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Initial time difference quasilinearization for Caputo Fractional Differential Equations
Advances in Difference Equations volume 2012, Article number: 92 (2012)
This paper deals with an application of the method of quasilinearization by not demanding the Hölder continuity assumption of functions involved and by choosing upper and lower solutions with initial time difference for nonlinear Caputo fractional differential equations. Thus, we construct monotone flows that are generated by solutions of linear fractional differential equations which converge uniformly and quadratically to the unique solution of the problem. Also, necessary comparison result concerning lower and upper solutions are proved without using Hölder continuity.
Mathematics subject classification: 34A12, 34A45, 34C11.
The method of quasilinearization is employed to provide an explicit analytic representation for the solution of nonlinear differential equations. In this technique, one gets monotone sequences whose iterates are the solutions of corresponding linear problems and furthermore these sequences converge uniformly and quadratically to the unique solution of the given nonlinear differential equations . This is a definite advantage of this constructive technique. Also this method has been generalized, refined and extended in several directions so as to be applicable to a much larger class of nonlinear problems by not demanding convexity or concavity property. Moreover, other possibilities that have been explored make the method of generalized quasilinearization universally useful in applications [2–8].
The concept of noninteger order derivative, popularly known as fractional derivative goes back to the seventeenth century [9, 10]. Since that time the fractional calculus has drawn the attention of many famous mathematicians. It is only a few decades ago, it was realized that the derivatives of arbitrary order provide an excellent framework for modeling the real world problems in a variety of disciplines. There has been a growing interest in this new area to study the concept of fractional differential equations and fractional dynamical systems [11–18].
The application of quasilinearization for fractional differential equations is a new research area. Depending on development in fractional order differential equations, this technique is reconsidered and similar results parallel to classical theory of differential equation with integer derivatives have been obtained. Recently, only a few papers were published in this direction [19–21].
We consider the following initial value problem
where f ∈ C([t0, T] × ℝ, ℝ) and cDq is the Caputo's fractional derivative of order q, 0 < q < 1.
The corresponding Volterra fractional integral equation is defined as
A large cycle of works have been done in the literature using local Hölder continuity assumption which is needed for comparison theorems, see [16, 17]. For instance, in  the solution of fractional differential equations is obtained by utilizing that condition. Obviously, of great interest is the study of the solution of nonlinear fractional differential equations without using Hölder continuity.
In this work, by not demanding Hölder continuity condition we employe the quasilinearization technique for the given nonlinear fractional order differential equation (1.1) in which upper and lower solutions will have different initial times and positions.
In this section, some basic definitions and theorems used throughout the paper are presented. First, we begin with the definition of the class C p [[t0, T], ℝ].
Definition 2.1. A function σ (t) is called a C p function if σ ∈ C [(t0, T], ℝ] and σ(t) (t - t0) p ∈ C [[t0, T], ℝ] with p = 1 - q.
Next, we give the definition of lower and upper solutions, respectively.
Definition 2.2. A function v ∈C p [[t0, T], ℝ], p = 1 - q, 0 < q < 1 is said to be a lower solution of (1.1) if
It is an upper solution if the inequalities are reversed.
Now, consider the following nonhomogeneous linear fractional differential equation,
where λ is a real number and f ∈ C p ([t0, T] × ℝ, ℝ). The equivalent Volterra fractional integral equation for t0 ≤ t ≤ T is
When we apply the method of successive approximations (see ) to find the solution x(t) = x(t, t0, x0) explicitly for the given nonhomogeneous IVP (2.1), we obtain
are Mittag-Leffler functions of one parameter and two parameters, respectively.
If f(t) ≡ 0, we get, as the solution of the corresponding homogeneous IVP
Remark 2.1. Let cDqu(t) ≤ Lu(t), u(t0) = u0 where u ∈ C p ([t0, T], ℝ+) and L is positive constant. Then we have the estimate
When q = 1, that result reduces to well known Gronwall's inequality. For the proof of this remark and further information about Gronwall's type inequality for fractional order differential equations, one can see .
If u0 = 0, then u (t) = 0 identically on [t0, T].
3 Comparison Theorem
In a recent study , the Hölder continuity assumption is relaxed to C p continuity of the functions involved in the Riemann-Liouville fractional differential equation. In the following we also prove a comparison result by not requiring the Hölder continuity with a different argument for Caputo fractional differential equations. It is obvious that this result is essential to extend the applicability of iterative techniques such as the monotone iterative technique and the method of quasilinearization.
Theorem 3.1. Let v(t), w (t) ∈ C p [[t0, T], ℝ] and f ∈ C[[t0, T] × ℝ, ℝ] and
Suppose further that the standard Lipschitz condition is satisfied
and L > 0.
Then v(t0) ≤ w(t0) implies
Proof. Suppose that v(t) ≤ w(t) for t0 ≤ t ≤ T is not true. Then, there exists a t1> t0 such that v(t1) > w(t1) and t0< t1 ≤ T. Since v(t0) ≤ w(t0) we encounter with two cases in view of the continuity of functions involved:
(i) If v(t0) < w(t0), then one can find a τ0 such that v(τ0) = w(τ0) and t0< τ0< t1. Thus, we have v(t) > w(t) on (τ0, t1].
(ii) If v(t0) = w(t0), then two situations are possible. Namely, one can get v(t) > w(t) on (t0, t1] or v(t) > w(t) on (τ0, t1] where t0< τ0< t1 and v(τ0) = w(τ0) as before.
In both cases, we can find an interval [t0, t1] or [τ0, t1] on which v(t) ≥ w(t).
Let us define
and assume that u (t) is defined on [τ0, t1] (or on [t0, t1]). Note that u (t) > 0 on (τ0, t1] and u (τ0) = 0. Taking Caputo's fractional derivative of both sides of (3.3), we get
By using the lower and upper properties of v (t) and w (t), we have
Since f is Lipschitz with L > 0 and v(t) ≥w(t) on [τ0, t1], we obtain
Therefore we get
which implies, in view of Remark 2.1,
which gives a contradiction. So, we have v(t) ≤ w(t) on [t0, T].
Corollary 3.1 The function f(t, u) = σ(t)u, where σ (t) ≤ L is admissible in Theorem 3.1 to yield u(t) ≤ 0 on t0 ≤ t ≤ T.
Observe that a dual result of corollary 2.1 is valid.
4 Quasilinearization with Initial Time Difference
The purpose of this section is to employ the quasilinearization technique for nonlinear Caputo's fractional order differential equation (1.1) by choosing lower and upper solutions with initial time difference and not imposing the Hölder continuity on functions involved. Also, we consider the function f (t, x) on the right hand side of the equation (1.1) which satisfies a weaker condition than convexity.
Theorem 4.1. Assume that
(i) α ∈ C p [[t0, t0 + T], ℝ], t0, T > 0, β ∈ C p [[τ0, τ0 + T], ℝ], τ0> 0, f ∈ C [[t0, τ0 + T] × ℝ, ℝ] and
with α (t0) ≤ x (s0) ≤ β(τ0) and t0< s0< τ0 where α(t) ≤ β (t + η1), t0 ≤ t ≤ t0 + T and η1 = τ0 - t0;
(ii) Suppose f x (t, x) exists and following relations hold
(iii) f (t, x) is nondecreasing in t for each x and f x (t, x) is nondecreasing in x for each t.
Then there exists monotone sequences and which converge uniformly and monotonically to the unique solution of (1.1) with x (s0) = x0 on [s0, s0 + T] and the convergence is quadratic.
Proof. Let and , t0 ≤ t ≤ t0 + T where η1 = τ0 - t0. Then we have
since f (t, x) is nondecreasing in t for each x, we get
Similarly, we can write
which shows is a lower solution of the problem.
Consider the following linear fractional equations
where η2 = s0 - t0. Note that unique solutions exist since the right hand side of the equations satisfy a Lipschitz condition.
We shall show that
First we must prove
Hence applying Corollary 3.1, we get on [t0, t0 + T]. Similarly, one can show that .
Now we must prove that on [t0, t0 + T]. To do so, we set , then
using the inequality in (ii) we get
this implies that
which because of Corollary 3.1 yields p (t) ≥ 0. Thus we have on [t0, t0 + T]. Hence (4.4) is proved.
Using mathematical induction with k > 1, we obtain
Now we need to show that
To prove this, we set so that utilizing equations in (4.1), (4.2) and the inequality in (ii), we have
Thus we obtain
Again using corollary 3.1 we get on [t0, t0 + T]. In a similar way, it can be shown that on [t0, t0 + T]. Next we must prove on [t0, t0 + T]. Let , then
Thus we have and p (t0) = 0. It follows from the Corollary 3.1 we reach on [t0, t0 +T]. Hence (4.6) is proved.
Employing standard techniques (see ), it can be easily shown that the monotone sequences and converge uniformly and monotonically to the unique solution of
Letting s = t + η2 and changing the variable, we have
Next we will prove that the convergence is quadratic. For this purpose, consider
Note that pn+1(t0) = 0. So we have
where . This inequality gives the estimate
where and E q, q is Mittag-Leffler function.
Thus we reach the desired result
Similarly, after using suitable computation, we get the quadratic convergence of such that
The proof is complete.
Corollary 4.1. If the assumptions of Theorem 4.1 hold with s0 = t0, then the conclusion remains the same.
Proof. For the proof, we let on [t0, t0 + T] and we proceed as we did in Theorem 4.1.
Corollary 4.2. If the assumptions of Theorem 4.1 hold with s0 = τ0, then the conclusion remains the same.
Proof. This time, we must set on [τ0, τ0 + T] proceed as we did in Theorem 4.1.
In case t0> τ0, a dual result of Theorem 4.1 can be proved with some suitable changes. Next result is given in this direction.
Theorem 4.2. Assume that
(i) α ∈ C p [[t0, t0 + T], ℝ], t0, T > 0, β ∈ C p [[τ0, τ0 + T], ℝ], τ0> 0, f ∈ C [[τ0, t0 + T] × ℝ, ℝ] and
with α(t0) ≤ x (s0) ≤ β(τ0) and τ0< s0< t0 where α(t + η1) ≤ β(t), τ0 ≤ t ≤ τ0 + T and η1 = t0 - τ0;
(ii) Suppose f x (t, x) exists and following relations hold
(iii) f (t, x) is nonincreasing in t for each x and f x (t, x) is nondecreasing in x for each t.
Then the conclusion of theorem 4.1 remains valid.
Proof. The proof being similar to theorem 4.1, we omit details.
In this work, the quasilinearization technique coupled with lower and upper solutions is employed to study Caputo fractional differential equations. We have observed that this technique is convenient even though initial functions α and β are given with initial times. In this way, by not requiring Hölder continuity condition, one gets monotone sequences whose iterates are solutions of corresponding linear problems and the sequences converge uniformly and quadratically to the unique solution of the given nonlinear problem.
Lakshmikantham V, Vatsala AS: Generalized Quasilinearization for Nonlinear Problems. Kluwer Academic Publisher, The Netherlands; 1998.
Köksal S, Yakar C: Generalized quasilinearization method with initial time difference simulation. Int J Electr Electron Phys Syst 2002., 24(5):
Yakar C, Yakar A: An extension of the quasilinearization method with initial time difference. Dynamics of continuous, discrete and impulsive systems. Ser A Math Anal 2007, 14(S2):275–279.
Yakar C, Yakar A: Further generalization of quasilinearization method with initial time difference. J Appl Funct Anal 2009, 4(4):714–727.
Maleknejad K, Najafi E: Numerical solution of nonlinear volterra integral equations using the idea of quasilinearization. Commun Nonlinear Sci Numer Simulat 2011, 16: 93–100. 10.1016/j.cnsns.2010.04.002
Amstera Pb, De Nápolia P: A quasilinearization method for elliptic problems with a nonlinear boundary condition. Nonlinear Anal 2007, 66: 2255–2263. 10.1016/j.na.2006.03.016
Kriveca R, Mandelzweig VB: Quasilinearization method and WKB. Comput Phys Commun 2006, 174: 119–126. 10.1016/j.cpc.2004.12.017
Hristova SG, Vatsala AS: Quasilinearization for the periodic boundary value problem for systems of impulsive differential equations. J Appl Math Stochastic Anal 2006, 2006: 25. Article ID 98318
Oldham KB, Spanier J: The Fractional Calculus. Academic Press, New York; 1974.
Samko S, Kilbas A, Marichev O: Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach, New York 1993, 1006.
Metzler R, Schick W, Kilian HG, Nonnenmacher TF: Relaxation in filled polymers: a fractional calculus approach. J Chem Phys 1995, 103(16):7180–7186. 10.1063/1.470346
Hilfe R (Ed): Applications of Fractional Calculus in Physics In World Scientific, River Edge 2000.
Kiryakova V: Generalized fractional calculus and applications. In Pitman Research Notes Mathematics Series. Volume 301. Longman-Wiley, New York; 1994.
Glockle WG, Nonnenmacher TF: A fractional calculus approach to self-similar protein dynamics. Biophys J 1995, 68(1):46–53. 10.1016/S0006-3495(95)80157-8
Podlubny I: Fractional Differential Equations. Academic Press, San Diego; 1999.
Lakshmikantham V, Leela S, Devi Vasundhara J: Theory of Fractional Dynamic Systems. Cambridge Academic Publishers, Cambridge; 2009.
Yakar A: Some generalizations of comparison results for fractional differential equations. Comput Math Appl 2011, 62: 3215–3220. 10.1016/j.camwa.2011.08.035
Yakar A, Koksal ME: Existence results for solutions of nonlinear fractional differential equations. Abst Appl Anal 2012, 2012: 12. Article ID 267108. doi:10.1155/2012/267108
Vasundhar Devi J, Suseela Ch: Quasilinearization for Caputo fractional differential equations. Commun Appl Anal 2008, 12(4):407–418.
Vasundhara Devi J, Mc Rae FA, Drici Z: Generalized quasilinearization for Caputo fractional differential equations. Comput Math Appl 2010, 59(3):1057–1062. 10.1016/j.camwa.2009.05.017
Yakar C, Yakar A: A refinement of quasilinearization method for Caputo's sense fractional order differential equations. Abst Appl Anal 2010., 2010: Article ID 704367. doi:10.1155/2010/704367
Yakar A: Method of Quasilinearization for Nonlinear Fractional Order Differential Equations. PhD Thesis, Gebze Institute of Technology 2010.
Denton Z, Vatsala AS: Fractional integral inequalities and applications. Comput Math Appl 2010, 59: 1087–1094. 10.1016/j.camwa.2009.05.012
Vasundhara Devi J, Mc Rae FA, Drici Z: Variational Lyapunov method for fractional differential equations. Comput Math Appl 2012. doi:10.1016/j.camwa.2012.01.070
This work has been supported by The Scientific and Technological Research Council of Turkey (TÜBİTAK).
The author declares that they have no competing interests.
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Yakar, A. Initial time difference quasilinearization for Caputo Fractional Differential Equations. Adv Differ Equ 2012, 92 (2012). https://doi.org/10.1186/1687-1847-2012-92
- Quasilinearization technique
- Caputo fractional differential equation
- Quadratic convergence