Abstract
In this paper, an analytical technique, namely the new iterative method (NIM), is applied to obtain an approximate analytical solution of the fractional Fornberg–Whitham equation. The obtained approximate solutions are compared with the exact or existing numerical results in the literature to verify the applicability, efficiency, and accuracy of the method.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
Introduction
In recent years, the fractional calculus used in many phenomena in engineering, physics, biology, fluid mechanics, and other sciences [1–8] can be described very successfully by models using mathematical tools from fractional calculus. Fractional derivatives provide an excellent instrument for the description of memory and hereditary properties of various materials and processes [9]. The fractional derivative has been occurring in many physical and engineering problems such as frequency-dependent damping behavior materials, signal processing and system identification, diffusion and reaction processes, creeping and relaxation for viscoelastic materials.
The new iterative method (NIM), proposed first by Gejji and Jafari [10], has proven useful for solving a variety of nonlinear equations such as algebraic equations, integral equations, ordinary and partial differential equations of integer, and fractional order and systems of equations as well. The NIM is simple to understand and easy to implement using computer packages and yield better results than the existing Adomain decomposition method [11], homotopy perturbation method [12], and variational iteration method [13].
In the present paper, we have to solve the nonlinear time-fractional Fornberg–Whitham equation by the NIM. This equation can be written in operator form as:
with the initial condition
where u(x, t) is the fluid velocity, α is constant and lies in the interval (0, 1), t is the time and x is the spatial coordinate. Subscripts denote the partial differentiation unless stated otherwise. Fornberg and Whitham obtained a peaked solution of the form u(x, t) = Ae −1/2(|x−4t/3|) where A is an arbitrary constant.
Preliminaries and notations
In this section, we set up notation and review some basic definitions from fractional calculus [14, 15].
Definition 2.1
A real function f(x), x > 0 is said to be in space C α , α ∊ R if there exists a real number p( > α), such that f(x) = x p f 1(x) where f 1(x) ∊ C[0, ∞).
Definition 2.2
A real function f(x), x > 0 is said to be in space C m α , m ∊ IN ∪ {0} if f (m) ∊ C α .
Definition 2.3
Let f ∊ C α and α ≥ −1, then the (left-sided) Riemann–Liouville integral of order μ, μ > 0 is given by:
Definition 2.4
The (left-sided) Caputo’s fractional derivative of f, f ∊ C m-1 , m ∊ IN ∪ {0}, is defined as:
Note that
Basic idea of new iterative method (NIM)
To describe the idea of the NIM, consider the following general functional equation [10, 16–20]:
where N is a nonlinear operator from a Banach space B → B and f is a known function. We are looking for a solution u of (3) having the series form:
The nonlinear operator N can be decomposed as follows:
From Eqs. (4) and (5), Eq. (3) is equivalent to:
We define the recurrence relation:
Then:
If N is a contraction, i.e.,
then:
and the series ∑ ∞ i=0 u i absolutely and uniformly converges to a solution of (3) [21], which is unique, in view of the Banach fixed point theorem [22]. The k term approximate solution of (3) and (4) is given by ∑ k−1 i=0 u i .
Convergence analysis of the new iterative method (NIM)
Now, we introduce the condition of convergence of the NIM, which is proposed by Daftardar-Gejji and Jafari in (2006) [10], also called (DJM) [23].
From (5), the nonlinear operator N is decomposed as follows:
N(u) = N(u 0) + [N(u 0 + u 1) − N(u 0)] + [N(u 0 + u 1 + u 2) − N(u 0 + u 1)] + ….
Let G 0 = N(u 0) and
Then N(u) = ∑ ∞ i=0 G i .
Set:
Then:
is a solution of the general functional Eq. (3). Also, the recurrence relation (7) becomes
Using Taylor series expansion for G i , i = 1, 2, …, n, we have
In general:
In the following theorem, we state and prove the condition of convergence of the method.
Theorem 3.1
If N is C (∞) in a neighborhood of u 0 and
for any n and for some real L > 0 and \(\left\| {u_{i} } \right\| \le M < \frac{1}{e}\,,\,\,i = 1,2, \ldots ,\) then the series ∑ ∞ n=0 G n is absolutely convergent, and moreover,
Proof
In view of (18)
\(\square\)
Thus, the series ∑ ∞ n=1 ‖G n ‖ is dominated by the convergent series LM(e − 1) ∑ ∞ n=1 (Me)n−1, where \(M < \frac{1}{e}\). Hence, ∑ ∞ n=0 G n is absolutely convergent, due to the comparison test. For more details, see [23].
Reliable algorithm of new iterative method (NIM) for solving the Linear and Nonlinear fractional partial differential equations
After the above presentation of the NIM, we introduce a reliable algorithm for solving nonlinear fractional PDEs using the NIM. Consider the following nonlinear fractional PDE of arbitrary order:
with the initial conditions
where A is a nonlinear function of u and ∂u (partial derivatives of u with respect to x and t) and B is the source function. In view of the integral operators, the initial value problem (22) is equivalent to the following integral equation
where
and
where I n t is an integral operator of n fold. We get the solution of (24) by employing the algorithm (7).
Solution of the problem
We first consider the following time-fractional Fornberg–Whitham equation [24, 25]:
with the initial condition:
Then, the exact solution is given by:
Note that Eq. (27) is equivalent to the integral equation
where \(f = e^{{\frac{x}{2}}}\) and \(N(u) = I_{t}^{\alpha } \left[ {D_{t}^{\alpha } u - D_{{_{xxt} }} u + D_{{_{x} }} u + uD_{{_{xxx} }} u - uD_{x} u + 3D_{x} uD_{xx} u} \right]\), using (7) we get
Therefore, the NIM series solution is:
Numerical results and discussion
In this section, we calculate numerical results of the displacement u(x, t) for different time-fractional Brownian motions α = 2/3, 3/4, 1 and for various values of t and x. The numerical results for the approximate solution (30) obtained using NIM and the exact solution for various values of t, x, and a are shown by Figs. 1, 2, 3, 4 and those for different values of t and α at x = 1 are depicted in Fig. 5. It is observed from Figs. 1, 2, 3, 4 that u(x, t) increases with the increase in both x and t for α = 2/3, 3/4 and α = 1. Figure 4 clearly show that, when α = 1, the approximate solution (30) obtained by the present method is very near to the exact solution. It is also seen from Fig. 5 that as the value of a increases, the displacement u(x, t) increases but afterward its nature is opposite. Finally, we remark that the approximate solution (30) is in full agreement with the results obtained homotopy perturbation method [24] and homotopy perturbation transform method [25]. In Table 1, we compute the absolute errors for differences between the exact solution (29) and the approximate solution (30) obtained by the NIM at some points.
Conclusion
In this paper, the new iterative method (NIM) has been applied for approximating the solution for the nonlinear fractional Fornberg–Whitham equation. The accuracy of the NIM for solving nonlinear fractional Fornberg–Whitham equation is good compared to the literature; however, it has the advantage of reducing the computations complexity presented in other perturbation techniques. In fact, in NIM, nonlinear problems are solved without using Adomian’s polynomials or He’s polynomials that appear in the decomposition methods. The numerical results show that the proposed method is reliable and efficient technique in finding approximate solutions for nonlinear differential equations.
References
Oldham, K.B., Spanier, J.: The fractional calculus. Academic Press, New York (1974)
Podlubny, I.: Fractional differential equations. Academic Press, New York (1999)
Kumar, S.: A numerical study for solution of time fractional nonlinear shallow-water equation in oceans. Zeitschrift fur Naturforschung A 68a, 1–7 (2013)
Kumar, S.: A new analytical modelling for telegraph equation via Laplace transform. Appl. Math. Model. 38(13), 3154–3163 (2014)
Kumar, S., Rashidi, M.M.: New analytical method for gas dynamics equation arising in shock fronts. Comput. Phys. Commun. 185(7), 1947–1954 (2014)
Kumar, S.: A fractional model to describing the Brownian motion of particles and its analytical solution. Adv. Mech. Eng. 7(12), 1–11 (2015)
Kumar, S.: A modified homotopy analysis method for solution of fractional wave equations. Adv. Mech. Eng. 7(12), 1–8 (2015)
Kumar, S.: An analytical algorithm for nonlinear fractional Fornberg–Whitham equation arising in wave breaking based on a new iterative method. Alex. Eng. J. 53(1), 225–231 (2014)
Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and applications of fractional differential equations. Elsevier, Amsterdam (2006)
Gejji, V.D., Jafari, H.: An iterative method for solving nonlinear functional equations. J. Math. Anal. Appl. 316(2), 753–763 (2006)
Adomian, G.: Solving Frontier problems of physics. The decomposition method. Kluwer, Boston (1994)
He, J.H.: Homotopy perturbation technique. Comput. Methods Appl. Mech. Eng. 178, 257–262 (1999)
He, J.H.: Variational iteration method-akind of nonlinear analytical technique: some examples. Int. J. Nonlinear Mech. 34, 699–708 (1999)
Podlubny, I.: Fractional differential equations. Academic Press, San Diego (1999)
Luchko, Yu., Gorenflo, R.: An operational method for solving fractional differential equations with the Caputo derivatives. Acta Math. Vietnamica 24, 207–233 (1999)
Al-luhaibi, M.S.: New iterative method for fractional gas dynamics and coupled burger’s equations, The Scientific World Journal, Volume 2015, Article ID 153124, 8 pages
Ramadan, M.A., Al-luhaibi, M.S.: New iterative method for Cauchy problems. J. Math. Comput. Sci. 5(6), 826–835 (2015)
Ramadan, M.A., Al-luhaibi, M.S.: New iterative method for solving the Fornberg–Whitham equation and comparison with homotopy perturbation transform method. Br. J. Math. Comput. Sci. 4(9), 1213–1227 (2014)
Hemeda, A.A., Al-luhaibi, M.S.: New iterative method for solving gas dynamic equation. Int. J. Appl. Math. Res. 3(2), 190–195 (2014)
Hemeda, A.A.: New iterative method: an application for solving fractional physical differential equations, Abstract and Applied Analysis Volume 2013, Article ID 617010, 9 pages
Cherruault, Y.: Convergence of Adomian’s method. Kybernetes 18(2), 31–38 (1989)
Jerri, A.J.: Introduction to integral equations with applications, 2nd edn. Wiley-Interscience, New York (1999)
Bhalekar, S., Gejji, V.D.: Convergence of the new iterative method. Int. J. Differ. Equ. 2011, Article ID 989065, 10 pages
Guptaa, P.K., Singh, M.: Homotopy perturbation method for fractional Fornberg–Whitham equation. Comput. Math Appl. 61, 250–254 (2011)
Singh, J., Kumar, D., Kumar, S.: New treatment of fractional Fornberg–Whitha equation via Laplace transform. Ain Shams Eng. J. 4, 557–562 (2013)
Acknowledgements
I am grateful to the prof. Mohamed A. Ramadan (Department of Mathematics, Faculty of Science, Menoufia University, Egypt) and referees for their helpful comments and suggestions that enhanced the paper.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Al-luhaibi, M.S. An analytical treatment to fractional Fornberg–Whitham equation. Math Sci 11, 1–6 (2017). https://doi.org/10.1007/s40096-016-0198-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40096-016-0198-5