Existence and Analytic Approximation of Solutions of Duffing Type Nonlinear Integro-Differential Equation with Integral Boundary Conditions

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

(1.1)

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

(2.1)

has a unique solution

(2.2)

where is the unique solution of the problem

(2.3)

and is given by

(2.4)

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

(2.5)

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

(2.6)

where

(2.7)

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

(2.8)

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

(2.9)

which leads to a contradiction. Thus, there exists such that Assuming (the case is similar), we can find such that and for every Then

(2.10)

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

(2.11)

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

(3.1)

We set

(3.2)

Observe that

(3.3)

and

(3.4)

Let us define

(3.5)

so that and

(3.6)

Now, we fix and consider the problem

(3.7)

Using , (3.3), (3.4), and (3.6), we obtain

(3.8)

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

(3.9)

Next, we consider

(3.10)

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

(3.11)

Continuing this process successively yields a sequence of solutions satisfying

(3.12)

where the element of the sequence is a solution of the problem

(3.13)

and is given by

(3.14)

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

(3.15)

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

(3.16)

where , is a bound on , is a bound on for provides a bound on Further, in view of (3.5), we have

(3.17)

where In view of , there exist and such that and Let and then

(3.18)

Using the estimates (3.16) and (3.18), we obtain

(3.19)

where provides a bound on and Taking the maximum over , we get

(3.20)

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

(4.1)

where We set

(4.2)

and note that

(4.3)

(4.4)

(4.5)

(4.6)

Letting we consider the problem

(4.7)

Using , (4.3), (4.4), and (4.5), we obtain

(4.8)

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

(4.9)

Continuing this process successively, we obtain a monotone sequence of solutions satisfying

(4.10)

where the element of the sequence is a solution of the problem

(4.11)

and is given by

(4.12)

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

(4.13)

Using Taylor's theorem, we find that

(4.14)

where is a bound for and is a bound on Again, by Taylor's theorem and using (4.2), we obtain

(4.15)

where

(4.16)

Making use of , we find that

(4.17)

Thus, we can find such that and consequently, we have

(4.18)

By virtue of (4.14) and (4.18), we have

(4.19)

where and is a bound on . Taking the maximum over and solving the above expression algebraically, we obtain

(4.20)

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:

(5.1)

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).