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Existence and Analytic Approximation of Solutions of Duffing Type Nonlinear Integro-Differential Equation with Integral Boundary Conditions
Journal of Inequalities and Applications volume 2009, Article number: 193169 (2009)
Abstract
A generalized quasilinearization technique is developed to obtain a sequence of approximate solutions converging monotonically and quadratically to a unique solution of a boundary value problem involving Duffing type nonlinear integro-differential equation with integral boundary conditions. The convergence of order for the sequence of iterates is also established. It is found that the work presented in this paper not only produces new results but also yields several old results in certain limits.
1. Introduction
Vascular diseases such as atherosclerosis and aneurysms are becoming frequent disorders in the industrialized world due to sedentary way of life and rich food. Causing more deaths than cancer, cardiovascular diseases are the leading cause of death in the world. In recent years, computational fluid dynamics () techniques have been used increasingly by researchers seeking to understand vascular hemodynamics.
methods possess the potential to augment the data obtained from in vitro methods by providing a complete characterization of hemodynamic conditions (blood velocity and pressure as a function of space and time) under precisely controlled conditions. However, specific difficulties in
studies of blood flows are related to the boundary conditions. It is now recognized that the blood flow in a given district may depend on the global dynamics of the whole circulation and the boundary conditions (e.g., the instantaneous velocity profile at the inlet section of the computed domain) for an in vitro blood flow computation need to be prescribed. Taylor et al. [1] assumed very long circular vessel geometry upstream the inlet section to obtain the analytic solution due to Womersley [2]. However, it is not always justified to assume a circular cross-section. In order to cope with this problem, an alternative approach prescribing integral boundary conditions is presented in [3]. The validity of this approach is verified by computing both steady and pulsated channel flows for Womersley number up to 15. In fact, the integral boundary conditions have various applications in other fields such as chemical engineering, thermoelasticity, underground water flow, population dynamics, and so forth, see for instance, [4–10] and references therein.
Integro-differential equations are encountered in many areas of science where it is necessary to take into account aftereffect or delay. Especially, models possessing hereditary properties are described by integro-differential equations in practice. Also, the governing equations in the problems of biological sciences such as spreading of disease by the dispersal of infectious individuals [11–13], the reaction-diffusion models in ecology to estimate the speed of invasion [14, 15], and so forth, are integro-differential equations. For the theoretical background of integro-differential equations, we refer the reader to the text [16].
In this paper, we study a boundary value problem for Duffing type nonlinear integro-differential equation (Duffing equation with both integral and nonintegral forcing terms of nonlinear type) with integral boundary conditions given by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193169/MediaObjects/13660_2008_Article_1915_Equ1_HTML.gif)
where are continuous functions,
, and
are nonnegative constants.
A generalized quasilinearization () technique due to Lakshmikantham [17, 18] is applied to obtain the analytic approximation of the solutions of the integral boundary value problem (1.1). In recent years, the
technique has been extensively developed and applied to a wide range of initial and boundary value problems for different types of differential equations, for instance, see [19–30] and the references therein. Section 2 contains some basic results. A monotone sequence of approximate solutions converging uniformly and quadratically to a unique solution of the problem (1.1) is obtained in Section 3. The convergence of order
for the sequence of iterates is established in Section 4 .
2. Preliminary Results
For any the nonhomogeneous linear problem
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193169/MediaObjects/13660_2008_Article_1915_Equ2_HTML.gif)
has a unique solution
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193169/MediaObjects/13660_2008_Article_1915_Equ3_HTML.gif)
where is the unique solution of the problem
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193169/MediaObjects/13660_2008_Article_1915_Equ4_HTML.gif)
and is given by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193169/MediaObjects/13660_2008_Article_1915_Equ5_HTML.gif)
Here, we note that the associated homogeneous problem has only the trivial solution and on
Definition 2.1.
A function is a lower solution of (1.1) if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193169/MediaObjects/13660_2008_Article_1915_Equ6_HTML.gif)
Similarly, is an upper solution of (1.1) if the inequalities in the definition of lower solution are reversed.
Now, we present some basic results which are necessary to prove the main results.
Theorem 2.2.
Let and
be lower and upper solutions of the boundary value problem (1.1), respectively, such that
If
and
are strictly decreasing in
for each
and for each
respectively, and
satisfy a one-sided Lipschitz condition:
then there exists a solution
of (1.1) such that
Proof.
We define by
and consider the modified problem
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193169/MediaObjects/13660_2008_Article_1915_Equ7_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193169/MediaObjects/13660_2008_Article_1915_Equ8_HTML.gif)
As and
are continuous and bounded, an application of Schauder's fixed point theorem [16] ensures the existence of a solution of the problem (2.6). We note that any solution
of the problem (2.6) such that
is also a solution of (1.1). Thus we need to show that
As
and
we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193169/MediaObjects/13660_2008_Article_1915_Equ9_HTML.gif)
which imply that are lower and upper solutions of (2.6), respectively, on
Now, we show that on
If the inequality
is not true on
then
will have a positive maximum at some
We first suppose that
Then
On the other hand, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193169/MediaObjects/13660_2008_Article_1915_Equ10_HTML.gif)
which leads to a contradiction. Thus, there exists such that
Assuming
(the case
is similar), we can find
such that
and
for every
Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193169/MediaObjects/13660_2008_Article_1915_Equ11_HTML.gif)
which can alternatively be written as Integrating from
to
and using
we obtain
for every
which together with
implies that
for every
Thus,
is nondecreasing on
which is a contradiction as
has a positive maximum value at
Hence
is an isolated maximum point. If
then
On the other hand, we find that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193169/MediaObjects/13660_2008_Article_1915_Equ12_HTML.gif)
If , then
which, on substituting in (2.11), yields
This contradicts that
If
then
Hence, in view of the fact that
satisfies a one-sided Lipschitz condition, we have
which leads to
a contradiction. For
we also get a contradiction
As before, there exists
such that
and
for every
which provides a contradiction that
is nondecreasing on
Thus,
is an isolated maximum point. In a similar manner,
yields a contradiction. Hence it follows that
. On the same pattern, it can be shown that
. Thus, we conclude that
This completes the proof.
Theorem 2.3.
Let and
be, respectively, lower and upper solutions of the boundary value problem (1.1). If
and
are strictly decreasing in
for each
and for each
respectively, and
satisfies a one-sided Lipschitz condition:
then
Proof.
We omit the proof as it follows the procedure employed in the proof of Theorem 2.2.
3. Analytic Approximation of the Solution with Quadratic Convergence
Theorem 3.1.
Assume that
() and
are, respectively, lower and upper solutions of (1.1) such that
() is such that
for each
and
where
for some continuous function
on
() is such that
for each
with
() is such that
and
Then, there exists a sequence of approximate solutions converging monotonically and quadratically to a unique solution of the problem (1.1).
Proof.
Let be defined by
so that
Using the generalized mean value theorem together with
, and
, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193169/MediaObjects/13660_2008_Article_1915_Equ13_HTML.gif)
We set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193169/MediaObjects/13660_2008_Article_1915_Equ14_HTML.gif)
Observe that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193169/MediaObjects/13660_2008_Article_1915_Equ15_HTML.gif)
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193169/MediaObjects/13660_2008_Article_1915_Equ16_HTML.gif)
Let us define
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193169/MediaObjects/13660_2008_Article_1915_Equ17_HTML.gif)
so that and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193169/MediaObjects/13660_2008_Article_1915_Equ18_HTML.gif)
Now, we fix and consider the problem
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193169/MediaObjects/13660_2008_Article_1915_Equ19_HTML.gif)
Using , (3.3), (3.4), and (3.6), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193169/MediaObjects/13660_2008_Article_1915_Equ20_HTML.gif)
which imply that and
are, respectively, lower and upper solutions of (3.7). It follows by Theorem 2.2 and 2.3 that there exists a unique solution
of (3.7) such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193169/MediaObjects/13660_2008_Article_1915_Equ21_HTML.gif)
Next, we consider
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193169/MediaObjects/13660_2008_Article_1915_Equ22_HTML.gif)
Using the earlier arguments, it can be shown that and
are lower and upper solutions of (3.10), respectively, and hence by Theorem 2.2 and 2.3, there exists a unique solution
of (3.10) such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193169/MediaObjects/13660_2008_Article_1915_Equ23_HTML.gif)
Continuing this process successively yields a sequence of solutions satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193169/MediaObjects/13660_2008_Article_1915_Equ24_HTML.gif)
where the element of the sequence
is a solution of the problem
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193169/MediaObjects/13660_2008_Article_1915_Equ25_HTML.gif)
and is given by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193169/MediaObjects/13660_2008_Article_1915_Equ26_HTML.gif)
Using the fact that is compact and the monotone convergence of the sequence
is pointwise, it follows by the standard arguments (Arzela Ascoli convergence criterion, Dini's theorem [21]) that the convergence of the sequence is uniform. If
is the limit point of the sequence, taking the limit
in (3.14), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193169/MediaObjects/13660_2008_Article_1915_Equ27_HTML.gif)
thus, is a solution of (1.1). Now, we show that the convergence of the sequence is quadratic. For that we set
In view of
, and (3.2), it follows by Taylor's theorem that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193169/MediaObjects/13660_2008_Article_1915_Equ28_HTML.gif)
where ,
is a bound on
,
is a bound on
for
provides a bound on
Further, in view of (3.5), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193169/MediaObjects/13660_2008_Article_1915_Equ29_HTML.gif)
where In view of
, there exist
and
such that
and
Let
and
then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193169/MediaObjects/13660_2008_Article_1915_Equ30_HTML.gif)
Using the estimates (3.16) and (3.18), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193169/MediaObjects/13660_2008_Article_1915_Equ31_HTML.gif)
where provides a bound on
and
Taking the maximum over
, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193169/MediaObjects/13660_2008_Article_1915_Equ32_HTML.gif)
where This establishes the quadratic convergence of the sequence of iterates.
4. Higher Order Convergence
Theorem 4.1.
Assume that
(B1) and
are, respectively, lower and upper solutions of (1.1) such that
(B2) is such that
and
with
for some continuous function
on
(B3) satisfies
and
(B4) is such that
and
where
Then, there exists a monotone sequence of approximate solutions converging uniformly and rapidly to the unique solution of the problem (1.1) with the order of convergence
Proof.
Using Taylor's theorem and the assumptions , we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193169/MediaObjects/13660_2008_Article_1915_Equ33_HTML.gif)
where We set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193169/MediaObjects/13660_2008_Article_1915_Equ34_HTML.gif)
and note that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193169/MediaObjects/13660_2008_Article_1915_Equ35_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193169/MediaObjects/13660_2008_Article_1915_Equ36_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193169/MediaObjects/13660_2008_Article_1915_Equ37_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193169/MediaObjects/13660_2008_Article_1915_Equ38_HTML.gif)
Letting we consider the problem
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193169/MediaObjects/13660_2008_Article_1915_Equ39_HTML.gif)
Using , (4.3), (4.4), and (4.5), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193169/MediaObjects/13660_2008_Article_1915_Equ40_HTML.gif)
Thus, it follows by definition that and
are, respectively, lower and upper solutions of (4.7). As before, by Theorem 2.2 and 2.3, there exists a unique solution
of (4.7) such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193169/MediaObjects/13660_2008_Article_1915_Equ41_HTML.gif)
Continuing this process successively, we obtain a monotone sequence of solutions satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193169/MediaObjects/13660_2008_Article_1915_Equ42_HTML.gif)
where the element of the sequence
is a solution of the problem
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193169/MediaObjects/13660_2008_Article_1915_Equ43_HTML.gif)
and is given by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193169/MediaObjects/13660_2008_Article_1915_Equ44_HTML.gif)
Employing the arguments used in the proof of Theorem 3.1, we conclude that the sequence converges uniformly to a unique solution
of (1.1).
In order to prove that the convergence of the sequence is of order we set
and
and note that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193169/MediaObjects/13660_2008_Article_1915_Equ45_HTML.gif)
Using Taylor's theorem, we find that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193169/MediaObjects/13660_2008_Article_1915_Equ46_HTML.gif)
where is a bound for
and
is a bound on
Again, by Taylor's theorem and using (4.2), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193169/MediaObjects/13660_2008_Article_1915_Equ47_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193169/MediaObjects/13660_2008_Article_1915_Equ48_HTML.gif)
Making use of , we find that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193169/MediaObjects/13660_2008_Article_1915_Equ49_HTML.gif)
Thus, we can find such that
and consequently, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193169/MediaObjects/13660_2008_Article_1915_Equ50_HTML.gif)
By virtue of (4.14) and (4.18), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193169/MediaObjects/13660_2008_Article_1915_Equ51_HTML.gif)
where and
is a bound on
. Taking the maximum over
and solving the above expression algebraically, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193169/MediaObjects/13660_2008_Article_1915_Equ52_HTML.gif)
This completes the proof.
5. Discussion
An algorithm for approximating the analytic solution of Duffing type nonlinear integro-differential equation with integral boundary conditions is developed by applying the method of generalized quasilinearization. A monotone sequence of iterates converging uniformly and quadratically (rapidly) to a unique solution of an integral boundary value problem (1.1) is presented. For the illustration of the results, let us consider the following integral boundary value problem:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193169/MediaObjects/13660_2008_Article_1915_Equ53_HTML.gif)
Let and
be, respectively, lower and upper solutions of (5.1). Clearly
and
are not the solutions of (5.1) and
Moreover, the assumptions
, and
of Theorem 3.1 are satisfied by choosing
Thus, the conclusion of Theorem 3.1 applies to the problem (5.1).
The results established in this paper provide a diagnostic tool to predict the possible onset of diseases such as cardiac disorder and chaos in speech by varying the nonlinear forcing functions , and
appropriately in (1.1). If the nonlinearity
in (1.1) is of convex type, then the assumption
in Theorem 3.1 reduces to
and
in Theorem 4.1 becomes
(that is,
in this case). The existence results for Duffing type nonlinear integro-differential equations with Dirichlet boundary conditions can be recorded by taking
and
in (1.1). Further, for
(
and
are constants) and
in (1.1), our results become the existence results for Duffing type nonlinear integro-differential equations with nonhomogeneous Dirichlet boundary conditions. If we take
in (1.1), our problem reduces to the Dirichlet boundary value problem involving Duffing type nonlinear integro-differential equations with integral boundary conditions. In case, we fix
in (1.1), the existence results for Duffing type nonlinear integro-differential equations with separated boundary conditions appear as a special case of our results. By taking
in (1.1), the results of [30] appear as a special case of our work. The results for forced Duffing equation involving a purely integral type of nonlinearity subject to integral boundary conditions follow by taking
in (1.1).
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Acknowledgment
This work was supported by Deanship of Scientific Research, King Abdulaziz University through Project no. 428/155.
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Alsaedi, A., Ahmad, B. Existence and Analytic Approximation of Solutions of Duffing Type Nonlinear Integro-Differential Equation with Integral Boundary Conditions. J Inequal Appl 2009, 193169 (2009). https://doi.org/10.1155/2009/193169
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Accepted:
Published:
DOI: https://doi.org/10.1155/2009/193169