1 − y0 ,
∆3 y0 = y3 − 3y2 + 3y1 − y0 ,
∆2 y0 = y2 − 2y1 + y0 ,
∆4 y0 = y4 − 4y3 + 6y2 − 4y1 + y0 .
We build a cubic spline of interpolation which corresponds to the following
conditions:
(3.2)
Φ(ti ) = yi , i = 0, n,
0
Φ (t0 ) = m0 ,
00
Φ (t0 ) = M0 .
A Numerical Method in Terms of
the Third Derivative for a Delay
Integral Equation from
Biomathematics
Alexandru Bica and Crăciun Iancu
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 7 of 33
J. Ineq. Pure and Appl. Math. 6(2) Art. 42, 2005
http://jipam.vu.edu.au
This spline function is s ∈ C 2 [−τ, 0] which, according to [5], have on the
each subinterval [ti−1 , ti ], i = 1, n, the expression:
(3.3) s(t) =
Mi − Mi−1
Mi−1
· (t − ti−1 )3 +
· (t − ti−1 )2
6hi
2
+ mi−1 · (t − ti−1 ) + yi−1 ,
for t ∈ [ti−1 , ti ] where yi = s(ti ), mi = s0 (ti ), Mi = s00 (ti ), i = 0, n. For
these values, according to [5], there exists the recurence relations:

yi − yi−1 6mi−1


−
− 2Mi−1

 Mi = 6 ·
h2
h
, i = 1, n.
(3.4)


1
y
−
y
i−1

 mi = 3 · i
− 2mi−1 − Mi−1 · h
h
2
Then, from [5] Lemma 2.1, there exists a unique cubic spline function of
interpolation, s, which satisfy the conditions:


s(ti ) = yi , ∀i = 0, n


s0 (t0 ) = m0
(3.5)


 s00 (t ) = M ,
0
0
and on the each subinterval of ∆n is defined by the relation (3.3).
Between the values of s0 and s00 on the knots there exists the relations:

yi − yi−1 − mi−1 · h


M
+
2M
=
6
·

i
i−1

h2
(3.6)
, i = 1, n.


2
(m
−
m
)
i
i−1

 Mi + Mi−1 =
h
A Numerical Method in Terms of
the Third Derivative for a Delay
Integral Equation from
Biomathematics
Alexandru Bica and Crăciun Iancu
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 8 of 33
J. Ineq. Pure and Appl. Math. 6(2) Art. 42, 2005
http://jipam.vu.edu.au
This spline function s approximates the solution Φ on the interval [−τ, 0] and
we have s(ti ) = Φ(ti ) = yi , ∀i = 0, n.
Remark 1. Since Φ ∈ C 2 [−τ, 0], we can estimate the error of this approximation, kΦ − sk, where k·k is the Čebyšev norm on the set of continuous functions
on an compact interval of the real axis: kuk = max{|u(t)| : t lies in an compact interval}, for any continuous function u on this interval. If we know the
R
21
0
00
00
2
value kΦ k2 = −τ [Φ (t)] dt , then for each t ∈ [−τ, 0] we have
00
3
2
|Φ(t) − s(t)| ≤ kΦ − sk ≤ kΦ k2 · h ≤
√
2
00
3
2
τ kΦ k · h ,
according to [4] page 127. Else, if we know only the values yi = Φ(ti ), i = 0, n
then |Φ(t) − s(t)| ≤ kΦ − sk , ∀t ∈ [−τ, 0], and for kΦ − sk we use the error
estimation from [7], since Φ ∈ C 2 [−τ, 0] and then Φ is a Lipschitzian function
on [−τ, 0].
In [7], it has been shown that
kΦ − sk ≤ max{ks − F1 k , ks − F2 k}
where
A Numerical Method in Terms of
the Third Derivative for a Delay
Integral Equation from
Biomathematics
Alexandru Bica and Crăciun Iancu
Title Page
Contents
JJ
J
II
I
Go Back
ks − F1 k = max{ai : i = 1, n}
ks − F2 k = max{bi : i = 1, n}
Close
Quit
Page 9 of 33
J. Ineq. Pure and Appl. Math. 6(2) Art. 42, 2005
http://jipam.vu.edu.au
and
ai = (s − F1 ) |[ti−1 ,ti ] ,
bi = (s − F2 ) |[t ,t ] ,
i−1 i
F1 (t) = sup{Φ(tk ) − kΦkL · |t − tk | : k = 0, n},
F2 (t) = inf{Φ(tk ) + kΦkL · |t − tk | : k = 0, n},
with
kΦ |∆n kL = max{| [ti−1 , ti ; Φ] |: i = 1, n}
if [ti−1 , ti ; Φ] = [Φ(ti ) − Φ(ti−1 )]/(ti − ti−1 ) is the divided difference of the
function Φ on the knots ti−1 , ti .
A Numerical Method in Terms of
the Third Derivative for a Delay
Integral Equation from
Biomathematics
Alexandru Bica and Crăciun Iancu
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 10 of 33
J. Ineq. Pure and Appl. Math. 6(2) Art. 42, 2005
http://jipam.vu.edu.au
4.
Main Results
From Theorem 2.1 follows that the equation (1.1) has a unique positive, bounded
and smooth solution on [−τ, T ]. Let ϕ be this solution, which, by virtue of Theorem 2.1, can be obtained by successive approximations method on [0, T ].
So, we have :

ϕm (t) = Φ(t) , for t ∈ [−τ, 0], ∀m ∈ N and



R0



ϕ
(t)
=
Φ(0)
=
b
=
f (s, Φ(s))ds,
0

−τ


R
Rt

t


 ϕ1 (t) = t−τ f (s, ϕ0 (s))ds = t−τ f (s, b)ds,
(4.1)
, for t ∈ [0, T ].

···




Rt



ϕ
(t)
=
f (s, ϕm−1 (s))ds,
m

t−τ




···
To obtain the sequence of successive approximations (4.1 ) we compute the
integrals using a quadrature rule.
We assume that there is l ∈ N∗ such that T = lτ. On each interval [iτ, (i +
1)τ ], i = 0, l − 1 we establish an equidistant partition. Then on the interval
[−τ, T ] we have q = l · n + n + 1 knots which realize the division:
(4.2) −τ = t0 < t1 < · · · < tn−1 < 0 = tn < tn+1 < · · · < tq−1 < tq = T,
having ti+1 − ti = h, ∀i = n, q − 1. We can see that tj − τ = tj−n , ∀j = n, q.
A Numerical Method in Terms of
the Third Derivative for a Delay
Integral Equation from
Biomathematics
Alexandru Bica and Crăciun Iancu
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 11 of 33
J. Ineq. Pure and Appl. Math. 6(2) Art. 42, 2005
http://jipam.vu.edu.au
In the aim to compute the integrals from (4.1) we use the following quadrature rule of N.S. Barnett and S.S. Dragomir in [1]:
Z
(4.3)
a
b
n−1
(b − a) X
F (t)dt =
[F (ti ) + F (ti+1 )]
2n i=0
−
where
(b − a)2 0
[F (a) − F 0 (b)] + Rn (F ),
12n2
b−a
ti = a + i ·
,
n
A Numerical Method in Terms of
the Third Derivative for a Delay
Integral Equation from
Biomathematics
i = 0, n,
and
(b − a)4
· kF 000 k , if F ∈ C 3 [a, b].
160n3
Here, we consider the function F defined by F (t) = f (t, x(t)), for t ∈ [−τ, T ].
Since the solution of (2.1) verify the relation,
|Rn (F )| ≤
x0 (t) = f (t, x(t)) − f (t − τ, x(t − τ )),
∀t ∈ [0, T ],
we point out the following connection between the smoothness of x and f.
Remark 2. In the conditions of Corollary 2.2, if Φ ∈ C 3 [−τ, 0] , f ∈ C 2 ([−τ, T ]×
[a, β]), and
d ∂f
∂f
000
Φ (0) = lim
(t, ϕ(t)) +
(t, ϕ(t))ϕ0 (t)
t>0,t→0 dt
∂t
∂x
∂f
∂f
0
(t − τ, ϕ(t − τ )) −
(t − τ, ϕ(t − τ )) ϕ (t − τ ) ,
−
∂t
∂x
Alexandru Bica and Crăciun Iancu
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 12 of 33
J. Ineq. Pure and Appl. Math. 6(2) Art. 42, 2005
http://jipam.vu.edu.au
then x ∈ C 3 [−τ, T ].
If x ∈ C 3 [−τ, T ] and f ∈ C 3 ([−τ, T ] × [a, β]) then F ∈ C 3 [−τ, T ] and
F (t) = [f (t, x(t))]000
t , ∀t ∈ [−τ, T ]. For this function F we apply the quadrature rule (4.3) and obtain the approximative values of the solution ϕ at the points
tk , k = n + 1, q, as in the following formula:
Z tk
(4.4) ϕm (tk ) =
f (s, ϕm−1 (s))ds
000
tk −τ
n−1
X
τ
=
2n
[f (tk+i − τ, ϕm−1 (tk+i − τ ))
i=0
−
+ f (tk+i+1 − τ, ϕm−1 (tk+i+1 − τ ))]
∂f
(tk , ϕm−1 (tk ))
∂t
τ2
12n2
∂f
+
(tk , ϕm−1 (tk )) · ϕ0m−1 (tk )
∂x
∂f
−
(tk − τ, ϕm−1 (tk − τ ))
∂t
∂f
(n)
0
− (tk − τ, ϕm−1 (tk − τ )) · ϕm−1 (tk − τ ) + rm,k (f ),
∂x
A Numerical Method in Terms of
the Third Derivative for a Delay
Integral Equation from
Biomathematics
Alexandru Bica and Crăciun Iancu
Title Page
Contents
JJ
J
II
I
Go Back
Close
∀ m ∈ N∗ and ∀k = n + 1, q, where,
ϕ0m−1 (t)
= f (t, ϕm−2 (t))−f (t−τ, ϕm−2 (t−τ )), ∀t ∈ [0, T ], ∀m ∈ N, m ≥ 2,
and ϕ00 (t) = 0, ϕ01 (t) = f (t, b) − f (t − τ, b), ∀t ∈ [0, T ].
Quit
Page 13 of 33
J. Ineq. Pure and Appl. Math. 6(2) Art. 42, 2005
http://jipam.vu.edu.au
To estimate the remainder we need to obtain an upper bound for the third
derivative [f (t, x(t))]000
t . After elementary calculus we have for t ∈ [−τ, T ]:
[f (t, x(t))]000
t =
∂3f
∂3f
(t,
x(t))
+
3
(t, x(t)) · x0 (t)
∂t3
∂t2 ∂x
∂3f
∂3f
2
3
0
+3
(t,
x(t))
·
[x
(t)]
+
(t, x(t)) [x0 (t)]
∂t∂x2
∂x3
∂2f
∂2f
+3
(t, x(t))x00 (t) + 3 2 (t, x(t))x0 (t)x00 (t)
∂t∂x
∂x
∂f
+
(t, x(t))x000 (t).
∂x
We denote
M0 = M = max {|f (t, x)| : t ∈ [−τ, T ], x ∈ [a, β]}
|α|
∂αf ∂ f (t, x) = max : t ∈ [−τ, T ], x ∈ [a, β], α1 + α2 = |α|
∂tα1 ∂xα2 ∂tα1 ∂xα2 ∂f ∂f M1 = max ∂t , ∂x ,
2 2 2 ∂ f ∂ f ∂ f M2 = max ∂t2 , ∂t∂x , ∂x2 ,
3 3 3 3 ∂ f ∂ f ∂ f ∂ f M3 = max ∂t3 , ∂t2 ∂x , ∂t∂x2 , ∂x3 .
A Numerical Method in Terms of
the Third Derivative for a Delay
Integral Equation from
Biomathematics
Alexandru Bica and Crăciun Iancu
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 14 of 33
J. Ineq. Pure and Appl. Math. 6(2) Art. 42, 2005
http://jipam.vu.edu.au
Consequently, we obtain the estimations:
[f (t, x(t)]000 ≤ M3 (1 + 6M0 + 12M02 + 8M03 )
t
+ M2 (8M1 + 32M0 M1 + 32M02 M1 )
+ 4M13 + 8M0 M3 = M 000 ,
∀t ∈ [−τ, T ],
and
τ 4 M 000
(n)
r
(f
)
,
∀m ∈ N∗ , ∀k = n + 1, q.
m,k ≤
160n3
Then, to compute the integrals (4.1), we can use the following algorithm:
A Numerical Method in Terms of
the Third Derivative for a Delay
Integral Equation from
Biomathematics
Alexandru Bica and Crăciun Iancu
ϕ1 (tk )
n−1
=
τ X
[f (tk+i − τ, ϕ0 (tk+i − τ ))
2n i=0
+f (tk+i+1 − τ, ϕ0 (tk+i+1 − τ ))]
τ
∂f
∂f
−
(tk , ϕ0 (tk )) +
(tk , ϕ0 (tk )) · ϕ00 (tk )
2
12n ∂t
∂x
∂f
∂f
0
−
(tk − τ, ϕ0 (tk − τ )) −
(tk − τ, ϕ0 (tk − τ )) · ϕ0 (tk − τ )
∂t
∂x
2
=:
(n)
+ r1,k (f )
(n)
ϕ
f1 (tk ) + r1,k (f ),
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 15 of 33
J. Ineq. Pure and Appl. Math. 6(2) Art. 42, 2005
http://jipam.vu.edu.au
ϕ2 (tk )
n−1
=
(4.5)
τ X
[f (tk+i − τ, ϕ1 (tk+i − τ ))
2n i=0
+f (tk+i+1 − τ, ϕ1 (tk+i+1 − τ ))]
τ
∂f
∂f
−
(tk , ϕ1 (tk )) +
(tk , ϕ1 (tk )) · ϕ01 (tk )
2
12n ∂t
∂x
∂f
∂f
0
− (tk − τ, ϕ1 (tk − τ )) −
(tk − τ, ϕ1 (tk − τ )) · ϕ1 (tk − τ )
∂t
∂x
2
(n)
+ r1,k (f )
n−1
τ Xh
(n)
=
f (tk+i − τ, ϕ
f1 (tk+i − τ ) + r1,k+i−n (f ))
2n i=0
i
(n)
+ f (tk+i+1 − τ, ϕ
f1 (tk+i+1 − τ ) + r1,k+i+1−n (f ))
τ2
∂f
(n)
−
·
(tk , ϕ
f1 (tk ) + r1,k (f ))
12n2
∂t
∂f
(n)
+
(tk , ϕ
f1 (tk ) + r1,k (f )) · (f (tk , b) − f (tk − τ, b))
∂x
∂f
(n)
−
(tk − τ, ϕ
f1 (tk − τ ) + r1,k−n (f ))
∂t
∂f
(n)
− (tk − τ, ϕ
f1 (tk − τ ) + r1,k−n (f ))(f (tk − τ, b) − f (tk − 2τ, b))
∂x
+
(n)
r2,k (f )
A Numerical Method in Terms of
the Third Derivative for a Delay
Integral Equation from
Biomathematics
Alexandru Bica and Crăciun Iancu
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 16 of 33
J. Ineq. Pure and Appl. Math. 6(2) Art. 42, 2005
http://jipam.vu.edu.au
n−1
τ X
=
[f (tk+i − −τ, ϕ
f1 (tk+i − τ )) + f (tk+i+1 − τ, ϕ
f1 (tk+i+1 − τ ))]
2n i=0
∂f
∂f
τ2
−
·
(tk , ϕ
f1 (tk )) +
(tk , ϕ
f1 (tk )) · (f (tk , b) − f (tk − τ, b))
2
12n
∂t
∂x
∂f
−
(tk − τ, ϕ
f1 (tk − τ ))
∂t
∂f
^
(n)
− (tk − τ, ϕ
f1 (tk − τ )) · (f (tk − τ, b) − f (tk − 2τ, b)) + r2,k (f )
∂x
^
(n)
^
=: ϕ
∀k = n + 1, q.
2 (tk ) + r2,k (f ),
We have the remainder estimation:
4
000
g
τ 2 M2 (1 + 2M )
(n)
r (f ) ≤ τ M
1 + τL +
,
2,k 160n3
6n2
Alexandru Bica and Crăciun Iancu
∀k = n + 1, q.
By induction, for m ≥ 3 we obtain:
(4.6)
A Numerical Method in Terms of
the Third Derivative for a Delay
Integral Equation from
Biomathematics
ϕm (tk )
n−1 τ X
(n) ^
=
f (tk+i − τ, ϕ]
m−1 (tk+i − τ ) + rm−1,k+i−n (f ))
2n i=0
+ f (tk+i+1 − τ, ϕ]
m−1 (tk+i+1 − τ,
i
^
ϕ]
m−1 (tk+i+1 − τ ) + rm−1,k+i+1−n (f ))
τ2
∂f
∂f
^
(n)
−
·
(t
,
ϕ
]
(t
)
+
r
(f
))
+
(tk , ϕ]
k
m−1
k
m−1 (tk )
m−1,k
12n2
∂t
∂x
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 17 of 33
J. Ineq. Pure and Appl. Math. 6(2) Art. 42, 2005
http://jipam.vu.edu.au
∂f
^
^
(n)
(n)
+ rm−1,k (f )) · ϕ0m−1 (tk ) −
(tk − τ, ϕ]
m−1 (tk − τ ) + rm−1,k−n (f ))
∂t
∂f
^
(n)
0
− (tk − τ, ϕ]
m−1 (tk − τ ) + rm−1,k−n (f )) · ϕm−1 (tk − τ )
∂x
n−1 τ X
(n) ^
=
f (tk+i − τ, ϕ]
m−1 (tk+i − τ ) + rm−1,k+i−n (f ))
2n i=0
i
^
+f (tk+i+1 − τ, ϕ]
(t
−
τ
)
+
r
(f
))
m−1 k+i+1
m−1,k+i+1−n
τ2
∂f
∂f
^
(n)
−
·
(tk , ϕ]
(tk , ϕ]
m−1 (tk ) + rm−1,k (f )) +
m−1 (tk )
2
12n
∂t
∂x
^
^
(n)
(n)
+ rm−1,k (f )) · (f (tk , ϕ]
m−2 (tk ) + rm−2,k (f ))
∂f
^
(n)
− f (tk − τ, ϕ]
(tk − τ, ϕ]
m−2 (tk − τ ) + rm−2,k−n (f )) −
m−1 (tk − τ )
∂t
+
^
(n)
rm−1,k−n (f ))
∂f
−
(tk − τ, ϕ]
m−1 (tk − τ )
∂x
^
^
(n)
(n)
+ rm−1,k−n (f )) · (f (tk − τ, ϕ]
m−2 (tk − τ ) + rm−2,k−n (f ))
^
(n)
(n)
− f (tk − 2τ, ϕ]
m−2 (tk − 2τ ) + rm−2,k−2n (f ))) + rm,k (f )
A Numerical Method in Terms of
the Third Derivative for a Delay
Integral Equation from
Biomathematics
Alexandru Bica and Crăciun Iancu
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 18 of 33
J. Ineq. Pure and Appl. Math. 6(2) Art. 42, 2005
http://jipam.vu.edu.au
n−1
τ X
=
f tk+i − τ, ϕ]
]
m−1 (tk+i − τ ) + f tk+i+1 − τ, ϕ
m−1 (tk+i+1 − τ )
2n i=0
∂f
τ2
∂f
−
·
tk , ϕ]
tk , ϕ]
]
m−1 (tk ) +
m−2 (tk )
m−1 (tk ) · (f tk , ϕ
2
12n
∂t
∂x
− f (tk − τ, ϕ]
m−2 (tk − τ )) −
∂f
tk − τ, ϕ]
m−1 (tk − τ )
∂t
∂f
tk − τ, ϕ]
]
m−1 (tk − τ ) · (f (tk − τ ), ϕ
m−2 (tk − τ )
∂x
^
(n)
− f tk − 2τ, ϕ]
m−2 (tk − 2τ ) ) +rm,k (f )
−
^
(n)
=: ϕf
m (tk ) + rm,k (f ),
∀m ∈ N , m ≥ 2 , ∀k = n + 1, q.
Remark 3. We can see that, for k = n, 2n we have tk − τ ∈ [−τ, 0] and then
ϕ0m−1 (tk − τ ) = Φ0 (tk − τ ) = s0 (tk − τ ) = mk−n
for m ∈ N, m ≥ 2 in (4.5) and (4.6).
A Numerical Method in Terms of
the Third Derivative for a Delay
Integral Equation from
Biomathematics
Alexandru Bica and Crăciun Iancu
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 19 of 33
J. Ineq. Pure and Appl. Math. 6(2) Art. 42, 2005
http://jipam.vu.edu.au
For the remainders we have the estimations:
g
(n)
r (f )
(4.7)
m,k 2
τ 4 M 000 ^
τ
M
(1
+
2M
)
2
(n)
≤
+ r
(f ) · τ L +
160n3 m−1,k 6n2
^
τ 2 LM2 ^
(n)
(n)
· r
+
r
(f
)
(f
)
3n2 m−1,k m−2,k τ 4 M 000
≤
1 + τ L + ... + τ m−1 Lm−1
3
160n
1
τ m Lm−2 M2 · (2M + 1) τ 4 M 000
+
·
+
O
6n2
160n3
n5
τ 4 M 000 (1 − τ m Lm ) τ 6 M 000 τ m−2 Lm−2 M2 · (2M + 1)
=
+
160n3 (1 − τ L)
960n5
1
+O
,
n5
A Numerical Method in Terms of
the Third Derivative for a Delay
Integral Equation from
Biomathematics
Alexandru Bica and Crăciun Iancu
Title Page
Contents
JJ
J
∀m ∈ N, m ≥ 3, ∀k = n + 1, q.
For instance, if m = 3 we have the estimation:
^
τ 4 M 000
(n)
(4.8) r3,k (f ) ≤
1 + τ L + τ 2 L2
3
160n
6
000
Go Back
Close
8
000
M22
· (2M + 1)
τ M M2 · (1 + 2τ L) (2M + 1) τ M
+
5
960n
5760n7
10
000 2
12
000 2
2
τ (M ) M2 L · (1 + τ L) τ (M ) M2 L · (2M + 1)
+
+
,
76800n8
460800n10
+
II
I
2
Quit
Page 20 of 33
J. Ineq. Pure and Appl. Math. 6(2) Art. 42, 2005
http://jipam.vu.edu.au
∀k = n + 1, q.
We obtain the following result:
Theorem 4.1. Considering the initial value problem (2.1) under the conditions
from Corollary 2.2 and Remark 2, if f ∈ C 3 ([−τ, T ] × [a, β]) , τ L < 1 and the
exact solution ϕ is approximated by the sequence (ϕf
m (tk ))k∈N∗ , k = 1, q , on
the equidistant points (3.1), through the successive approximation method (4.1),
combined with the quadrature rule (4.3), then the following error estimation
holds :
(4.9)
τ m · Lm
τ 4 M 000
|ϕ(tk ) − ϕf
· kϕ0 − ϕ1 kC[0,T ] +
m (tk )| ≤
1 − τL
160n3 (1 − τ L)
1
τ 6 M 000 τ m−2 Lm−2 M2 · (2M + 1)
+
+O
,
5
960n
n5
∀m ∈ N∗ , m ≥ 2, ∀k = n + 1, q.
Proof. We have
Alexandru Bica and Crăciun Iancu
Title Page
Contents
∗
|ϕ(tk ) − ϕf
f
m (tk )| ≤ |ϕ(tk ) − ϕm (tk )|+|ϕm (tk ) − ϕ
m (tk )| , ∀m ∈ N , ∀k = n, q.
From Banach’s fixed point principle we have
|ϕ(tk ) − ϕm (tk )| ≤ kϕ − ϕm k ≤
∀m ∈ N∗ , ∀k = n + 1, q. Also,
g
(n)
|ϕm (tk ) − ϕf
m (tk )| ≤ rm,k (f ) ,
A Numerical Method in Terms of
the Third Derivative for a Delay
Integral Equation from
Biomathematics
τ m · Lm
· kϕ0 − ϕ1 kC[0,T ] ,
1 − τL
JJ
J
II
I
Go Back
Close
Quit
∀m ∈ N∗ , m ≥ 2, ∀k = n + 1, q.
Using the remainder estimation (4.7), the proof is completed.
Page 21 of 33
J. Ineq. Pure and Appl. Math. 6(2) Art. 42, 2005
http://jipam.vu.edu.au
If F 000 is Lipschitzian we can use a recent formula of N.S. Barnett and S.S.
Dragomir from Corollary 1 in [2],
Z b
(b − a)2 0
(b − a)
F (t)dt =
[F (a) + F (b)] −
[F (b) − F 0 (a)] + R(F ),
2
12
a
5
with |R(F )| ≤ L(b−a)
, where L is the Lipschitz constant. A composite quadra720
ture formula can be easy obtained, considering an uniform partition of [a, b]
with the knots ti = a + i · b−a
, i = 0, n and the step h = b−a
,
n
n
Z b
n−1
(b − a)2 0
(b − a) X
F (t)dt =
[F (ti ) + F (ti+1 )]−
[F (a) − F 0 (b)]+Rn (F ),
2
2n
12n
a
i=0
having the remainder estimation,
Alexandru Bica and Crăciun Iancu
L (b − a)5
.
|Rn (F )| ≤
720n4
Here, we use these formulas for F (t) = f (t, x(t)) and obtain,
Z tk
(4.10)
f (t, x(t))dt
tk −τ
Z tk
=
F (t)dt
tk −τ
n−1
X
τ
=
2n
A Numerical Method in Terms of
the Third Derivative for a Delay
Integral Equation from
Biomathematics
Title Page
Contents
JJ
J
II
I
Go Back
Close
[F (tk+i − τ ) + F (tk+i+1 − τ )]
Quit
i=0
2
τ
−
[F 0 (tk ) − F 0 (tk − τ )] + Rn (F ),
12n2
Page 22 of 33
∀k = n + 1, q.
J. Ineq. Pure and Appl. Math. 6(2) Art. 42, 2005
http://jipam.vu.edu.au
with
τ 5 L3
,
720n4
if ∃L3 > 0 such that |F 000 (u) − F 000 (v)| ≤ L3 |u − v| , ∀u, v ∈ [−τ, T ]. From
(4.7) and (4.3) we see that the relations (4.5) and (4.6) remains unchanged in
this case, and for the remainders the estimations (4.7) becomes:
|Rn (F )| ≤
τ 5 L3
(n) ,
∀k = n + 1, q
r1,k (f ) ≤
720n4
2
5
g
r(n) (f ) ≤ τ L3 1 + τ L + τ M2 (1 + 2M ) ,
∀k = n + 1, q
2,k 720n4
6n2
A Numerical Method in Terms of
the Third Derivative for a Delay
Integral Equation from
Biomathematics
Alexandru Bica and Crăciun Iancu
g
(n)
(4.11) rm,k (f )
5
^
τ L3
τ 2 M2 (1 + 2M )
(n)
≤
+ r
(f ) · τ L +
720n4 m−1,k 6n2
^
τ 2 LM2 ^
(n)
(n)
· r
+
r
(f
)
(f
)
m−1,k
m−2,k
3n2
τ 5 L3 (1 − τ m Lm ) τ 7 L3 τ m−2 Lm−2 M2 · (2M + 1)
1
≤
+
+O
,
4
6
720n (1 − τ L)
4320n
n6
∀m ∈ N∗ , ∀k = n + 1, q.
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 23 of 33
J. Ineq. Pure and Appl. Math. 6(2) Art. 42, 2005
http://jipam.vu.edu.au
For instance, the estimation (4.8) becomes:
5
g
(n) τ 2 M2 (1 + 2M )
r(n) (f ) ≤ τ L3 + rg
3,k 720n4 2,k (f ) · τ L +
6n2
(n) τ 2 LM2 g
(n)
· r (f )
+
r
(f
)
3n2 2,k 1,k
τ 5 L3
2 2
L
1
+
τ
L
+
τ
≤
720n4
τ 7 L3 M2 · (1 + 2τ L) (2M + 1) τ 9 L3 M22 · (2M + 1)2
+
+
4320n6
25920n8
14
2
2
12
2
τ (L3 ) M2 L · (1 + τ L) τ (L3 ) M2 L · (2M + 1)
+
+
,
1825200n10
10951200n12
A Numerical Method in Terms of
the Third Derivative for a Delay
Integral Equation from
Biomathematics
Alexandru Bica and Crăciun Iancu
∀k = n + 1, q.
Title Page
In this way we obtain the following result:
3
Contents
3
Theorem 4.2. If f ∈ C ([−τ, T ] × [α, β]), Φ ∈ C [−τ, 0], having,
∂f
d ∂f
(t, ϕ(t)) +
(t, ϕ(t))ϕ0 (t)
Φ (0) = lim
t>0,t→0 dt
∂t
∂x
∂f
∂f
0
(t − τ, ϕ(t − τ )) −
(t − τ, ϕ(t − τ )) ϕ (t − τ ) ,
−
∂t
∂x
000
3
∂3f
, ∂f
∂t3 ∂t∂x2
∂3f
∂x3
and the functions
and
are Lipschitzian in t and x, then ϕ ∈
3
000
C [−τ, T ], F is Lipschitzian with a Lipschitz constant L3 > 0 and the fol-
JJ
J
II
I
Go Back
Close
Quit
Page 24 of 33
J. Ineq. Pure and Appl. Math. 6(2) Art. 42, 2005
http://jipam.vu.edu.au
lowing estimation holds:
τ m · Lm
τ 5 L3
· kϕ0 − ϕ1 kC[0,T ] +
1 − τL
160n3 (1 − τ L)
7
m−2 m−2
1
τ L3 τ
L
M2 · (2M + 1)
+
+O
,
6
4320n
n6
∀m ∈ N∗ , m ≥ 2, ∀k = n + 1, q.
|ϕ(tk ) − ϕf
m (tk )| ≤
Proof. From Remark 2, we infer that ϕ ∈ C 3 [−τ, T ] and because f ∈ C 3 ([−τ, T ]
×[α, β]) we see that F ∈ C 3 [−τ, T ] and ∂f
is Lipschitzian in t and x. For this
∂x
reason there exist L01 , L001 > 0 such that
∂f
∂f
(u, x) −
≤ L01 |u − v|
(v,
x)
∂x
∂x
∂f
(u, x) − ∂f (u, y) ≤ L001 |x − y| ,
∂x
∂x
3
∀u, v ∈ [−τ, T ], ∀x, y ∈ [α, β]. Since ∂∂t3f is Lipschitzian in t and x there exist
L30 , L030 > 0 such that
3
3
∂ f
∂
f
(u,
x(u))
−
(v,
x(v))
∂t3
∂t3
3
3
∂ f
∂ f
∂3f
∂3f
≤ 3 (u, x(u)) − 3 (v, x(u)) + 3 (v, x(u)) − 3 (v, x(v))
∂t
∂t
∂t
∂t
0
≤ L30 |u − v| + L30 |x(u) − x(v)|
≤ L30 |u − v| + L030 · kx0 k |u − v|
≤ (L30 + 2L030 M ) |u − v| ,
A Numerical Method in Terms of
the Third Derivative for a Delay
Integral Equation from
Biomathematics
Alexandru Bica and Crăciun Iancu
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 25 of 33
J. Ineq. Pure and Appl. Math. 6(2) Art. 42, 2005
http://jipam.vu.edu.au
∀u, v ∈ [−τ, T ]. We have x0 (t) = f (t, x(t)) − f (t − τ, x(t − τ )), ∀t ∈ [0, T ]
∂3f
∂3f
and kx0 k ≤ 2M. Similar, since ∂t∂x
2 and ∂x3 are Lipschitzian in t and x there
exist L12 , L012 > 0 and L03 , L003 > 0 such that we have:
3
∂ f
∂3f
≤ (L12 + L012 · kx0 k) |u − v| ,
(u,
x(u))
−
(v,
x(v))
∂t∂x2
∂t∂x2
3
∂ f
∂3f
0
0
∂x3 (u, x(u)) − ∂x3 (v, x(v)) ≤ (L03 + L03 · kx k) |u − v| ,
∀u, v ∈ [−τ, T ]. Now, we can state that
∂f
(u, x(u)) − ∂f (v, x(v)) ≤ (L01 + L001 · kx0 k) |u − v| ,
∂x
∂x
∀u, v ∈ [−τ, T ], and since x ∈ C 3 [−τ, T ] we have that x0 and x00 are Lipschitzian. Also, we have F 000 (t) = [f (t, x(t))]000
t and
[f (t, x(t))]000
t =
∂3f
∂3f
(t,
x(t))
+
3
(t, x(t))x0 (t)
∂t3
∂t2 ∂x
∂3f
∂3f
2
3
0
+3
(t,
x(t))
[x
(t)]
+
(t, x(t)) [x0 (t)]
2
3
∂t∂x
∂x
2
∂2f
∂
f
+3
(t, x(t))x00 (t) + 3 2 (t, x(t))x0 (t)x00 (t)
∂t∂x
∂x
∂f
+
(t, x(t))x000 (t).
∂x
A Numerical Method in Terms of
the Third Derivative for a Delay
Integral Equation from
Biomathematics
Alexandru Bica and Crăciun Iancu
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 26 of 33
J. Ineq. Pure and Appl. Math. 6(2) Art. 42, 2005
http://jipam.vu.edu.au
From all the above we can see that:
|F 000 (u) − F 000 (v)|
≤ (L30 +
2L030 M ) |u
3 ∂ f 0
− v| + 3 ∂t2 ∂x · kx k |u − v|
2
3
+ 3(L12 + L012 · kx0 k) · kx0 k |u − v| + (L03 + L003 · kx0 k) · kx0 k · |u − v|
2 2 ∂ f ∂ f 000
· kx00 k2 |u − v|
·
kx
k
|u
−
v|
+
3
+ 3
2
∂t∂x
∂x + (L01 + L001 · kx0 k) · kx000 k |u − v|
≤ L3 |u − v| , ∀u, v ∈ [−τ, T ].
Here we have
L3 = L30 + 2L030 M + 6M1 M3 (2M + 1) + 12M 2 (L12 + 2L012 M )
+ 8M 3 (L03 + 2L003 M ) + 6M2 (2M + 1) M2 (2M + 1) + 2M12
+ 12M2 M12 (2M + 1)2 + 2(L01 + 2L001 M )
× (2M + 1)[M2 (2M + 1) + 2M12 ] > 0,
since kx00 k ≤ 2M1 (2M + 1) and kx00 k ≤ 2(2M + 1)[M2 (2M + 1) + 2M12 ],
having for t ∈ [0, T ]
x00 (t) =
∂f
∂f
(t, x(t)) +
(t, x(t))x0 (t)
∂t
∂x
∂f
∂f
−
(t − τ, x(t − τ )) −
(t − τ, x(t − τ )) x0 (t − τ ),
∂t
∂x
A Numerical Method in Terms of
the Third Derivative for a Delay
Integral Equation from
Biomathematics
Alexandru Bica and Crăciun Iancu
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 27 of 33
J. Ineq. Pure and Appl. Math. 6(2) Art. 42, 2005
http://jipam.vu.edu.au
and
∂2f
∂2f
(t,
x(t))
+
(t, x(t))x0 (t)
∂t2
∂t∂x
2
∂ f
∂2f
0
0
+ x (t) ·
(t, x(t)) + 2 (t, x(t))x (t)
∂t∂x
∂x
2
∂f
∂ f
+ x00 (t) (t, x(t)) − 2 (t − τ, x(t − τ ))
∂x
∂t
∂2f
−
(t − τ, x(t − τ )) · x0 (t − τ )
∂t∂x
2
∂ f
0
+ x (t − τ ) ·
(t − τ, x(t − τ ))
∂t∂x
∂2f
0
+ 2 (t − τ, x(t − τ )) · x (t − τ )
∂x
∂f
+ x00 (t − τ ) (t − τ, x(t − τ )).
∂x
000
Then, F is Lipschitzian, and so we can apply the quadrature rule (4.9) from
[2] and we have the inequality (4.11). Using the estimation from Banach’s fixed
point principle we obtain the desired estimation.
x000 (t) =
Remark 4. To approximate the solution ϕ on [0, T ] we can use the cubic spline
of interpolation s, defined by the interpolatory conditions:


s(tk ) = ϕf
∀k = n + 1, q
m (tk ),


s(tn ) = yn


 s00 (t ) = M = 0, s00 (t ) = M = 0
n
n
q
q
A Numerical Method in Terms of
the Third Derivative for a Delay
Integral Equation from
Biomathematics
Alexandru Bica and Crăciun Iancu
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 28 of 33
J. Ineq. Pure and Appl. Math. 6(2) Art. 42, 2005
http://jipam.vu.edu.au
for a fixed m ∈ N∗ , on the knots tk , k = n, q, as in [4, p. 119–128]:
s : [0, T ] → R, s(t) = si (t), ∀t ∈ [ti−1 , ti ],
∀i = n + 1, q
where,
si (t) =
Mi (t − ti−1 )3 + Mi−1 (ti − t)3 (6ϕi−1 − Mi−1 h2i ) · (ti − t)
+
6hi
6hi
2
(6ϕi − Mi hi ) · (t − ti−1 )
+
, ∀t ∈ [ti−1 , ti ], i = n + 1, q,
6hi
and ϕk = s(tk ), Mk = s00 (tk ), ∀k = n, q. The values Mk , k = n, q are obtained
from the relations
hi Mi−1 Mi (hi + hi+1 ) hi+1 Mi+1
+
+
6
3
6
ϕi+1 − ϕi ϕi − ϕi−1
=
−
,
hi+1
hi
A Numerical Method in Terms of
the Third Derivative for a Delay
Integral Equation from
Biomathematics
Alexandru Bica and Crăciun Iancu
Title Page
i = n + 1, q − 1,
where Mn = Mq = 0. Since f ∈ C 3 ([−τ, T ] × [α, β]) we infer that ϕ ∈
C 4 [0, T ]. If Φ ∈ C 4 [−τ, 0] and Φ(0) = lim ϕ(t) then ϕ ∈ C 4 [−τ, T ]. In
t>0,t→0
these hypothesis, for t ∈ [0, T ]\{ tk , k = n, q}, we have the following error
estimation:
g
τ m · Lm
5 (n)
|ϕ(t) − s(t)| ≤
· kϕ0 − ϕ1 k + rm,k (f ) +
· ϕIV · h4 ,
1 − τL
384
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 29 of 33
J. Ineq. Pure and Appl. Math. 6(2) Art. 42, 2005
http://jipam.vu.edu.au
where h =
T
q−n−1
= nτ . We have
xIV (t)
∂3f
∂3f
∂3f
2
= 3 (t, x(t)) + 3 2 (t, x(t)) · x0 (t) + 3
(t, x(t)) [x0 (t)]
2
∂t
∂t ∂x
∂t∂x
2
∂3f
∂
f
3
+ 3 (t, x(t)) [x0 (t)] + 3
(t, x(t))x00 (t)
∂x
∂t∂x
∂f
∂2f
+ 3 2 (t, x(t)) · x0 (t) · x00 (t) +
(t, x(t)) · x000 (t)
∂x
∂x
∂3f
∂3f
− 3 (t − τ, x(t − τ )) − 3 2 (t − τ, x(t − τ )) · x0 (t − τ )
∂t
∂t ∂x
3
∂ f
2
−3
(t − τ, x(t − τ )) · [x0 (t − τ )]
2
∂t∂x
∂3f
3
− 3 (t − τ, x(t − τ )) · [x0 (t − τ )]
∂x
∂2f
∂2f
−3
(t − τ ))x00 (t − τ ) − 3 2 (t − τ, x(t − τ )) · x0 (t − τ )x00 (t − τ )
∂t∂x
∂x
∂f
−
(t − τ, x(t − τ ))x000 (t − τ ),
∂x
A Numerical Method in Terms of
the Third Derivative for a Delay
Integral Equation from
Biomathematics
Alexandru Bica and Crăciun Iancu
Title Page
Contents
JJ
J
II
I
Go Back
∀t ∈ [0, T ]. Since
Close
0
00
k f k≤ M, kϕ k ≤ 2M, kϕ k ≤ 2M1 (1 + 2M ),
kϕ000 k ≤ 2M2 (1 + 2M )2 + 4M12 (1 + 2M ),
Quit
Page 30 of 33
J. Ineq. Pure and Appl. Math. 6(2) Art. 42, 2005
http://jipam.vu.edu.au
we have the following estimation:
IV ϕ ≤ 2M3 + 6M3 kϕ0 k + 6M3 kϕ0 k2 + 2M3 kϕ0 k3
+ 6M2 kϕ00 k + 6M2 kϕ0 k · kϕ00 k + 2M1 kϕ000 k
≤ 2M3 (1 + 2M )3 + 16M1 M2 (1 + 2M )2 + 8M13 (1 + 2M ).
A Numerical Method in Terms of
the Third Derivative for a Delay
Integral Equation from
Biomathematics
Alexandru Bica and Crăciun Iancu
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 31 of 33
J. Ineq. Pure and Appl. Math. 6(2) Art. 42, 2005
http://jipam.vu.edu.au
References
[1] N.S. BARNETT AND S.S. DRAGOMIR, Perturbed trapezoidal inequality
in terms of the third derivative and applications, RGMIA Res. Rep. Coll.,
4(2) (2001), 221–233.
[2] N.S. BARNETT AND S.S. DRAGOMIR, Perturbed trapezoidal inequality
in terms of the fourth derivative and applications, Korean J. Comput. Appl.
Math., 9(1) (2002), 45–60.
[3] D. GUO AND V. LAKSMIKANTHAM, Positive solutions of nonlinear
integral equations arising in infectious diseases, J. Math. Anal. Appl., 134
(1988), 1–8.
A Numerical Method in Terms of
the Third Derivative for a Delay
Integral Equation from
Biomathematics
Alexandru Bica and Crăciun Iancu
[4] C. IACOB (Ed.), Classic and Modern Mathematics, Ed. Tehnică, Bucureşti, 1983 (in Romanian).
[5] C. IANCU, On the cubic spline of interpolation, Seminar of Functional
Analysis and Numerical Methods, Cluj-Napoca, Preprint 4, 1981, 52–71.
[6] C. IANCU, A numerical method for approximating the solution of an integral equation from biomathematics, Studia Univ. "Babeş-Bolyai", Mathematica, XLIII(4) (1998).
[7] C. IANCU AND C. MUSTAŢĂ, Error estimation in the approximation of
functions by interpolation cubic splines, Mathematica- Rev. Anal. Numér.
Théor. Approx., 29(52)(1) (1987), 33–39.
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 32 of 33
J. Ineq. Pure and Appl. Math. 6(2) Art. 42, 2005
http://jipam.vu.edu.au
[8] R. PRECUP, Positive solutions of the initial value problem for an integral
equation modelling infectious diseases, Seminar of Fixed Point Theory,
Cluj-Napoca, Preprint 3, 1991, 25–30.
[9] R. PRECUP, Periodic solutions for an integral equation from biomathematics via Leray-Schauder principle, Studia Univ. ”Babes-Bolyai”, Mathematica, XXXIX(1) (1994).
[10] I.A. RUS, Principles and Applications of the Fixed Point Theory, Ed. Dacia, Cluj-Napoca, 1979 (in Romanian).
[11] I.A. RUS, A delay integral equation from biomathematics, Seminar of Differential Equations, Cluj-Napoca, Preprint 3, 1989, 87–90.
A Numerical Method in Terms of
the Third Derivative for a Delay
Integral Equation from
Biomathematics
Alexandru Bica and Crăciun Iancu
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 33 of 33
J. Ineq. Pure and Appl. Math. 6(2) Art. 42, 2005
http://jipam.vu.edu.au