Abstract
In this paper, we investigate the existence and approximation of the solutions of a nonlinear nonlocal threepoint boundary value problem involving the forced Duffing equation with mixed nonlinearities. Our main tool of the study is the generalized quasilinearization method due to Lakshmikantham. Some illustrative examples are also presented.
Mathematics Subject Classification (2000): 34B10, 34B15.
Keywords:
Duffing equation; nonlocal boundary value problem; quasilinearization; quadratic convergence1 Introduction
The Duffing equation plays an important role in the study of mechanical systems. There are multiple forms of the Duffing equation, ranging from dampening to forcing terms. This equation possesses the qualities of a simple harmonic oscillator, a nonlinear oscillator, and has indeed an ability to exhibit chaotic behavior. Chaos can be defined as disorder and confusion. In physics, chaos is defined as behavior so unpredictable as to appear random, allowing great sensitivity to small initial conditions. The chaotic behavior can emerge in a system as simple as the logistic map. In that case, the "route to chaos" is called perioddoubling. In practice, one would like to understand the route to chaos in systems described by partial differential equations such as flow in a randomly stirred fluid. This is, however, very complicated and difficult to treat either analytically or numerically. The Duffing equation is found to be an appropriate candidate for describing chaos in dynamic systems. The advantage of a pseudochaotic equation like the Duffing equation is that it allows control of the amount of chaos it exhibits. Chaotic oscillators are important tools for creating and testing models that are more realistic. This is why the Duffing equation is of great interest. The use of the Duffing equation aids in the dynamic behavior of chaos and bifurcation, which studies how small changes in a function can cause a sudden change in behavior [1]. Another important application of the Duffing equation is in the field of the prediction of diseases. A careful measurement and analysis of a strongly chaotic voice has the potential to serve as an early warning system for more serious chaos and possible onset of disease. This chaos is with the help of the Duffing equation. In fact, the success at analyzing and predicting the onset of chaos in speech and its simulation by equations such as the Duffing equation has enhanced the hope that we might be able to predict the onset of arrhythmia and heart attacks someday [2].
The Duffing equation is a mathematical representation of the oscillator. Both the equation and oscillator are prone to many output waveforms. One of the simplest waveforms includes simple harmonic motion like a pendulum. Other waveforms are considerably more complex and can quickly be described as shear oscillatory chaos. The Duffing equation can be a forced or unforced damped chaotic harmonic oscillator. Exact solutions of secondorder nonlinear differential equations like the forced Duffing equation are rarely possible due to the possible chaotic output. There do exist a number of powerful procedures for obtaining approximate solutions of nonlinear problems such as Galerkin's method, expansion methods, dynamic programming, iterative techniques, the method of upper and lower bounds, and Chapligin method to name a few. The monotone iterative technique coupled with the method of upper and lower solutions [3] manifests itself as an effective and flexible mechanism that offers theoretical as well as constructive existence results in a closed set, generated by the lower and upper solutions. In general, the convergence of the sequence of approximate solutions given by the monotone iterative technique is at most linear. To obtain a sequence of approximate solutions converging quadratically, we use the method of quasilinearization. The origin of the quasilinearization lies in the theory of dynamic programming [4,5]. Agarwal [6] discussed quasilinearization and approximate quasilinearization for multipoint boundary value problems. In fact, the quasilinearization technique is a variant of Newton's method. This method applies to semilinear equations with convex (concave) nonlinearities and generates a monotone scheme whose iterates converge quadratically to a solution of the problem at hand. The nineties brought new dimensions to this technique when Lakshmikantham [7,8] generalized the method of quasilinearization by relaxing the convexity assumption. This development was so significant that it attracted the attention of many researchers, and the method was extensively developed and applied to a wide range of initial and boundary value problems for different types of differential equations. A detailed description of the quasilinearization method and its applications can be found in the monograph [9] and the papers [1026] and the references therein.
In this paper, we study a nonlinear nonlocal threepoint boundary value problem of the forced Duffing equation with mixed nonlinearities given by
where N(t, x) ∈ C[J × ℝ, ℝ] is such that
and g_{i}: ℝ → ℝ (i = 1,2) are given continuous functions. The details of such a decomposition can be found in Section 1.5 of the text [9]. In (1.3), it is assumed that f(t,x) is nonconvex, k(t,x) is nonconcave, and H(t,x) is a Lipschitz function:
A quasilinearization technique due to Lakshmikantham [9] is applied to obtain an analytic approximation of the solution of the problem (1.11.2). In fact, we obtain sequences of upper and lower solutions converging monotonically and quadratically to a unique solution of the problem at hand. It is worth mentioning that the forced Duffing equation with mixed nonlinearities has not been studied so far.
2 Preliminaries
As argued in [12], the solution x(t) of the problem (1.11.2) can be written in terms of the Green's function as
where
Observe that G(t,s) < 0 on [0,1] × [0,1].
Definition 2.1. We say that α ∈ C^{2}[J, ℝ] is a lower solution of the problem (1.11.2) if
and β ∈ C^{2}[J, ℝ] will be an upper solution of the problem (1.11.2) if the inequalities are reversed in the definition of lower solution.
Now we state some basic results that play a pivotal role in the proof of the main result. We do not provide the proof as the method of proof is similar to the one described in the text [9].
Theorem 2.1. Let α and β be lower and upper solutions of (1.11.2), respectively. Assume that
(i) f_{x}(t,x) + k_{x}(t,x)  L > 0 for every (t,x) ∈ J × ℝ.
(ii) g_{1 }and g_{2 }are continuous on ℝ satisfying the onesided Lipschitz condition:
Then α(t) ≤ β(t), t ∈ J.
Theorem 2.2. Let α and β be lower and upper solutions of (1.11.2), respectively, such that α(t) ≤ β(t), t ∈ J. Then, there exists a solution x(t) of (1.11.2) such that α(t) ≤ x(t) ≤ β(t), t ∈ J.
3 Main result
Theorem 3.1. Assume that
(A_{1}) α_{0}, β_{0 }∈ C^{2}[J, ℝ] are lower and upper solutions of (1.11.2), respectively.
(A_{2}) N ∈ C[J × ℝ, ℝ] be such that
where f_{x}(t, x), k_{x}(t, x), f_{xx}(t, x), k_{xx}(t, x) exist and are continuous, and for continuous functions ϕ, χ,(f_{xx}(t, x) + ϕ_{xx}(t, x)) ≥ 0, (k_{xx}(t, x) + χ_{xx}(t, x)) ≤ 0 with ϕ_{xx }≥ 0, χ_{xx }≤ 0 for every (t, x) ∈ S, where S = {(t, x) ∈ J × ℝ: α_{0}(t) ≤ x(t) ≤ β_{0}(t)}. H(t, x) satisfies the onesided Lipschitz condition:
where L > 0 is a Lipschitz constant and f_{x}(t, x) + k_{x}(t, x)  L > 0 for every (t, x) ∈ S.
(A_{3}) For
Then, there exist monotone sequences {α_{n}} and {β_{n}} that converge in the space of continuous functions on J quadratically to a unique solution x(t) of the problem (1.11.2).
Proof. Let us define F: J × ℝ → ℝ by F(t, x) = f(t, x) + ϕ(t, x), K: J × ℝ → ℝ by K(t, x) = k(t, x) + χ(t, x), G_{i}: ℝ → ℝ by G_{i}(x) = g_{i}(x) + ψ_{i}(x), i = 1, 2. By the assumption (A_{2}) and the generalized mean value theorem, we get
Interchanging x and y, (3.1) and (3.2) take the form
By the assumption (A_{3}), we obtain
which, on interchanging x and y yields
We set
and for i = 1,2,
Observe that
and
Now, we consider the problem
Using (A_{1}), (3.7) and (3.8), we obtain
and
which imply that α_{0 }and β_{0 }are, respectively, lower and upper solutions of (3.113.12). Thus, by Theorems 2.1 and 2.2, there exists a solution α_{1 }for the problem (3.113.12) such that
Next, consider the problem
Using (A_{1}), (3.9) and (3.10), we get
and
which imply that α_{0 }and β_{0 }are, respectively, lower and upper solutions of (3.143.15). Again, by Theorems 2.1 and 2.2, there exists a solution β_{1 }of (3.143.15) satisfying
Now we show that α_{1}(t) ≤ β_{1}(t). For that, we prove that α_{1}(t) is a lower solution and β_{1}(t) is an upper solution of (1.11.2). Using the fact that α_{1}(t) is a solution of (3.113.12) satisfying α_{0}(t) ≤ α_{1}(t) ≤ β_{0}(t) and (3.73.8), we obtain
By the above inequalities, it follows that α_{1 }is a lower solution of (1.11.2).
In view of the fact that β_{1}(t) is a solution of (3.143.15) together with (3.9), we get
and by virtue of (3.10), we have
Thus, β_{1 }is an upper solution of (1.11.2). Hence, by Theorem 2.1, it follows that
Combining (3.13, 3.16) and (3.17) yields
Now, by induction, we prove that
For that, we consider the boundary value problems
and
Assume that for some n > 1, α_{0}(t) ≤ α_{n}(t) ≤ β_{n}(t) ≤ β_{0}(t) and we will show that α_{n+1}(t) ≤ β_{n+1}(t).
Using (3.7), we have
By (3.8), we obtain
which yields
Thus, α_{n }is a lower solution of (3.183.19). In a similar manner, we find that β_{n }is an upper solution of (3.183.19). Thus, by Theorems 2.1 and 2.2, there exists a solution α_{n+1}(t) of (3.183.19) such that α_{n}(t) ≤ α_{n+1}(t) ≤ β_{n}(t), t ∈ J. Similarly, it can be proved that α_{n}(t) ≤ β_{n+1}(t) ≤ β_{n}(t), t ∈ J, where β_{n+1}(t) is a solution of (3.203.21) and α_{n}(t), β_{n}(t) are lower and upper solutions of (3.203.21), respectively. Next, we show that α_{n+1}(t) ≤ β_{n+1}(t).
For that, we have to show that α_{n+1}(t) and β_{n+1}(t) are lower and upper solutions of (1.11.2), respectively. Using (3.7, 3.8) together with the fact that α_{n+1}(t) is a solution of (3.183.19), we get
which implies that α_{n+1 }is a lower solution of (1.11.2). Employing a similar procedure, it can be proved that β_{n+1 }is an upper solution of (1.11.2). Hence, by Theorem 2.1, it follows that α_{n+1}(t) ≤ β_{n+1}(t). Therefore, by induction, we have
Since [0,1] is compact and the monotone convergence is pointwise, it follows that {α_{n}} and {β_{n}} are uniformly convergent with
such that α_{0}(t) ≤ x(t) ≤ y(t) ≤ β_{0}(t), where
and
By the uniqueness of the solution (which follows by the hypotheses of Theorem 2.1), we conclude that x(t) = y(t). This proves that the problem (1.11.2) has a unique solution x(t) given by
In order to prove that each of the sequences {α_{n}}, {β_{n}} converges quadratically, we set z_{n}(t) = β_{n}(t)  x(t) and r_{n}(t) = x(t)  α_{n}(t), and note that z_{n }≥ 0, r_{n }≥ 0. We will only prove the quadratic convergence of the sequence {r_{n}} as that of {z_{n}} is similar. By the mean value theorem, we find that
where α_{n }≤ ζ_{5}, ζ_{8 }≤ β_{n}, α_{n }≤ ζ_{6}, ζ_{7 }≤ x, and
Now we define
and obtain
where α_{n }≤ γ_{1}, δ_{1}, ρ_{1}, σ_{1 }≤ x, α_{n }≤ γ_{2 }≤ x, and α_{n }≤ δ_{2}, ρ_{2}, σ_{2 }≤ α_{n+1}. Letting
where
4 Examples
Example 4.1. Consider the problem
Here f(t, x) = 2x  tcos(πx/2), k(t, x) ≡ 0,
Example 4.2. Consider the nonlinear boundary value problem given by
where f(t, x) = 7x + sin(πxt/2), k(t, x) = tcos(πx/2),
and
such that f_{xx}(t, x) + ϕ_{xx}(t, x) = π^{2}t^{2}[2M_{1 } cos(πtx/2)]/4 ≥ 0, k_{xx }+ χ_{xx }= π^{2}[8M_{2 } tcos(πx/2)]/4 ≤ 0. Clearly,
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
Both authors, AA and MHA, contributed to each part of this work equally and read and approved the final version of the manuscript.
Acknowledgements
The authors thank the referees for their useful comments. This research was partially supported by Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, Saudi Arabia.
References

Yolasi, CK, Kyprianidis, IM, Stovbouios, I: Experimental study of a nonlinear circuit described by Duffing's equation. J Istanbul Kültür Univ. 4, 45–54 (2006). PubMed Abstract

Vaidya, PG, Winkel, CR: Analysis of induced chaos in Duffing's equation, using Caseygrams. J Acoust Soc Am. 99(4), 2496–2500 (1996)

Ladde, GS, Lakshmikantham, V, Vatsala, AS: Monotone Iterative Techniques for Nonlinear Differential Equations. Pitman, Boston (1985)

Bellman, R, Kalaba, R: Quasilinearization and Nonlinear Boundary Value Problems. Amer Elsevier New York (1965)

Lee, ES: Quasilinearization and Invariant Embedding. Academic Press, New York (1968)

Agarwal, RP: Quasilinearization and approximate quasilinearization for multipoint boundary value problems. J Math Anal Appl. 107, 317–330 (1985). Publisher Full Text

Lakshmikantham, V: An extension of the method of quasilinearization. J Optim Theory Appl. 82, 315–321 (1994). Publisher Full Text

Lakshmikantham, V: Further improvement of generalized quasilinearization. Nonlinear Anal. 27, 223–227 (1996). Publisher Full Text

Lakshmikantham, V, Vatsala, AS: Generalized Quasilinearization for Nonlinear Problems, Mathematics and its Applications. Kluwer, Dordrecht (1998)

Ahmad, B, Nieto, JJ, Shahzad, N: The BellmanKalabaLakshmikantham quasilinearization method for Neumann problems. J Math Anal Appl. 257, 356–363 (2001). Publisher Full Text

Mandelzweig, VB, Tabakin, F: Quasilinearization approach to nonlinear problems in physics with application to nonlinear ODEs. Comput Phys Comm. 141, 268–281 (2001). Publisher Full Text

Eloe, PW, Gao, Y: The method of quasilinearization and a threepoint boundary value problem. J Korean Math Soc. 39, 319–330 (2002)

Ramos, JI: Piecewise quasilinearization techniques for singular boundaryvalue problems. Comput Phys Comm. 158, 12–25 (2004). Publisher Full Text

Ahmad, B: A quasilinearization method for a class of integrodifferential equations with mixed nonlinearities. Nonlinear Anal Real World Appl. 7, 997–1004 (2006). Publisher Full Text

Alsaedi, A: Monotone iteration scheme for a forced Duffing equation with nonlocal threepoint conditions. Commun Korean Math Soc. 22(1), 53–64 (2007). Publisher Full Text

Amster, P, De Napoli, P: A quasilinearization method for elliptic problems with a nonlinear boundary condition. Nonlinear Anal. 66, 2255–2263 (2007). Publisher Full Text

O'Regan, D, ElGebeily, M: A quasilinearization method for a class of second order singular nonlinear differential equations with nonlinear boundary conditions. Nonlinear Anal Real World Appl. 8, 174–186 (2007). Publisher Full Text

Ahmad, B, Alsaedi, A, Alghamdi, B: Analytic approximation of solutions of the forced Duffing equation with integral boundary conditions. Nonlinear Anal Real World Appl. 9, 1727–1740 (2008). Publisher Full Text

Ahmad, B, Nieto, JJ: Existence and approximation of solutions for a class of nonlinear impulsive functional differential equations with antiperiodic boundary conditions. Nonlinear Anal. 69, 3291–3298 (2008). Publisher Full Text

Ahmad, B, Alghamdi, B: Approximation of solutions of the nonlinear Duffing equation involving both integral and nonintegral forcing terms with separated boundary conditions. Comput Phys Comm. 179, 409–416 (2008). Publisher Full Text

Pei, M, Chang, SK: A quasilinearization method for secondorder fourpoint boundary value problems. Appl Math Comp. 202, 54–66 (2008). Publisher Full Text

Pei, M, Chang, SK: A quasilinearization method for secondorder fourpoint boundary value problems. Appl Math Comp. 202, 54–66 (2008). Publisher Full Text

O'Regan, D, ElGebeily, M: Existence, upper and lower solutions and quasilinearization for singular differential equations. IMA J Appl Math. 73, 323–344 (2008)

Ahmad, B, Alsaedi, A: Existence of approximate solutions of the forced Duffing equation with discontinuous type integral boundary conditions. Nonlinear Anal Real World Appl. 10, 358–367 (2009). Publisher Full Text

Nieto, JJ, Ahmad, B: Approximation of solutions for an initial and terminal value problem for the forced Duffing equation with nonviscous damping. Appl Math Comput. 216, 2129–2136 (2010). Publisher Full Text

Alsaedi, A: Approximation of solutions for secondorder mpoint nonlocal boundary value problems via the method of generalized quasilinearization. Bound Value Probl. 17 (2011) (Art. ID 929061)