# Numerical Methods for Fractional-Order Fornberg-Whitham Equations in the Sense of Atangana-Baleanu Derivative

Authors: Naveed Iqbal; Humaira Yasmin; Akbar Ali; Abdul Bariq; M. Mossa Al-Sawalha; Wael W. Mohammed
Journal: Journal of Function Spaces
 (2021)

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## Abstract

In this paper, we investigate the numerical solution of the Fornberg-Whitham equations with the help of two powerful techniques: the modified decomposition technique and the modified variational iteration technique involving fractional-order derivatives with Mittag-Leffler kernel. To confirm and illustrate the accuracy of the proposed approach, we evaluated in terms of fractional order the projected models. Furthermore, the physical attitude of the results obtained has been acquired for the fractional-order different value graphs. The results demonstrated that the future method is easy to implement, highly methodical, and very effective in analyzing the behavior of complicated fractional-order linear and nonlinear differential equations existing in the related areas of applied science.


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## Body

## 1. Introduction

The analysis of the Fornberg-Whitham equation (FWE) is a significant mathematical equation of mathematical physics. The Fornberg-Whitham equation is defined as [1, 2]
(1)DIμ−DζζIμ+Dζμ=μDζζζμ−μDζμ+3DζμDζζμ.This model was invented to evaluate the nonlinear breaking dispersive ocean waves. The Fornberg-Whitham equation is shown to yield peakon solutions as a physical equation for waves of restricting height and the occurrences of wave breaking. Fractional calculus is now widely used and accepted, owing to its well-known uses in a variety of fields of seemingly disparate sectors of science and engineering [3, 4]. Many scholars, including Gupta and Singh [5] and Alderremy et al. [6], have examined the fractional of the Fornberg-Whitham equation relevant to the fractional Caputo derivative, Sunthrayuth et al. [7], Singh et al. [8], etc. Because of the singular kernel of the fractional Caputo derivative, its implementations are limited. Caputo and Fabrizio [9] created derivatives of any (real or complex) order with a nonsingular kernel. Caputo and Fabrizio’s derivative has been used to a variety of real-world situations, including fractional nonhomogeneous heat models [10], El Nino-Southern fractional oscillations models [11], and arbitrary-order system of smoking models [12]. Atangana and Baleanu [13] devised a novel fractional-order derivative called the Atangana-Baleanu (AB) fractional derivative, which has the kernel of a Mittag-Leffler-type function. Kumar et al. [14] investigated the regularised long-wave equation with a Mittag-Leffler-type kernel incorporating the fractional operator. A Mittag-Leffler-type kernel is used in the chemical kinetics system connected with a fractional derivative which was investigated by Singh et al. [15]. Baleanu et al. [16] recently proposed optimal fractional models with nonsingular Mittag-Leffler kernels. As we all know, the Mittag-Leffler function is more beneficial in expressing physical difficulties than the power function or the exponential function; as a result, the AB fractional derivative is well suited to unravelling material heterogeneities and structures or media with different scales.George Adomian was introduced to and established a technique for “solving integro-differential, differential equations, delay differential, and partial differential equations” [17, 18]. The result is discovered as an infinite sequence that quickly converges to precise solution. This method has been shown to be effective in solving both linear and nonlinear models. The method for solving a nonlinear operator problem is to use decomposition equations in a series of functions. Each expression’s sequence is derived from a polynomial derived from the expansion of an approximate solution into power sequences. The Adomian decomposition method technique is really simple in theory, but the difficulty arises when it comes to determining polynomials and illustrating the convergence of a series of functions [19]. Lesnic [20] analyzed the convergent of the Adomian decomposition method when using heat and wave models for both backward and forward time evolution. Gaber and El-Sayed used the Adomian method of solving fractal-order partial differential equations on a finite domain in [21]. Ghoreishi et al. [22] investigated the Adomian decomposition method’s ability to investigate nonlinear wave problems with changing coefficients, demonstrating that the Adomian decomposition method can solve these equations without the need for dissertation, linearization, transformation, or perturbation.The variational iteration approach [23, 24] was published in the late 1990s to solve a seepage flow with fractional derivatives and a nonlinear oscillator, and it has since been widely utilized as a primary analytical tool for solving a variety of nonlinear problems. It has fully grown into a fully fledged mathematical approach as a result of considerable research by a number of authors, including He [25, 26], Ganji and Sadighi [27], Ozis and Yildirim [28], and Noor and Mohyud-Din [29]. On November 24, 2018, we searched Clarivate’s Web of Science for “variational iteration approach” and got 3761 hits. The technique’s identification of the Lagrange multiplier necessitates the understanding of variational theory [30], and the technique’s sophisticated identification process may limit its implementation to real-world issues.


## 2. Preliminary Concepts

Definition 1.
The Caputo fractional derivative is given as [31]
(2)D0CIγfζ,I=1Γn−γ∫0II−θn−γ−1fnζ,θdθ,n−1<γ≤n.Definition 2.
The Laplace transformation connected with fractional Caputo derivativeLCDIγfI is expressed by [31]
(3)LLCDIγfIs=sγLfx,Is−sγ−1fx,0.Definition 3.
In Caputo sense, the Atangana-Baleanu derivative is defined as [31]
(4)ABCDIγfI=Aγ1−γ∫aIf′kEγ−γ1−γ1−kγdk,where Aγ is a normalization function such that A0=A1=1,f∈H1a,b,b>a,γ∈0,1, and Eγ represent the Mittag-Leffler function.Definition 4.
The Atangana-Baleanu derivative in the Riemann-Liouville sense is defined as [31]
(5)ABCDIγfI=Aγ1−γddI∫aIfkEγ−γ1−γ1−kγdk.Definition 5.
The Laplace transform connected with the Atangana-Baleanu operator is defined as [31]
(6)ABDIγfIs=AγsγLfIs−sγ−1f01−γsγ+γ/1−γ.Definition 6.
Consider0<γ<1, and f is a function of γ; then, the fractional-order integral operator of γ is given as [31]
(7)ABCIIγfI=1−γAγfI+γAγΓγ∫aIfkI−kγ−1dk.


## 3. The Methodology of Variational Iteration Method

This section introduces the solution of fractional partial differential equations with the help of the variational iteration method.(8)ABCDIγνζ,I+G¯ζ,I+Nζ,I−Pζ,I=0,ϕ−1<γ≤ϕ.The initial condition is
(9)νζ,0=gζ,where ABCDIγ=∂γ/∂Iγ is the fractional derivative Caputo order γ, G¯ and N are linear and nonlinear terms, respectively, and P is the source function.The Laplace transformation is applied to equation (8); we get
(10)LABCDIγνζ,I+LG¯ζ,I+Nζ,I−Pζ,I=0.The Lagrange multiplier iterative method is(11)LABCDIγνζ,I+LG¯ζ,I+Nζ,I−Pζ,I=0.A Lagrange multiplier is as(12)λs=−sγ1−γ+γsγ.Applying inverse Laplace transformL−1, equation (11) can be written as
(13)νϕ+1ζ,I=νϕζ,I−L−1sγ1−γ+γsγ−LG¯ζ,I+Nζ,I−LPζ,I.


## 4. The Conceptualization of MDM

In this section, we discuss the solution of fractional partial differential equations with the help of the modified decomposition method.(14)ABCDIγνζ,I+G¯ζ,I+Nζ,I−Pζ,I=0,m−1<γ≤m.The initial condition is
(15)νζ,0=gζ,where ABCDIγ=∂γ/∂Iγ is the fractional derivative of Caputo order γ, G¯ and N are linear and nonlinear terms, respectively, and P is the source term.Using Laplace transformation to equation (14), we get
(16)LABCDIγνζ,I+LG¯ζ,I+Nζ,I−Pζ,I=0.Taking the Laplace transform of differentiation property, we have(17)Lνζ,I=1sνζ,0+sγ1−γ+γsγLPζ,I−sγ1−γ+γsγLG¯ζ,I+Nζ,I.MDM result of infinite seriesνζ,I,
(18)νζ,I=∑ϕ=0∞νϕζ,I.N nonlinear function is defined as
(19)Nζ,I=∑ϕ=0∞Aϕ.The nonlinear terms can be analyzed with the aid of Adomian polynomials. So the Adomian polynomial formula is expressed as(20)Aϕ=1j!∂ϕ∂λϕN∑ϕ=0∞λϕνϕλ=0.Then, put equations (18) and (19) into (17), which gives
(21)L∑ϕ=0∞νϕζ,I=1sνζ,0+sγ1−γ+γsγLPζ,I−sγ1−γ+γsγLG¯∑ϕ=0∞νϕ+∑ϕ=0∞Aϕ.Applying the inverse Laplace transformation to equation (21), we get
(22)∑ϕ=0∞νϕζ,I=L−11sνζ,0+sγ1−γ+γsγLPζ,I−sγ1−γ+γsγLG¯∑ϕ=0∞νϕ+∑ϕ=0∞Aϕ.Define the terms as follows:(23)ν0ζ,I=L−11sνζ,0+sγ1−γ+γsγLPζ,I,ν1ζ,I=−L−1sγ1−γ+γsγLG¯1ν0+A0.In general,ϕ≥1 is defined as
(24)νϕ+1ζ,I=−L−1sγ1−γ+γsγLG¯νϕ+Aϕ.


## 5. Application of Techniques

Example 7.
Consider the time-fractional nonlinear Fornberg-Whitham equation(25)DIγν−DζζIν+Dζν=νDζζζν−νDζν+3DζνDζζν,0<γ≤1,with the initial condition
(26)νζ,0=eζ/2.Taking Laplace transformation of (25),
(27)sγsγ1−γ+γLνζ,I−1sνζ,0=LDζζIν−Dζν+νDζζζν−νDζν+3DζνDζζν.Using inverse Laplace transformation(28)νζ,I=L−1νζ,0s−sγ1−γ+γsγLDζζIν−Dζν+νDζζζν−νDζν+3DζνDζζν.Applying Adomian procedure, we have(29)ν0ζ,I=L−1νζ,0s=L−1eζ/2s,ν0ζ,I=eζ/2,∑ϕ=0∞νϕ+1ζ,I=L−1sγ1−γ+γsγL∑ϕ=0∞DζζIνϕ−∑ϕ=0∞Dζνϕ+∑ϕ=0∞Aϕ−∑ϕ=0∞Bϕ+3∑ϕ=0∞Cϕ,ϕ=0,1,2,⋯,A0νDζζζν=ν0Dζζζν0,B0νDζν=ν0Dζν0,A1νDζζζν=ν0Dζζζν1+ν1Dζζζν0,B1νDζν=ν0Dζν1+ν1Dζν0,A2νDζζζν=ν1Dζζζν2+ν1Dζζζν1+ν2Dζζζν0,B2νDζν=ν1Dζν2+ν1Dζν1+ν2Dζν0,C0DζνDζζν=Dζν0Dζζν0,C1DζνDζζν=Dζν0Dζζν1+Dζν1Dζζν0,C2DζνDζζν=Dζν1Dζζν2+Dζν1Dζζν1+Dζν2Dζζν0,for ϕ=1,
(30)ν1ζ,I=L−1sγ1−γ+γsγLDζζIν0−Dζν0+A0−B0+3C0=−12eζ/21−γ+γIγΓγ+1,for ϕ=2,
(31)ν2ζ,I=L−1sγ1−γ+γsγLDζζIν1−Dζν1+A1−B1+3C1,ν2ζ,I=−18eζ/2I2γ−1Γ2γ+14eζ/21−γ2+γ2I2γΓ2γ+1+21−γγIγΓγ+1,for ϕ=3,
(32)ν3ζ,I=L−1sγ1−γ+γsγLDζζIν2−Dζν2+A2−B2+3C2,ν3ζ,I=−132eζ/2I3γ−2Γ3γ−1+18eζ/2I3γ−1Γ3γ−18eζ/21−γ3+γ1−γ1+γ+2γ2IγΓγ+1+3γ21−γI2γΓ2γ+1+γ3Γ2γ+1I3γΓ3γ+1.The modified decomposition method solution of example (1) is
(33)νζ,I=eζ/2−12eζ/21−γ+γIγΓγ+1−18eζ/2I2γ−1Γ2γ+14eζ/21−γ2+γ2I2γΓ2γ+1+21−γγIγΓγ+1−132eζ/2I3γ−2Γ3γ−1+18eζ/2γ3γ−1Γ3γ−18eζ/21−γ3+γ1−γ1+γ+2γ2IγΓγ+1+3γ21−γI2γΓ2γ+1+γ3Γ2γ+1I3γΓ3γ+1−⋯.The simplification of equation (33)
(34)νζ,I=eζ/21−21−γ+γIγΓγ+1−18I2γ−1Γ2γ+141−γ2+γ2I2γΓ2γ+1+21−γγIγΓγ+1−132I3γ−2Γ3γ−1+18I3γ−1Γ3γ−181−γ3+γ1−γ1+γ+2γ2IγΓγ+1+3γ21−γI2γΓ2γ+1+γ3Γ2γ+1I3γΓ3γ+1+⋯.Apply the variational method to obtain series form solution. The iteration formulas for equation (25), we get
(35)νϕ+1ζ,I=νjζ,I−L−1sγ1−γ+γsγLDζζIνϕ+Dζνϕ−νϕDζζζνϕ+νϕDζνϕ−3DζνϕDζζνϕ,where(36)ν0ζ,I=eζ/2.Forϕ=0,1,2,⋯,
(37)ν1ζ,I=ν0ζ,I−L−1sγ1−γ+γsγLDζζIν0+Dζν0−ν0Dζζζν0+ν0Dζν0−3Dζν0Dζζν0,ν1ζ,I=eζ/2−12eζ/21−γ+γIγΓγ+1,ν2ζ,I=ν1ζ,I−L−1sγ1−γ+γsγLDζζIν1+Dζν1−ν1Dζζζν1+ν1Dζν1−3Dζν1Dζζν1,ν2ζ,I=eζ/2−12eζ/21−γ+γIγΓγ+1−18eζ/2I2γ−1Γ2γ+14eζ/21−γ2+γ2I2γΓ2γ+1+21−γγIγΓγ+1,ν3ζ,I=ν2ζ,I−L−1sγ1−γ+γsγLDζζIν2+Dζν2−ν2Dζζζν2+ν2Dζν2−3Dζν2Dζζν2,ν3ζ,I=eζ/2−12eζ/21−γ+γIγΓγ+1−18eζ/2I2γ−1Γ2γ+14eζ/21−γ2+γ2I2γΓ2γ+1+21−γγIγΓγ+1−132eζ/2I3γ−2Γ3γ−1+18eζ/2I3γ−1Γ3γ−18eζ/21−γ3+γ1−γ1+γ+2γ2IγΓγ+1+3γ21−γI2γΓ2γ+1+γ3Γ2γ+1I3γΓ3γ+1,νζ,I=eζ/2−12eζ/21−γ+γIγΓγ+1−18eζ/2I2γ−1Γ2γ+14eζ/21−γ2+γ2I2γΓ2γ+1+21−γγIγΓγ+1−132eζ/2I3γ−2Γ3γ−1+18eζ/2γ3γ−1Γ3γ−18eζ/21−γ3+γ1−γ1+γ+2γ2IγΓγ+1+3γ21−γI2γΓ2γ+1+γ3Γ2γ+1I3γΓ3γ+1−⋯.The exact solution of equation (25) at γ=1,
(38)νζ,I=eζ/2−2I/3.In Figure1, the analytical results of MDM/MVITM example 1 graphs show close contact with each other at γ=1 and 0.8. It is investigated that analytical results are in close relation with the actual results of example 1. In Figure 2, the results of example 1 at different fractional-order of the derivative are plotted at γ=0.6 and 0.4. Figure 3 shows the different fractional of two and three dimensional. The graphical representation has shown the convergence phenomena of fractional-order results towards the result at integer-order of example 1.Example 8.
Consider the time-fractional nonlinear Fornberg-Whitham equation is given as(39)DIγν−DζζIν+Dζν=νDζζζν−νDζν+3DζνDζζν,I>0,0<γ≤1.The initial condition is
(40)νζ,0=cosh2ζ4.Figure 1
The solution graph of MDM/MVITM atγ=1 and 0.8 of example 1.Figure 2
The solutions graph of MDM/MVITM atγ=0.6 and 0.4 of example 1.Figure 3
The different fractional-order graph of MDM/MVITM of example 1.Applying Laplace transformation of (39), we get
(41)sγsγ1−γ+γLνζ,I−1sνζ,0=LDζζIν−Dζν+νDζζζν−νDζν+3DζνDζζν.Using inverse Laplace transformation,(42)νζ,I=L−1νζ,0s−sγ1−γ+γsγLDζζIν−Dζν+νDζζζν−νDζν+3DζνDζζν.Using ADM procedure, we get(43)ν0ζ,I=L−1νζ,0s=L−1expcosh2ζ/4s,ν0ζ,I=cosh2ζ4,∑ϕ=0∞νϕ+1ζ,I=L−1sγ1−γ+γsγL∑ϕ=0∞DζζIνϕ−∑ϕ=0∞Dζνϕ+∑ϕ=0∞Aϕ−∑ϕ=0∞Bϕ+3∑ϕ=0∞Cϕ,ϕ=0,1,2,⋯,for ϕ=0,
(44)ν1ζ,I=L−1sγ1−γ+γsγLDζζIν0−Dζν0+A0−B0+3C0=−1132sinhζ41−γ+γIγΓγ+1,for ϕ=1,
(45)ν2ζ,I=L−1sγ1−γ+γsγLDζζIν1−Dζν1+A1−B1+3C1,ν2ζ,I=−1128sinhζ41−γ+γIγΓγ+1+1211024coshζ41−γ2+γ2I2γΓ2γ+1+21−γγIγΓγ+1,for ϕ=2,
(46)ν3ζ,I=L−1sγ1−γ+γsγLDζζIν2−Dζν2+A2−B2+3C2,ν3ζ,I=−11512sinhζ41−γ+γIγΓγ+1+1212048coshζ41−γ2+γ2I2γΓ2γ+1+21−γγIγΓγ+1−133149152sinhζ41−γ3+γ1−γ1+γ+2γ2IγΓγ+1+3γ21−γI2γΓ2γ+1+γ3Γ2γ+1I3γΓ3γ+1.The MDM solution of example (8) is
(47)νζ,I=ν0ζ,I+ν1ζ,I+ν2ζ,I+ν3ζ,I+ν4ζ,I+⋯,νζ,I=cosh2ζ4−1132sinhζ41−γ+γIγΓγ+1−1128sinhζ41−γ+γIγΓγ+1+1211024coshζ41−γ2+γ2I2γΓ2γ+1+21−γγIγΓγ+1−11512sinhζ41−γ+γIγΓγ+1+1212048coshζ41−γ2+γ2I2γΓ2γ+1+21−γγIγΓγ+1−133149152sinhζ41−γ3+γ1−γ1+γ+2γ2IγΓγ+1+3γ21−γI2γΓ2γ+1+γ3Γ2γ+1I3γΓ3γ+1⋯.Apply the variational method to find the analytical solution.The iteration formulas for equation (39), we get
(48)νϕ+1ζ,I=νjζ,I−L−1sγ1−γ+γsγLDζζIνϕ+Dζνϕ−νϕDζζζνϕ+νϕDζνϕ−3DζνϕDζζνϕ,where
(49)ν0ζ,I=cosh2ζ4,for ϕ=0,1,2,⋯,
(50)ν1ζ,I=ν0ζ,I−L−1sγ1−γ+γsγLDζζIν0+Dζν0−ν0Dζζζν0+ν0Dζν0−3Dζν0Dζζν0,ν1ζ,I=cosh2ζ4−1132sinhζ41−γ+γIγΓγ+1,ν2ζ,I=ν1ζ,I−L−1sγ1−γ+γsγLDζζIν1+Dζν1−ν1Dζζζν1+ν1Dζν1−3Dζν1Dζζν1,ν2ζ,I=cosh2ζ/4−11/32sinhζ/41−γ+γIγ/Γγ+1−11/28sinhζ/41−γ+γIγ/Γγ+1+121/1024coshζ/41−γ2+γ2I2γ/Γ2γ+1+21−γγIγ/Γγ+1,(51)ν3ζ,I=ν2ζ,I−L−1sγ1−γ+γsγLDζζIν2+Dζν2−ν2Dζζζν2+ν2Dζν2−3Dζν2Dζζν2,ν3ζ,I=cosh2ζ4−1132sinhζ41−γ+γIγΓγ+1−1128sinhζ41−γ+γIγΓγ+1+1211024coshζ41−γ2+γ2I2γΓ2γ+1+21−γγIγΓγ+1−11512sinhζ41−γ+γIγΓγ+1+1212048coshζ41−γ2+γ2I2γΓ2γ+1+21−γγIγΓγ+1−133149152sinhζ41−γ3+γ1−γ1+γ+2γ2IγΓγ+1+3γ21−γI2γΓ2γ+1+γ3Γ2γ+1I3γΓ3γ+1,νζ,I=cosh2ζ4−1132sinhζ41−γ+γIγΓγ+1−1128sinhζ41−γ+γIγΓγ+1+1211024coshζ41−γ2+γ2I2γΓ2γ+1+21−γγIγΓγ+1−11512sinhζ41−γ+γIγΓγ+1+1212048coshζ41−γ2+γ2I2γΓ2γ+1+21−γγIγΓγ+1−133149152sinhζ41−γ3+γ1−γ1+γ+2γ2IγΓγ+1+3γ21−γI2γΓ2γ+1+γ3Γ2γ+1I3γΓ3γ+1+⋯.The exact solution of equation (39) at γ=1,
(52)νζ,I=cosh2ζ4−11I24.In Figure4, the analytical results of MDM/MVITM example 2 graphs show that close contact with each other at γ=1 and 0.8. It is investigated that analytical results are in close relation with the actual results of example 2. In Figure 5, the results of example 2 at different fractional-order of the derivative are plotted at γ=0.6 and 0.4. Figure 6 shows the different fractional of two and three dimensional. The graphical representation has shown the convergence phenomena of fractional-order results towards the result at integer-order of example 2.Figure 4
The solutions graph of MDM/MVITM atγ=1 and 0.8 of example 2.Figure 5
The solutions graph of MDM/MVITM atγ=0.6 and 0.4 of example 2.Figure 6
The different fractional-order graph of MDM/MVITM of example 2.


## 6. Conclusion

In this paper, we have been successfully applied two modified methods to investigate the approximate solutions of fractional Fornberg-Whitham equations. Agreement between numerical results obtained by the modified decomposition method and modified variational iteration method involving fractional-order derivatives with Mittag-Leffler kernel with exact result appears very appreciable by means of illustrative results in figures. The proposed techniques are easy to implement, effective, and suitable for achieving the results of nonlinear fractional Fornberg-Whitham equations. Moreover, both the modified decomposition method and variational iteration method provide the convergent series results with easily calculated components without applying any linearization, perturbation, or limiting assumptions. Finally, we can conclude the suggested methods are more accurate and highly methodical and which can be applied to investigate nonlinear models that arise in applied sciences.



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