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In this study, the closed form of series solutions of the original and modified nonlinear time-fractional Fornberg-Whitham equations are derived by means of the Laplace decomposition method (LDM). The fractional order derivatives are expressed in the sense of Caputo. For the specific choice of parameters, the obtained solutions are compared with the exact solutions to validate the accuracy of this method. Numerical solutions are represented graphically which illustrate the behavior of the solutions. Further, the computations express that the above method is straightforward, and it desires the smaller size of computation.

The fractional order nonlinear differential equations (NDE) have been the focus of many studies due to their numerous applications in the areas of mathematical physics and engineering [

Nonlinear partial differential equations (PDE) widely express complex mechanical and physical phenomenon. Many nonlinear PDE, such as the Korteweg de Vries equation, Camassa Holm equation and Degasperis-Procesi equation have different types of traveling wave solutions. The Fornberg-Whitham equation is one such type of equation which was first introduced for illustrating the qualitative behaviour of wave breaking [

The original Fornberg-Whitham equation with time-fractional of the form [

u t α − u x x t + u x = u u x x x − u u x + 3 u x u x x , (1)

with the initial condition

u ( x , 0 ) = e x / 2 , x > 0 ,

where u ( x , t ) is the fluid velocity. For α = 1 the Equation (1) becomes a classical Fornberg-Whitham equation. This equation was utilized to study the qualitative behaviours of wave-breaking and obtained a peaked limiting travelling wave solution of the form

u ( x , t ) = A e − 1 / 2 ( | x − 4 / 3 t | ) , (2)

where A is constant [

The modified time-fractional Fornberg-Whitham equation is considered by changing the nonlinear term u u x in Equation (1) to u 2 u x as follows [

u t α − u x x t + u x = u u x x x − u 2 u x + 3 u x u x x , (3)

with the initial condition

u ( x ,0 ) = a sech 2 ( b x ) .

For α = 1 , using the bifurcation theory method which is widely used to solve the nonlinear differential equations, the explicit peakon and soliton solutions are obtained by the form

u ( x , t ) = a sech 2 ( b ( x + 4 a / 3 t ) ) (4)

where a = − 3 / 4 ( 5 − 15 ) and b = 1 / 20 10 ( 5 − 15 ) [

In the present study, we consider the original and modified time-fractional Fornberg-Whitham equation to find the series solutions through Laplace decompasition method (LDM). The LDM is combined with the Adomian decomposition method and the Laplace transform. Our manuscript is structured as follows: In Section II, we list some basic definitions and properties of the fractional calculus theory. A brief summary of Laplace decomposition method is given in Section III and the implementation of LDM is given in Section IV. Then the following two subsections, the series solutions of the time-fractional Fornberg-Whitham equation and modified Fornberg-Whitham equation are provided. Finally, the concluding remark to the proposed results is given in Section V.

We provided some definitions and basic properties of fractional calculus theory and then we give the basics of Laplace transformation which will be applied later in this work.

Definition 2.1 Let a function f ∈ C μ , μ ≥ − 1 and α ≥ 0 , then the Riemann-Liouville fractional integral of order α is defined by

J α f ( x ) = 1 Γ ( α ) ∫ 0 x ( x − t ) α − 1 f ( t ) d t , x > 0 , (5)

Some basic properties of the operator J α are as following [

1) J α ( J β f ( x ) ) = J β ( J α f ( x ) ) ,

2) J α ( J β f ( x ) ) = J α + β f ( x ) ,

3) J α x γ = Γ ( γ + 1 ) Γ ( α + γ + 1 ) x α + γ ,

where α , β ≥ 0 and γ > − 1 .

Caputo and Mainardi [

Definition 2.2 The Caputo fractional derivative of order α > 0 is defined by

D α f ( x ) = J m − α D m f ( x ) = 1 Γ ( m − α ) ∫ 0 x ( x − t ) m − α − 1 f ( m ) ( t ) d t , (6)

where m − 1 < α ≤ m , m ∈ N , x > 0 , f ∈ C − 1 m .

Also the derivative satisfies the following properties:

1) D α ( J α f ( x ) ) = f ( x )

2) J α ( D α f ( x ) ) = f ( x ) − ∑ k = 0 m − 1 f ( k ) ( 0 + ) x k k ! , x > 0

Details study of fractional derivatives and integrals can be found in [

Definition 2.3 The Laplace transform of a given function f ( t ) is defined by

F ( s ) = L { f ( t ) } = ∫ 0 ∞ f ( t ) e − s t d t , (7)

whenever the improper integral converges.

Some properties of the Laplace transform are given below:

1) L { f ( n ) ( t ) } = s n L { f ( t ) } − ∑ k = 0 n − 1 s n − 1 − k f ( k ) ( 0 ) ,

2) L { t n f ( t ) } = ( − 1 ) n F ( n ) ( s ) .

In order to explain the central idea of this method, first we consider a general fractional nonlinear PDE with the initial condition of the following form:

D t α u ( x , t ) + R u ( x , t ) + N u ( x , t ) = g ( x , t ) (8)

u ( x ,0 ) = f ( x )

where D t α u ( x , t ) denotes the Caputo fractional derivative of u ( x , t ) , R is a linear operator, N is a nonlinear operator and g is a non-homogeneous term.

We take the Laplace transform on both sides of Equation (8) and then using the differentiation property of Laplace transform, we get,

L [ u ( x , t ) ] = f ( x ) s + 1 s α L [ g ( x , t ) ] − 1 s α L [ R u ( x , t ) + N u ( x , t ) ] . (9)

Now, taking the inverse Laplace transform to Equation (9)

u ( x , t ) = G ( x , t ) − L − 1 [ 1 s α L [ R u ( x , t ) + N u ( x , t ) ] ] . (10)

where G ( x , t ) is source term and give initial condition.

The series solution of the Adomian decomposition method is given by,

u ( x , t ) = ∑ n = 0 ∞ u n ( x , t ) , (11)

with the nonlinear term N u ( x , t ) can be decomposed as

N u ( x , t ) = ∑ n = 0 ∞ A n ( u 0 , u 1 , ⋯ , u n ) , (12)

Adomian polynomials A n can be derived by particular algorithms constructed by Adomian [

A n = 1 n ! d n d λ n [ g ( ∑ k = 0 n λ k u k ) ] λ = 0 . (13)

Substituting (11) and (12) into Equation (10), we get,

∑ n = 0 ∞ u n ( x , t ) = G ( x , t ) − L − 1 [ 1 s α L [ R ∑ n = 0 ∞ u n ( x , t ) + ∑ n = 0 ∞ A n ( u 0 , u 1 , ⋯ , u n ) ] ] , (14)

From the equation (14) we get,

{ u 0 ( x , t ) = G ( x , t ) , u n + 1 ( x , t ) = − L − 1 [ 1 s α L [ R u n ( x , t ) + A n ( u 0 , u 1 , ⋯ , u n ) ] ] , n = 0 , 1 , 2 , ⋯ (15)

From (15), the series solution of (8) can be found

u ( x , t ) ≈ ∑ n = 0 k u n ( x , t ) , where l i m k → ∞ ∑ n = 0 k u n ( x , t ) = u ( x , t ) . (16)

In generally, the above series is convergence rapidly [

First, consider the generalized Fornberg-Whitham equation with time-fractional of the form

u t α − u x x t + u x = u u x x x − u n u x + 3 u x u x x , t > 0 , 0 < α , (17)

u ( x , 0 ) = u 0 ( x ) ,

where n = 1 and 2. Comparing (17) with Equation (8) we get g ( x , t ) = 0 and linear term R turn out to be

R u = − u x x t + u x , (18)

whereas the nonlinear term

N u = u n u x − u u x x x − 3 u x u x x . (19)

Using Equation (17), we get

{ u 0 ( x , t ) = u 0 ( x ) , u n + 1 ( x , t ) = − L − 1 [ 1 s α L [ − u n x x t ( x , t ) + u n x ( x , t ) + A n ] ] , n = 0 , 1 , 2 , ⋯ (20)

Also

N 1 u = u n u x − u u x x x − 3 u x u x x = ∑ n = 0 ∞ A n (21)

Adomian polynomials { A n } n = 0 ∞ can be calculated using the procedure stated in [

A 0 = u 0 n u 0 x − u 0 u 0 x x x − 3 u 0 x u 0 x x ,

A 1 = n u 0 n + 1 u 1 u 0 x + u 0 n u 1 x − u 1 u 0 x x x − u 0 u 1 x x x − 3 u 1 x u 0 x x − 3 u 0 x u 1 x x ,

A 2 = 1 2 n 2 u 0 n − 2 u 1 2 u 0 x − 1 2 n u 0 n − 2 u 1 2 u 0 x + n u 0 n − 1 u 2 u 0 x + n u 0 n − 1 u 1 u 1 x + u 0 n u 2 x − u 2 u 0 x x x − u 1 u 1 x x x − u 0 u 2 x x x − 3 u 2 x u 0 x x − 3 u 1 x u 1 x x − 3 u 0 x u 2 x x ,

A 3 = 1 6 n 3 u 0 n − 3 u 1 3 u 0 x + 1 3 n u 0 n − 3 u 1 3 u 0 x − 1 2 n 2 u 0 n − 3 u 1 3 u 0 x − 1 2 n u 0 n − 2 u 1 2 u 1 x + 1 2 n 2 u 0 n − 2 u 1 2 u 1 x − n u 0 n − 2 u 1 u 2 u 0 x + n 2 u 0 n − 2 u 1 u 2 u 0 x + n u 0 n − 1 u 3 u 0 x + n u 0 n − 1 u 2 u 1 x + n u 0 n − 1 u 1 u 2 x + u 0 n u 3 x − u 3 u 0 x x x − u 2 u 1 x x x − u 1 u 2 x x x − u 0 u 3 x x x − 3 u 3 x u 0 x x − 3 u 2 x u 1 x x − 3 u 1 x u 2 x x − 3 u 0 x u 3 x x ,

and so on. Using the above recursive relationships with Adomian polynomials the decomposition series (16) can be found as follows:

u 0 ( x , t ) = u 0 ,

u 1 ( x , t ) = L − 1 [ 1 s L { u 0 x x t − a u 0 x + A 0 } ] ,

u 2 ( x , t ) = L − 1 [ 1 s L { u 1 x x t − a u 1 x + A 1 } ] ,

u 3 ( x , t ) = L − 1 [ 1 s L { u 2 x x t − a u 2 x + A 2 } ] ,

⋮

u n ( x , t ) = L − 1 [ 1 s L { u n − 1 x x t − a u n − 1 x + A n − 1 } ] ,

and so on. In the following two subsection, we utilize the above results for two initial value problems associated with the original and modified time-fractional Fornberg-Whitham equations in order to find the series solutions.

For n = 1 , the Equation (17) reduces to the original time-fractional Fornberg-Whitham equation as follows;

u t α − u x x t + u x = u u x x x − u u x + 3 u x u x x , t > 0 , 0 < α ≤ 1 , (22)

u ( x , 0 ) = e x / 2 , x > 0.

For α = 1 , the exact solution of peaked limiting form of the travelling wave is given by [

u ( x , t ) = A e − 1 / 2 ( | x − 4 / 3 t | ) . (23)

By using the above recursive relationships and Adomian polynomials A n , we find the terms of decomposition series (16) as follows:

u 0 ( x , t ) = e x / 2 ,

u 1 ( x , t ) = − e x / 2 t α 2 Γ [ 1 + α ] ,

u 2 ( x , t ) = + e x / 2 t 2 α 4 Γ [ 1 + 2 α ] − e x / 2 t − 1 + 2 α 8 Γ [ 2 α ] ,

u 3 ( x , t ) = − e x / 2 t 3 α 8 Γ [ 1 + 3 α ] + e x / 2 t − 1 + 3 α 8 Γ [ 3 α ] − e x / 2 t − 2 + 3 α 32 Γ [ − 1 + 3 α ] ,

u 4 ( x , t ) = + e x / 2 t 4 α 16 Γ [ 1 + 4 α ] − 3 e x / 2 t − 1 + 4 α 32 Γ [ 4 α ] + 3 e x / 2 t − 2 + 4 α 64 Γ [ − 1 + 4 α ] − e x / 2 t − 3 + 4 α 128 Γ [ − 2 + 4 α ] ,

and so on. Consequently, the approximate analytical solution of (22) is found in the closed form of series as follows:

u ( x , t ) = l i m k → ∞ [ e x / 2 + ∑ n = 1 k ∑ i = 0 n − 1 ( n − 1 i ) ( − 1 2 ) n + i e x / 2 t n α − i Γ [ n α + 1 − i ] ] . (24)

To verify the accuracy of the LDM, we compare the numerical values of approximate solutions with the exact solutions in

In

For n = 2 , the Equation (17) reduces to the modified time-fractional Fornberg-Whitham equation as follows;

t | x | α = 0.2 | α = 0.7 | α = 1 | Exact Solution | Abs. error | u E x a c t − u L D M | |
---|---|---|---|---|---|---|

0.1 | 0.5 | 0.55330756 | 1.0548937 | 1.2014201 | 1.2012147 | 0.00020536966 |

1.0 | 0.71046097 | 1.3545103 | 1.542654 | 1.5423903 | 0.00026369986 | |

1.5 | 0.91224994 | 1.7392257 | 1.9808069 | 1.9804683 | 0.00033859733 | |

2.0 | 1.1713521 | 2.23321 | 2.5434064 | 2.5429716 | 0.00043476757 | |

0.5 | 0.5 | 0.78942785 | 0.83412887 | 0.91975844 | 0.92004441 | 0.00028597406 |

1.0 | 1.0136454 | 1.0710427 | 1.1809932 | 1.1813604 | 0.00036719796 | |

1.5 | 1.3015465 | 1.375246 | 1.5164253 | 1.5168968 | 0.00047149152 | |

2.0 | 1.6712188 | 1.7658508 | 1.9471286 | 1.94773404 | 0.00060540709 | |

0.9 | 0.5 | 0.78469782 | 0.7174914 | 0.70407427 | 0.70468809 | 0.00061381852 |

1.0 | 1.0075719 | 0.92127719 | 0.90404926 | 0.90483742 | 0.00078815858 | |

1.5 | 1.293748 | 1.1829433 | 1.1608222 | 1.1618342 | 0.0010120157 | |

2.0 | 1.6612053 | 1.5189293 | 1.4905252 | 1.4918247 | 0.0012994538 |

u t α − u x x t + u x = u u x x x − u 2 u x + 3 u x u x x , t > 0 , 0 < α ≤ 1 , (25)

u ( x ,0 ) = a sech 2 ( b x ) ,

For α = 1 , the explicit peakon and soliton solution of Equation (25) is given by [

u ( x , t ) = a sech 2 ( b ( x + 4 a / 3 t ) ) . (26)

By using the above recursive relationships and Adomian polynomials A n , we find the terms of decomposition series (16) as follows:

u 0 ( x , t ) = a sech 2 ( b x ) ,

u 1 ( x , t ) = c 1 0 a sech 2 ( b x ) t α Γ [ 1 + α ] ,

u 2 ( x , t ) = + c 2 0 a sech 2 ( b x ) t 2 α 4 Γ [ 1 + 2 α ] − c 2 1 a sech 2 ( b x ) t − 1 + 2 α Γ [ 2 α ] ,

u 3 ( x , t ) = − c 3 0 a sech 2 ( b x ) t 3 α Γ [ 1 + 3 α ] + c 3 1 a sech 2 ( b x ) t − 1 + 3 α Γ [ 3 α ] + c 3 2 a sech 2 ( b x ) t − 2 + 3 α Γ [ − 1 + 3 α ] ,

u 4 ( x , t ) = c 4 0 a sech 2 ( b x ) t 4 α 16 Γ [ 1 + 4 α ] − c 4 1 a sech 2 ( b x ) t − 1 + 4 α Γ [ 4 α ] + c 4 2 a sech 2 ( b x ) t − 2 + 4 α Γ [ − 1 + 4 α ] − c 4 3 a sech 2 ( b x ) t − 3 + 4 α Γ [ − 2 + 4 α ] ,

and so on. Consequently, as the above case, the approximate analytical solution of (25) is found in the closed form of series as follows:

u ( x , t ) = l i m k → ∞ [ a sech 2 ( b x ) + ∑ n = 1 k ∑ i = 0 n − 1 c n i a sech 2 ( b x ) t n α − i Γ [ n α + 1 − i ] ] , (27)

where,

c 1 0 = 2 b ( 1 + a ( a + 22 b 2 − 8 b 2 cosh ( 2 b x ) ) sech ( b x ) 4 ) tanh ( b x ) ,

c 2 0 = 2 b 2 ( 2 + sech ( b x ) 2 ( − 3 − 128 a b 2 + a sech ( b x ) 2 ( 4 ( 130 b 2 + a ( 3 + 864 b 4 ) ) − 2 ( 7 a + 10 ( 21 + 32 a 2 ) b 2 + 13280 a b 4 sech ( b x ) 2 + 2 a ( 5 a 2 + 960 a b 2 + 26172 b 4 ) sech ( b x ) 4 − 11 a ( a + 30 b 2 ) ( a + 90 b 2 ) sech ( b x ) 6 ) ) ) ,

c 2 1 = b 3 ( − 26 − 304 a 2 − 7968 a b 2 + ( − 33 + 16 a ( 9 a + 326 b 2 ) ) cosh ( 2 b x ) − 2 ( 3 + 128 a b 2 ) cosh ( 4 b x ) + cosh [ 6 b x ] ) sech ( b x ) 6 tanh ( b x ) ,

and so on. In

Similarly as in the previous case, the approximate solutions are the same behavior with the exact solution. This is illustrated by

t | x | α = 0.2 | α = 0.7 | α = 1 | Exact Solution | Abs. error | u E x a c t − u L D M | |
---|---|---|---|---|---|---|

0.1 | 0.5 | −0.84015035 | −0.84719431 | −0.84200369 | −0.84170019 | 0.00030349875 |

1.0 | −0.79140244 | −0.83591408 | −0.82723018 | −0.82678633 | 0.00044385734 | |

1.5 | −0.74342766 | −0.81235381 | −0.8015035 | −0.8010345 | 0.00046899813 | |

2.0 | −0.70107965 | −0.77787419 | −0.76618668 | −0.76580255 | 0.00038413104 | |

0.5 | 0.5 | −0.81320446 | −0.84990096 | −0.84802561 | −0.84516644 | 0.0028591668 |

1.0 | −0.80726139 | −0.84820141 | −0.84379648 | −0.84074121 | 0.0030552675 | |

1.5 | −0.7933088 | −0.83324893 | −0.82726913 | −0.82471511 | 0.0025540116 | |

2.0 | −0.77275166 | −0.8062717 | −0.79944392 | −0.79796266 | 0.001481259 | |

0.9 | 0.5 | −0.78865494 | −0.83899061 | −0.84577223 | −0.83899395 | 0.0067782736 |

1.0 | −0.78842909 | −0.84297517 | −0.85111767 | −0.84525761 | 0.0058600603 | |

1.5 | −0.78148415 | −0.83403147 | −0.84330532 | −0.83966942 | 0.003635898 | |

2.0 | −0.76844313 | −0.81333057 | −0.82306334 | −0.82253964 | 0.00052370169 |

In this study, the Laplace-Adomian decomposition method has been successfully applied to find a closed form of series solution of the original and modified time-fractional Fornberg-Whitham equations. For some specific choice of parameters, the approximate solutions are compared with the exact solutions. Further, the behaviours of the solutions are presented by 3D and 2D graphs. The graphs showed that the behaviors of the approximate solutions are the same behaviors of the exact solutions. As a result, we realize that the approximate solutions are rapidly convergent series as the exact solutions. Further, the computations show that the described method is easy to apply and it needs the smaller size of computation than the existing methods. Therefore, it is obvious that the LDM is very powerful, efficient and accurate for solving the nonlinear fractional differential equations.

The authors declare no conflicts of interest regarding the publication of this paper.

Mathanaranjan, T. and Vijayakumar, D. (2021) Laplace Decomposition Method for Time-Fractional Fornberg-Whitham Type Equations. Journal of Applied Mathematics and Physics, 9, 260-271. https://doi.org/10.4236/jamp.2021.92019