An investigation with Hermite Wavelets for accurate solution of Fractional Jaulent–Miodek equation associated with energy-dependent Schrödinger potential

doi:10.1016/j.amc.2015.08.058

Applied Mathematics and Computation

Volume 270, 1 November 2015, Pages 458-471

https://doi.org/10.1016/j.amc.2015.08.058 Get rights and content

Abstract

In the present paper, a wavelet method based on the Hermite wavelet expansion along with operational matrices of fractional derivative and integration is proposed for finding the numerical solution to a coupled system of nonlinear time-fractional Jaulent–Miodek (JM) equations. Consequently, the approximate solutions of fractional Jaulent–Miodek equations acquired by using Hermite wavelet technique were compared with those derived by using optimal homotopy asymptotic method (OHAM) and exact solutions. The present proposed numerical technique is easy, expedient and powerful in computing the numerical solution of coupled system of nonlinear fractional differential equations like Jaulent–Miodek equations.

Introduction

The calculus of integrals and derivatives of arbitrary real or complex order (generally known as fractional calculus) has attained widespread popularity in the past few decades or so, mostly due to its demonstrated implementation in various fields of engineering and science.

With the progress of science and engineering, nonlinear fractional differential equations had been used as the models to describe real physical phenomena in solid state physics, plasma waves, fluid mechanics, chemical physics and so forth. Thus, for the last few decades, huge attention has been focused for finding the solutions (both analytical and numerical) of these problems.

Consider the following time-fractional coupled Jaulent–Miodek (JM) equations $\frac{\partial^{α} u}{\partial t^{α}} + \frac{\partial^{3} u}{\partial x^{3}} + \frac{3}{2} v \frac{\partial^{3} v}{\partial x^{3}} + \frac{9}{2} \frac{\partial v}{\partial x} \frac{\partial^{2} v}{\partial x^{2}} - 6 u \frac{\partial u}{\partial x} - 6 u v \frac{\partial v}{\partial x} - \frac{3}{2} \frac{\partial u}{\partial x} v^{2} = 0,$ $\frac{\partial^{α} v}{\partial t^{α}} + \frac{\partial^{3} v}{\partial x^{3}} - 6 \frac{\partial u}{\partial x} v - 6 u \frac{\partial v}{\partial x} - \frac{15}{2} \frac{\partial v}{\partial x} v^{2} = 0,$ which is associated with energy-dependent Schrödinger potential [1], [2], [3]. Here 0 < α ≤ 1 , is the parameter representing the order of the fractional derivative, deemed in the Caputo sense.

Systems of nonlinear partial differential equations [4], [5] come up in lots of scientific physical models. In contemporary years, significant research has been done to study the classical Jaulent–Miodek equations. Various methods such as unified algebraic method [6], Adomian decomposition method [7], tanh-sech method [8], homotopy perturbation method [9], Exp-function method [10], and homotopy analysis method [11] had been implemented for solving of coupled Jaulent–Miodek equations. But in keeping with the available information, the comprehensive analysis of the nonlinear fractional order coupled Jaulent–Miodek equation is only an initiation.

This present article is dedicated to study the fractional coupled Jaulent–Miodek equation. Hence, it emphasizes on the implementation of two-dimensional Hermite wavelet method to solve the problem of time-fractional coupled Jaulent–Miodek equation. With a view to exhibit the capabilities of the methods, we employ these methods to deal with fractional order coupled Jaulent–Miodek equations. The approximate solutions attained via Hermite wavelet technique were compared with exact solutions and those derived by using OHAM in case of fractional order.

The structure of the paper is as follows: a formal outline to fractional calculus has been supplied in Section 2 for the particular intend of the paper. In Section 3, the mathematical introductions of Hermite wavelet is proposed. The function approximation utilizing Hermite wavelet and the operational matrices are illustrated in Sections 4 and 5 respectively. In Section 6, fundamental concept of optimal homotopy asymptotic method (OHAM) is studied. In Section 7, two-dimensional Hermite wavelet approach has been implemented to determine the numerical approximate solutions for the fractional coupled Jaulent–Miodek equations. Evaluation of proposed procedure with regard to OHAM for solution of fractional coupled Jaulent–Miodek equations has been mentioned in Section 8. The convergence of Hermite wavelet has been studied in Section 9. The numerical results and discussions are examined in Section 10 and Section 11 accomplishes the paper.

Section snippets

Fractional derivative and integration

Definition

The Riemann–Liouville fractional integral operator J^α (α > 0) of a function f(t), is defined as [12], [13] $J^{α} f (t) = \frac{1}{Γ (α)} \int_{0}^{t} {(t - τ)}^{α - 1} f (τ) d τ, α > 0 and α \in ℜ^{+}$

Some of the properties of the operator J^α are as follows $J^{α} t^{γ} = \frac{Γ (1 + γ)}{Γ (1 + γ + α)} t^{α + γ}, (γ > - 1)$ $J^{α} J^{β} f (t) = J^{α + β} f (t), (α > 0, β > 0)$

Definition

The Caputo fractional derivative ₀D_t^α of a function f(t) is defined as [12], [13] $_{0} {D_{t}}^{α} f (t) = \frac{1}{Γ (n - α)} \int_{0}^{t} \frac{f^{(n)} (τ)}{{(t - τ)}^{α - n + 1}} d τ, (n - 1 < α \leq n, n \in N)$

Some properties of the Caputo fractional derivative are as follows: $_{0} {D_{t}}^{α} t^{β} = \frac{Γ (1 + β)}{Γ (1 + β - α)} t^{β - α}, 0 < α < β + 1, β$

Hermite wavelets

The Hermite polynomials H_m(x) of order m are defined on the interval $(- \infty, \infty)$ , and can be deduced with the assistance of the following recurrence formulae: $\begin{matrix} H_{0} (x) & = & 1, \\ H_{1} (x) & = & 2 x, \\ H_{m + 1} (x) & = & 2 x H_{m} (x) - 2 m H_{m - 1} (x), m = 1, 2, 3, \dots \end{matrix}$

The Hermite polynomials H_m(x) are orthogonal with respect to the weight function $e^{- x^{2}}$ .

The Hermite wavelets are defined on interval [0, 1) by [14] $ψ_{n, m} (x) = {\begin{matrix} 2^{k / 2} \sqrt{\frac{1}{n! 2^{n} \sqrt{π}}} H_{m} (2^{k} x - \hat{n}), & for \frac{\hat{n} - 1}{2^{k}} \leq x < \frac{\hat{n} + 1}{2^{k}} \\ 0, & otherwise \end{matrix}$ where $k = 1, 2, \dots$ , is the level of resolution, $n = 1, 2, \dots, 2^{k - 1}, \hat{n} = 2 n - 1$ is the translation

Function approximation

A function f(x, t) defined over [0, 1) × [0, 1)can be extended in terms of Hermite wavelet as [15] $f (x, t) = \sum_{n = 1}^{\infty} \sum_{i = 0}^{\infty} \sum_{l = 1}^{\infty} \sum_{j = 0}^{\infty} c_{n, i, l, j} ψ_{n, i, l, j} (x, t)$

If the infinite series in Eq. (4.1) is truncated, then it can be written as $f (x, t) ≅ \sum_{n = 1}^{2^{k_{1} - 1}} \sum_{i = 0}^{M_{1} - 1} \sum_{l = 1}^{2^{k_{2} - 1}} \sum_{j = 0}^{M_{2} - 1} c_{n, i, l, j} ψ_{n, i, l, j} (x, t) = Ψ^{T} (x) C Ψ (t),$ where Ψ(x) and Ψ(t) are $2^{k_{1} - 1} M_{1} \times 1$ and $2^{k_{2} - 1} M_{2} \times 1$ matrices respectively, given by $\begin{matrix} Ψ (x) & \equiv & {[ψ_{1, 0} (x), ψ_{1, 1} (x), \dots, ψ_{1, M_{1} - 1} (x), ψ_{2, 0} (x), \dots, ψ_{2, M_{1} - 1} (x), \dots, ψ_{2^{k_{1} - 1}, 0} (x), \dots, ψ_{2^{k_{1} - 1}, M_{1} - 1} (x)]}^{T} \\ Ψ (t) & \equiv & [ψ_{1, 0} (t), ψ_{1, 1} (t), \dots, ψ_{1, M_{2} - 1} (t), \end{matrix}$

Operational matrix of the general order integration [16]

The integration of $Ψ (t) \equiv {[ψ_{1, 0} (t), \dots, ψ_{1, M - 1} (t), ψ_{2, 0} (t), \dots, ψ_{2, M - 1} (t), \dots, ψ_{2^{k - 1}, 0} (t), \dots, ψ_{2^{k - 1}, M - 1} (t)]}^{T}$ can also be approximated via $\int_{0}^{t} Ψ (τ) d τ ≅ Q Ψ (t),$ where Q is known as the Hermite wavelet operational matrix of integration. To derive the Hermite wavelet operational matrix of the general order of integration, let us recall Eq. (2.1), the fractional integral of order α( > 0)which is defined by Podlubny [12].

The Hermite wavelet operational matrix Q^α for integration of the general order α is given by $Q^{α} Ψ (t) =$

Fundamental concept of optimal homotopy asymptotic method (OHAM)

To demonstrate the elementary notion of optimal homotopy asymptotic method [17], [18], [19], we consider the general nonlinear differential equation as follows $A (u (x, t)) + g (x, t) = 0, x \in Ω$ with boundary conditions $B (u, \frac{\partial u}{\partial t}) = 0, x \in Γ$ where u(x, t) is an unknown function, g(x, t) is a known analytic function, A is a differential operator, B is a boundary operator and Γ is the boundary of the domain Ω.

The operator A can be disseminated as $A = L + N$ where L is the linear and N is the nonlinear operator.

A homotopy ϕ

Two-dimensional Hermite wavelets for solving time-fractional coupled Jaulent–Miodek equations

To demonstrate the accuracy and efficiency of the proposed numerical technique, we consider time-fractional coupled Jaulent–Miodek equation. The numerical approximate solutions thus achieved are compared with the exact solutions in case of classical order and with the solutions obtained by OHAM in case of fractional order respectively.

Consider the nonlinear time-fractional coupled Jaulent–Miodek equation $\frac{\partial^{α} u}{\partial t^{α}} + \frac{\partial^{3} u}{\partial x^{3}} + \frac{3}{2} v \frac{\partial^{3} v}{\partial x^{3}} + \frac{9}{2} \frac{\partial v}{\partial x} \frac{\partial^{2} v}{\partial x^{2}} - 6 u \frac{\partial u}{\partial x} - 6 u v \frac{\partial v}{\partial x} - \frac{3}{2} \frac{\partial u}{\partial x} v^{2} = 0,$ $\frac{\partial^{α} v}{\partial t^{α}} + \frac{\partial^{3} v}{\partial x^{3}} - 6 \frac{\partial u}{\partial x} v - 6 u \frac{\partial v}{\partial x} -$

To compare with OHAM for solution of nonlinear time-fractional coupled Jaulent–Miodek equation

Implementing optimal homotopy asymptotic method [19], the homotopy for Eqs. (7.1) and (7.2) can be written as $\begin{matrix} (1 - p) L (ϕ (x, t; p)) = H (p) [\frac{\partial^{α} ϕ (x, t; p)}{\partial t^{α}} + \frac{\partial^{3}}{\partial x^{3}} ϕ (x, t; p) + \frac{3}{2} ψ (x, t; p) \frac{\partial^{3} ψ (x, t; p)}{\partial x^{3}} \\ + \frac{9}{2} \frac{\partial ψ (x, t; p)}{\partial x} \frac{\partial^{2} ψ (x, t; p)}{\partial x^{2}} - 6 ϕ (x, t; p) \frac{\partial ϕ (x, t; p)}{\partial x} \\ - 6 ϕ (x, t; p) ψ (x, t; p) \frac{\partial ψ (x, t; p)}{\partial x} - \frac{3}{2} \frac{\partial ϕ (x, t; p)}{\partial x} {(ψ (x, t; p))}^{2}] \end{matrix}$ $\begin{matrix} (1 - p) L (ψ (x, t; p)) = \tilde{H} (p) [\frac{\partial^{α} ψ (x, t; p)}{\partial t^{α}} + \frac{\partial^{3}}{\partial x^{3}} ψ (x, t; p) - 6 \frac{\partial ϕ (x, t; p)}{\partial x} ψ (x, t; p) \\ - 6 ϕ (x, t; p) \frac{\partial ψ (x, t; p)}{\partial x} - \frac{15}{2} \frac{\partial ψ (x, t; p)}{\partial x} {(ψ (x, t; p))}^{2}] \end{matrix}$ where $ϕ (x, t; p) = u_{0} (x, t) + \sum_{i = 1}^{\infty} u_{i} (x, t) p^{i}$ $ψ (x, t; p) = v_{0} (x, t) + \sum_{i = 1}^{\infty} v_{i} (x, t) p^{i}$ $H (p) = p C_{1} + p^{2} C_{2} + p^{3} C_{3} + \dots$ $\tilde{H} (p) = p$

Convergence of Hermite wavelet

Theorem 9.1

(Convergence theorem)

If a continuous function u(x, t) ∈ L²(ℜ × ℜ) defined on [0, 1) × [0, 1) be bounded, i.e. |u(x, t)| ≤ K, then the Hermite wavelets expansion of u(x, t) converges uniformly to it.

Proof

Let u(x, t) be a function defined on [0, 1) × [0, 1) and|u(x, t)| ≤ K, where K is a positive constant.

The Hermite wavelet coefficients of continuous functions u(x, t) are defined as $\begin{matrix} c_{i j} & = & \int_{0}^{1} \int_{0}^{1} u (x, t) ψ_{i} (x) ψ_{j} (t) d x d t \\ = & 2^{\frac{k_{1}}{2}} \sqrt{\frac{1}{n_{1}! 2^{n_{1}} \sqrt{π}}} \int_{0}^{1} \int_{I_{1}} u (x, t) H_{m_{1}} (2^{k_{1}} x - {\hat{n}}_{1}) ψ_{j} (t) d x d t, \end{matrix}$ where $I_{1} = [\frac{{\hat{n}}_{1} - 1}{2^{k_{1}}}, \frac{{\hat{n}}_{1} + 1}{2^{k_{1}}})$

Now by

Numerical results and discussion

The comparison of the absolute errors for nonlinear time-fractional coupled Jaulent–Miodek Eqs. (7.1) and (7.2) have been exemplified in Tables 1 and 2 that are generated through the results attained by two-dimensional Hermite wavelet method and OHAM at various points of x and t taking $α = 1$ . In the present study, in order to inspect the accuracy and reliability of Hermite wavelets for solving fractional order nonlinear coupled system of Jaulent–Miodek equations, we compare the numerical

Conclusion

In this article, the time-fractional coupled Jaulent–Miodek equations have been solved with the aid of making use of two-dimensional Hermite wavelet process. The results thus obtained are then compared with exact solutions as well as with optimal homotopy asymptotic method (OHAM). The obtained results demonstrate the efficiency, accuracy and reliability of the proposed algorithm based on two-dimensional Hermite wavelet approach and its applicability to nonlinear time-fractional coupled

Acknowledgments

This research work was financially supported by DST, Government of India under Grant No. SR/S4/MS.:722/11. Again the authors would like convey their sincere thanks to learned anonymous reviewers for their useful comments for the improvement and betterment of the manuscript.

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An investigation with Hermite Wavelets for accurate solution of Fractional Jaulent–Miodek equation associated with energy-dependent Schrödinger potential

Abstract

Introduction

Section snippets

Fractional derivative and integration

Hermite wavelets

Function approximation

Operational matrix of the general order integration [16]

Fundamental concept of optimal homotopy asymptotic method (OHAM)

Two-dimensional Hermite wavelets for solving time-fractional coupled Jaulent–Miodek equations

To compare with OHAM for solution of nonlinear time-fractional coupled Jaulent–Miodek equation

Convergence of Hermite wavelet

Numerical results and discussion

Conclusion

Acknowledgments

Chaos Soliton. Fract.

Chaos Soliton. Fract.

Phys. Lett. A

Phys. Lett. A

Phys. Lett. A

Appl. Math. Lett.

Comput. Fluids

Appl. Math. Comput.

A direct perturbation method: nonlinear Schrodinger equation with loss

Chin. Phys. Lett.

Nonlinear fractional Jaulent-Miodek and Whitham-Broer-Kaup equations with Sumudu transform

Abstr. Appl. Anal.

Partial differential equations and solitary waves theory