Theoretical and numerical analysis of nonlinear Boussinesq equation under fractal fractional derivative

Obaid J. Algahtani

doi:10.1515/nleng-2022-0338

Artikel Open Access

Theoretical and numerical analysis of nonlinear Boussinesq equation under fractal fractional derivative

Obaid J. Algahtani

Veröffentlicht/Copyright: 24. Oktober 2023

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Abstract

A nonlinear Boussinesq equation under fractal fractional Caputo’s derivative is studied. The general series solution is calculated using the double Laplace transform with decomposition. The convergence and stability analyses of the model are investigated under Caputo’s fractal fractional derivative. For the numerical illustrations of the obtained solution, specific examples along with suitable initial conditions are considered. The single solitary wave solutions under fractal fractional derivative are attained by considering small values of time ( t ) . The wave propagation has a symmetrical form. The solitary wave’s amplitude diminishes over time, and its extended tail expands over a long distance. It is observed that the fractal fractional derivatives are an extremely constructive tool for studying nonlinear systems. An error analysis is also carried out for compactness.

Keywords: fractal-fractional operator; double-Laplace transform; Boussinesq equation; decomposition technique; power law kernel; convergence and stability analysis

1 Introduction

In recent years, partial differential equations (PDEs) and fractional partial differential equations (FPDEs) have played a significant role in several areas of science and technology, such as heat and mass transfer, viscoelasticity, image processing and computer vision, cosmology, astronomy, biological models, fluid mechanics, electro-magnetics, finance, environmental sciences, biomedical engineering, and control systems [1–3]. The applications of PDEs and FPDEs extend over numerous disciplines and have a great impact on scientific research and technological developments [4,5].

The fractional order derivatives and integrals are the generalisations in integer-orders that accurately describe physical phenomena [6,7]. It has been investigated that extending integer-order differentiation and integration to fractional orders offers an extensive medium for investigating systems where conventional methods degenerate [8,9]. It is worth mentioning that FPDEs are difficult to solve explicitly using classical operators. Therefore, several operators have been constructed in fractional calculus to study real-world phenomena described by the compactness of power-law kernels [10–12]. These operators help to study the global dynamics of physical phenomena due to the traditional properties and recall explanation of FPDEs, which are more useful in modelling actual phenomena as compared to classical order differentiations. It should be noted that classical differentiation describes the dynamics in the neighbourhood of a point in real-world problems, while fractional order defines such dynamics in an interval [13,14].

The concepts of combining fractal and fractional derivatives have developed novel operators known as fractal-fractional derivatives and integrations, which offer an unconventional approach for investigating a variety of linear and nonlinear systems [15]. This concept allows extracting classical derivatives when the fractal dimension becomes one [16,17]. These innovative operators offer a distinctive opportunity to study a wide range of real-world phenomena in various disciplines of science and engineering [18–20]. Numerous numerical and analytical schemes, such as Riccati polynomials, Jacobi polynomials, and Chebyshev cardinal functions, have been developed to investigate the fractal-fractional theory in various systems [21,22].

The study of linear and nonlinear waves in dispersion-prone mediums has been given great attention in recent years, as it plays a key role to describe the dynamics of shallow water waves. For this, several mathematical models, such as Hirota, Klein–Gordon, Korteweg–de Vries (KdV), sine-Gordon, and Boussinesq equations, are investigated to analyse the behaviour of nonlinear waves called solitons. The Boussinesq equation considered herein describes the propagation of long waves with small amplitudes on the surface of shallow water. It reveals both decaying and growing localised modes in both linear and nonlinear mediums, making it a good choice for physical experiments in the fields of nonlinear engineering modelling and applications. The Boussinesq equation also offers nonlinear dispersion stability, which can guide the evolution of solitons. In addition, it has received substantial interest in several other areas, such as magnetic sound waves in plasma [23], electromagnetic waves in nonlinear dielectrics [24,25], and wave formation in two directions [26]. In addition, water dynamics in water supply systems have been approximated analytically and numerically under both permanent and short regimes using one-dimensional Boussinesq equation [27].

There are mainly two types of Boussinesq model equations depending on the values of a and b , which can be uniquely determined by the depth and velocity of the wave propagation, governed by

(1) ψ t t − ψ x x − a ψ x x x x − b ψ x x 2 = 0 , x ∈ R , t > 0 .

When a > 0 , Eq. (1) is referred to as the nonlinear beam or “good” Boussinesq equation, while for a < 0 is called “bad” Boussinesq equation [28]. For a = b = 1 , Eq. (1) represents a solitary wave solution. It is also worth mentioning that Eq. (1) shows similar behaviour to KdV and cubic Schrödinger equations, which allow the existence of solitary wave solutions. However, it differs from these models in different aspects. For instance, it has a limited range of velocities for the solitary waves, exhibits interactions that can result in singular solutions, and has a probability of coupled waves being changed into a single soliton solution.

The Boussinesq equation was first investigated by Hirota for the multi-soliton solution [28]. Since then, numerous solutions like positions and complexitons, negations, and solitons have been studied by applying the Hirota bilinear form and the Wronskian method [29,30]. This model has also been studied for several physical phenomena using different numerical schemes. For instance, many linearised methods have been applied to investigate single and double solitary wave solutions in finite-difference schemes [31]. A robust class of spectral method has been proposed for the numerical solution of the Boussinesq equation that results in single solitary waves in the nonlinear instability and stability in orbits [32]. Some other methods, such as indirect finite-difference recursive scheme [33], a recursive relation with three successive stages [34,35], and the techniques using regularisation and filtering methods, have also been developed [36].

Here, we study a Boussinesq equation with constant coefficients with Caputo fractional derivative in the following form [37]:

(2) D t α , β 0 F F P ψ − ψ x x − ψ x x x x − ψ x x 2 = 0 , 1 < α ≤ 2 , 0 < β ≤ 1 .

This study focuses on evaluating the approximate solution of Eq. (2) using the modified double Laplace decomposition method (MDLDM). We also study the convergence and stability of the considered model with Caputo’s fractional derivative and fractal dimensions.

2 Preliminaries

Definition 1

Let ψ = ψ ( x , t ) lie in the x t -plane, then the integral described by

(3) L x L t [ ψ ( x , t ) ] = ∫ 0 ∞ e − ξ 1 x ∫ 0 ∞ e − ξ 2 t ψ ( t ) d t d x , x > 0 , t > 0 ,

with ξ 1 , ξ 2 ∈ C is called double Laplace transform of ψ ( x , t ) .

Definition 2

For partial derivatives, the double Laplace can be described as

(4) L x L t ∂ n ψ ( x , t ) ∂ x n = ξ 1 n ψ ¯ ( ξ 1 , ξ 2 ) − ∑ j = 0 n − 1 ξ 1 n − 1 − j L t ∂ j ψ ( 0 , t ) ∂ x j

and

(5) L x L t ∂ m ψ ( x , t ) ∂ t m = ξ 2 m ψ ¯ ( ξ 1 , ξ 2 ) − ∑ j = 0 m − 1 ξ 2 m − 1 − j L x ∂ j ψ ( x , 0 ) ∂ t j ,

where n , m = 1 , 2 , 3 , … . Applying the above formulation to a fractional derivative with parameters x and t in the Caputo sense gives

(6) L x L t { D x γ C ψ ( x , t ) } = ξ 1 γ ψ ¯ ( ξ 1 , ξ 2 ) − ∑ j = 0 n − 1 ξ 1 γ − 1 − j L t ∂ j ψ ( 0 , t ) ∂ x j ,

(7) L x L t { D t β C ψ ( x , t ) } = ξ 2 β ψ ¯ ( ξ 1 , ξ 2 ) − ∑ j = 0 m − 1 ξ 2 β − 1 − j L x ∂ j ψ ( x , 0 ) ∂ t j ,

where [ γ ] + 1 = n , [ β ] + 1 = m .

Definition 3

[3,18] Let α ∈ ( 0 , 1 ] and ψ ∈ C [ a , c ] , then the derivative described by

(8) D a C ψ ( t ) = ψ m ( t ) α = m , 1 Γ ( m − α ) ∫ a t ( t − ξ ) m − a − 1 ψ ́ ( ξ ) d ξ ∀ α ∈ ( m − 1 , m ] ,

is called a Caputo fractional derivative. When α = 1 , it turns into a classical derivative.

Definition 4

[18] Let ψ ( t ) be fractal-fractional derivative in ( a , b ) , then

D t α , β a F F P ψ = 1 Γ ( m − α ) ∫ a t ( t − ξ ) m − a − 1 d d t β ψ ( ξ ) d ξ , m − 1 < α , β ≤ m ,

with

d ψ ( t ) d t β = lim t → ξ ψ ( t ) − ψ ( ξ ) t β − ξ β .

Generally, one may also describe as

D t α , β a F F P ψ ( t ) = 1 Γ ( m − α ) ∫ a t ( t − ξ ) m − a − 1 d γ d t β ψ ( ξ ) d ξ , 0 < m − 1 < α , β , γ ≤ m ,

with

d γ ψ ( t ) d t β = lim t → ξ ψ γ ( t ) − ψ γ ( ξ ) t β − ξ β .

Definition 5

[18] The fractal fractional integral can be described as

I t α 0 F = β Γ ( α ) ∫ 0 t ξ α − 1 ψ ( ξ ) ( t − ξ ) α − 1 d ξ .

3 Analytical scheme

The MDLDM has emerged as a powerful analytical tool for investigating complex physical systems [38]. The MDLDM combines the Laplace decomposition method with suitable modifications to address the challenges posed by highly nonlinear and time-varying problems [39]. The method has proven to be effective in providing accurate solutions and deep insights into the behaviour of plasma models, making it an attractive choice for researchers in various fields [40,41]. Recent investigations have demonstrated the versatility of MDLDM in solving electron-acoustics solitary potentials in streaming plasma [42]. To demonstrate the proposed method, let us consider non-linear system in the form

(9) L ψ ( x , t ) + R ψ ( x , t ) + N ψ ( x , t ) = ϕ ( x , t ) , ∀ t ∈ R ,

where L = D α ( x , T ) = ∂ α ψ ( x , t ) ∕ ∂ x α , N is nonlinear, R contains the linear terms, and ϕ ( x , t ) is an external function. Using a double Laplace transform to Eq. (9), we may write

(10) L x L t { D α ψ } + R L x L t { ψ } + N L x L t { ψ } = L x L t { ϕ } .

Applying double Laplace on n th-derivative, we obtain

(11) L x L t { ψ } = G ( ξ 1 , ξ 2 ) − 1 ξ 2 α R L x L t { ψ } − 1 ξ 2 α N L x L t { ψ } ,

where

(12) G ( ξ 1 , ξ 2 ) = 1 ξ 2 α F ( ξ 1 , ξ 2 ) + 1 ξ 2 α ∑ i = 0 α − 1 ξ 2 α − 1 − i L x ∂ i ψ ( x , 0 ) ∂ x i .

Here, we emphasise on the solution of the form:

ψ = ∑ n = 0 ∞ ψ n ,

where the non-linear terms can be divided as

N ψ = ∑ n = 0 ∞ B n ,

while B n are known as Adomian polynomials [39] for the hierarchy of ψ ( i ) , provided by the subsequent rule

(13) B n = 1 n ! d n d λ n ∑ k = 0 n λ k ψ k ( x , t ) λ = 0 .

Finally, using Eq. (13) and applying inverse double Laplace to Eq. (11), one may write

∑ n = 0 ∞ ψ n = L x − 1 L t − 1 G ( ξ 1 , ξ 2 ) − L x − 1 L t − 1 1 ξ 2 α R L x L t { ψ ( x , t ) } − L x − 1 L t − 1 1 ξ 2 α L x L t ∑ n = 0 ∞ B n .

Equating the expression for ψ i gives

ψ 0 = L x − 1 L t − 1 [ G ( ξ 1 , ξ 2 ) ] , ψ 1 = − L x − 1 L t − 1 1 ξ 2 α L x L t [ R ψ 0 ] − L x − 1 L t − 1 1 ξ 2 α L x L t [ B 0 ] , ψ 2 = − L x − 1 L t − 1 1 ξ 2 α L x L t [ R ψ 1 ] − L x − 1 L t − 1 1 ξ 2 α L x L t [ B 1 ] , ψ 3 = − L x − 1 L t − 1 1 ξ 2 α L x L t [ R ψ 2 ] − L x − 1 L t − 1 1 ξ 2 α L x L t [ B 2 ] , ⋮ ψ n + 1 = − L x − 1 L t − 1 1 ξ 2 α L x L t [ R ψ n ] − L x − 1 L t − 1 1 ξ 2 α L x L t [ B n ] .

Finally,

(14) ψ = ∑ n = 0 ∞ ψ ( n ) .

3.1 Approximate solution

Here, we use the MDLDM to obtain the approximate solution of the proposed model with fractal fractional derivative. The fractal fractional (FF) operator with Caputo’s fractional derivative can be written as

(15) D t α , β a F F P ψ ( t ) = ϕ ( x ) ,

with ϕ ( x ) an exterior function of the form:

(16) D t α a C ψ ( t ) = β t β − 1 ϕ ( x ) .

The proposed model with fractal fractional derivative can be described as

(17) D t α , β a F F P ψ ( t ) = ψ x x + ψ x x x x + ψ x x 2 ,

with suitable initial condition

(18) ψ ( x , 0 ) = ϕ ( x ) , ψ t ( x , 0 ) = Φ ( x ) .

From Eqs (15)–(16)

(19) D t α c ψ ( t ) = β t β − 1 [ ψ x x + ψ x x x x + ψ x x 2 ] .

Using double Laplace transform

(20) L x L t ψ ( x , t ) = 1 s L x L t ϕ ( x ) + 1 ξ 2 L x L t Φ ( x ) + 1 ξ α L x L t β t β − 1 [ ψ x x + ψ x x x x + ψ x x 2 ] .

Now consider

(21) ψ ( x , t ) = ψ n ( x , t ) .

The non-linear expressions can be decomposed as

(22) ψ x x 2 = ∑ n = 0 ∞ B n ,

where B n is obtained in previous section in the form of Eq. (13). Using inverse double Laplace

(23) ∑ n = 0 ∞ ψ ( x , t ) = ϕ ( x ) + t Φ ( x ) + L x − 1 L t − 1 × 1 ξ α L x L t β t β − 1 ∑ n = 0 ∞ ψ n x x + ∑ n = 0 ∞ ψ n x x x x + ∑ n = 0 ∞ B n .

Comparing terms on both sides

ψ 0 = ϕ ( x ) + t Φ ( x ) , ψ 1 = L x − 1 L t − 1 1 ξ α L x L t { β t β − 1 { ψ 0 x x + ψ 0 x x x x + B 0 } } , ψ 2 = L x − 1 L t − 1 1 ξ α L x L t { β t β − 1 { ψ 1 x x + ψ 1 x x x x + B 1 } } , ψ 3 = L x − 1 L t − 1 1 ξ α L x L t { β t β − 1 { ψ 2 x x + ψ 2 x x x x + B 2 } } , ψ 4 = L x − 1 L t − 1 1 ξ α L x L t { β t β − 1 { ψ 3 x x + ψ 3 x x x x + B 3 } } , ψ 5 = L x − 1 L t − 1 1 ξ α L x L t { β t β − 1 { ψ 4 x x + ψ 4 x x x x + B 4 } } .

The final solution can be obtained in the form:

(24) ψ ( x , t ) = ∑ n = 0 ∞ ψ n ( x , t ) .

4 Theoretical investigation

In this section, we study the convergence and stability of the proposed model. To check the existence and uniqueness of Boussinesq equation, we prove the following theorems.

Theorem 1

The following Lipschitz condition satisfies for FF Caputo’s derivative with fractal dimension β

max t ∈ [ 0 , 1 ] ∥ D t α , β a F F P ψ ( t ) − D t α , β a F F P φ ( t ) ∥ ≤ λ ‖ ψ − φ ∥ .

Proof

max t ∈ [ 0 , 1 ] ∥ D t α , β a F F P ψ ( t ) − D t α , β a F F P φ ( t ) ∥ = 1 Γ [ n − α ] ∫ a t d ( t − ζ ) n − α − 1 d ζ β ψ ( ζ ) d ζ − 1 Γ [ n − α ] ∫ a t d ( t − ζ ) n − α − 1 d ζ β φ ( ζ ) d ζ = 1 Γ [ n − α ] ∫ a t d ( t − ζ ) n − α − 1 d ζ β ψ ( ζ ) d ζ − ∫ a t d ( t − ζ ) n − α − 1 d ζ β φ ( ζ ) d ζ ≤ μ Γ [ n − α ] ∥ ψ − φ ∥ ≤ λ ∥ ψ − φ ∥ .

To investigate the boundedness, existence, and uniqueness of the solution to Eq. (2), one may write

D t α , β 0 F F P ψ ( x , t ) = ϕ ( x , t ; ψ ) = ψ x x + ψ x x x x + ψ x x 2 .

Using FF integral to Eq. (2)

(25) ψ ( x , t ) − ψ ( x , 0 ) = β Γ α ∫ 0 t ζ α − 1 ϕ ( x , t ; ψ ) ( t − ζ ) α − 1 d ζ .

We consider the bounded functions ψ ( x , t ) and φ ( x , t ) and illustrate that the kernel ϕ ( x , t ; ψ ) satisfies Lipschitz’s condition. For this, one may proceed as

∥ ϕ ( x , t ; ψ ) − ϕ ( x , t ; φ ) ∥ = ∥ ψ x x + ψ x x x x + ψ x x 2 − φ x x − φ x x x x − φ x x 2 ∥ = ∥ ( ψ x x − φ x x ) + ( ψ x x 2 − φ x x 2 ) + ( ψ x x x x − φ x x x x ) ∥ ≤ ∂ 2 ∂ x 2 ( ψ − φ ) + ∂ 4 ∂ x 4 ( ψ − φ ) + ∂ 2 ∂ x ( ψ 2 − φ 2 ) ≤ B ∥ ψ − φ ∥ + C ∥ ψ − φ ∥ + D ∥ ψ − φ ∥ ∥ ψ + φ ∥ ≤ ( B + C + D ( ψ + φ ) ) ∥ ψ − φ ∥ , ‖ ψ ‖ ≤ ψ , ‖ φ ‖ ≤ φ ≤ λ ‖ ψ − φ ‖ ,

where λ ≤ 0 fulfils the aforementioned condition for ϕ ( x , t ; ψ ) . Using the FF integral

ψ n + 1 = β Γ ( α ) ∫ 0 t ζ α − 1 ψ ( ζ ) ( t − ζ ) α − 1 d ζ ,

and considering that ψ 0 = ψ ( x , 0 ) , we have

Y n ( x , t ) = ψ n − ψ n − 1 = β Γ ( α ) ∫ 0 t ( ϕ ( x , ζ ; ψ n − 1 ) − ϕ ( x , ζ ; ψ n − 2 ) ) d ζ

and

(26) ψ n ( x , t ) = ∑ i = 0 n Y i ( x , t ) .

Theorem 2

For time t, the function ψ ( x , t ) is bounded, then

∥ Y n ( x , t ) ∥ ≤ λ β t Γ ( α ) n ∥ ψ ( x , 0 ) ∥ .

Proof

To prove the theorem, we use mathematical induction. Let the above inequality be true for n = 1

∥ Y 1 ( x , t ) ∥ = ∥ ψ 1 ( x , t ) − ψ 0 ( x , t ) ∥ = β Γ ( α ) ∫ 0 t ∥ ϕ ( x , ζ ; ψ 0 ) − ϕ ( x , ζ ; ψ 1 ) ∥ d ζ ≤ β Γ ( α ) ∫ 0 t λ ∥ ψ 0 − ψ 1 ∥ d ζ = λ β t Γ ( α ) ∥ ψ ( x , 0 ) ∥ .

Let us suppose that, the above inequality is true for n = k , then

∥ Y k ∥ ≤ λ β t Γ ( α ) k ∥ ψ ∥ ,

which implies that the result is true for n = k + 1 . For this

∥ Y k + 1 ( x , t ) ∥ ≤ λ β t Γ ( α ) k + 1 ∥ ψ ( x , t ) ∥ .

To demonstrate this, we proceed as

∥ Y k + 1 ∥ = ∥ ψ k + 1 − ψ k ∥ ≤ β Γ ( α ) ∫ 0 t ∥ ϕ ( x , ζ ; ψ k ) − ϕ ( x , ζ ; ψ k − 1 ) ∥ d ζ ≤ λ β Γ ( α ) ∫ 0 t ∥ ψ k − ψ k − 1 ∥ d ζ ≤ λ β Γ ( α ) ∫ 0 t ∥ Y k ( x , ζ ) ∥ d ζ ≤ λ β t Γ ( α ) k ∥ ψ ( x , 0 ) ∥ λ β Γ ( α ) ∫ 0 t d ζ ≤ λ β t Γ ( α ) k + 1 ∥ ψ ( x , 0 ) ∥ .

Hence proved.

Theorem 3

For t = t 0 , if 0 ≤ λ β t 0 < Γ ( α ) , subsequently solution of the proposed system with power law kernel exists.

Proof

Considering Eq. (26) by writing

∥ ψ n ( x , t ) ∥ ≤ ∑ j = 0 n ∥ Y n ( x , t ) ∥ ≤ ∑ j = 0 n β λ t Γ ( α ) j ∥ ψ n ( x , 0 ) ∥ ,

at t = t 0 , we have

∥ ψ n ( x , t ) ∥ ≤ ∥ ψ ( x , 0 ) ∥ ∑ j = 0 n β λ t 0 Γ ( α ) j .

Consequently,

lim n → ∞ ∥ ψ n ( x , t ) ∥ ≤ ∥ ψ ( x , 0 ) ∥ β λ t 0 Γ ( α ) j .

Since

0 ≤ β λ t 0 < Γ ( α ) ,

which is convergent series. Therefore, ψ n ( x , t ) exists and is bounded.

Theorem 4

For t = t 0 , the Boussinesq equation with a power law kernel has a unique solution if the condition

0 ≤ β λ t 0 < Γ ( α )

is satisfied.

Proof

Using contradiction, consider that the solution of the considered equation is not unique. Then we suppose that ψ and φ are other two solutions of the proposed system, such that

ψ − φ = β Γ ( α ) ∫ 0 t [ ψ ( x , ζ ; ψ ) − ϕ ( x , ζ ; φ ) ] d ζ ∥ ψ − φ ∥ ≤ β Γ ( α ) ∫ 0 t λ ∥ ψ − φ ∥ d ζ ≤ λ β t Γ ( α ) ∥ ψ − φ ∥ ,

but 0 ≤ β λ t 0 < Γ ( α ) , which contradicts our supposition. Hence

∥ ψ − φ ∥ = 0

concludes that the proposed model has a unique solution.

For stability analysis, we use Picard’s stability of the proposed model.

Theorem 5

Let ζ be a self-mapping described by

ζ ( ψ n + 1 ) = ψ n + L x − 1 L t − 1 × 1 ξ α L x L t { β t β − 1 { ψ n x x + ψ n x x x x + ψ n x x 2 } } ,

then the iteration in Caputo fractional derivative case is ζ -stable in L 1 ( a , b ) , if condition ( B + C + D ( ψ + φ ) ) ∥ ψ − φ ∥ < 1 is fulfilled.

Proof

Using Banach contraction theorem, first, we demonstrate that the mapping has a single fixed point, ζ . Let the bounded iteration for ( m , n ) ε A ∈ A , and suppose that

ζ ψ m − ζ ψ n = ψ m − ψ n + L x − 1 L t − 1 1 ξ α L x L t { β t β − 1 ( ψ m x x + ψ m x x x x + ψ m x x 2 ) } ] − L x − 1 L t − 1 1 ξ α L x L t { β t β − 1 ( ψ n x x + ψ n x x x x + ψ n x x 2 ) } ] , = ψ m − ψ n + L x − 1 L t − 1 1 ξ α L x L t { β t β − 1 ( ψ m x x + ψ m x x x x + ψ n x x 2 − ψ n x x − ψ n x x x x − ψ n x x 2 ) } , = ψ m − ψ n + L x − 1 L t − 1 1 ξ α L x L t { β t β − 1 ( B ( ψ m − ψ n ) + C ( ψ m − ψ n ) + ( ψ m x x 2 − ψ n x x 2 ) ) } .

Using triangular inequality

‖ ζ ψ m − ζ ψ n ‖ ≤ ‖ ψ m − ψ n ‖ + L x − 1 L t − 1 1 ξ α L x L t { β t β − 1 ( B ( ψ m − ψ n ) + C ( ψ m − ψ n ) + ( ψ m x x 2 − ψ n x x 2 ) ) } , ≤ B ‖ ( ψ m − ψ n ) ‖ + C ‖ ( ψ m − ψ n ) ‖ + D ‖ ψ − φ ‖ ‖ ψ + φ ‖ .

Now using boundedness of ψ m and ψ n

‖ ζ ψ m − ζ ψ n ‖ ≤ ( B + C + D ( ψ + φ ) ) ∥ ψ − φ ∥ ,

from the above result

∣ ζ ψ m − ζ ψ n ‖ ≤ λ ‖ ( ψ m − ψ n ) ‖ , when λ < 1 .

It should be noted that A i are the functions obtained from

L x − 1 L t − 1 1 ξ α L x L t β t β − 1 ( * ) .

By assumption, the mapping ζ fulfils the contraction. Hence by Banach fixed point theorem, the mapping ζ has a unique fixed point.

5 Numerical examples and discussion

Here, we illustrate our result with the help of examples. Suppose the following fractal-fractional order Boussinesq’s equation

Example 1

(27) D t α , β 0 F F P ψ ( x , t ) + ψ x x ( x , t ) − ψ x x x x ( x , t ) − ψ x x 2 ( x , t ) = 0 , 1 < α ≤ 2 , 0 < β ≤ 1 x , t ∈ R .

with initial conditions

(28) ψ ( x , 0 ) = 3 2 ( ω − 1 ) sech 2 1 2 1 − ω 2 ( x + x 0 ) .

The exact solution of Eq. (27) is

ψ ( x , t ) = 3 2 ( ω − 1 ) sech 2 1 2 1 − ω 2 ( x + x 0 − ω t ) .

Applying MDLDM

(29) A 0 = ψ 0 2 ψ 0 x x 2 , A 1 = 2 ψ 0 x x ψ 1 x x , A 2 = 2 ψ 0 x x ψ 2 x x + ψ 1 x x 2 .

Using the method used in Section 3 with the Adomian polynomials obtained in Eq. (29), we obtain

ψ 0 = 3 2 ( ω − 1 ) sech 2 1 2 Δ ,

ψ 1 = β Γ ( β + 1 ) t α + β + 1 4 Γ ( α + β ) a ( ω 2 − 1 ) 2 sech 2 1 2 Δ × 9 a ( ω 2 − 1 ) 2 sech 6 1 2 Δ − 6 ( ω 2 − 1 ) ( 2 a ( ω 2 − 1 ) − 5 ) × sech 4 1 2 Δ + ( 4 a ( ω 2 − 1 ) 2 − 30 ω 2 + 36 ) × sech 2 1 2 Δ + 4 ( ω 2 − 2 ) 8 ,

ψ 2 = β Γ ( β + 2 ) t α + β + 2 4 Γ ( 2 α + 2 β ) a ( ω 2 − 1 ) 3 × [ r 1 F 1 ( θ ) ω 8 + 128 a 2 cosh ( θ 2 ) r 2 F 1 ( θ ) ω 6 + 65,480 a F 1 ( θ ) ω 6 − 512 a 2 cosh ( θ 2 ) ω 6 − 5,264 a F 2 ( θ ) ω 6 + 72 a cosh ( r 3 F 1 ( θ ) ω 4 − 197,840 a F 1 ( θ ) ω 4 − 32,810 F 1 ( θ ) ω 4 + 768 a 2 F 2 ( θ ) ω 4 + 16,352 a F 2 ( θ ) ω 4 + 12,606 F 2 ( θ ) ω 4 − 240 a cosh ( 5 Δ ) ω 4 − 498 F 3 ( θ ) ω 4 + cosh ( 6 Δ ) r 4 F 1 ( θ ) ω 2 + 199,240 a F 1 ( θ ) ω 2 + 59,720 F 1 ( θ ) ω 2 − r 6 cosh ( Δ ) + ( 33,536 a 2 ( ω 2 − 1 ) 4 − 64 a ( 2,960 ω 2 − 2,887 ) ( ω 2 − 1 ) 2 − 15 ( 7,407 ω 4 − 14,524 ω 2 + 7,100 ) ) cosh ( 2 Δ ) − 3,840 a 2 F 1 ( θ ) − 66,880 a F 1 ( θ ) − 27,080 F 1 ( θ ) + 128 a 2 F 2 ( θ ) + 5,824 a F 2 ( θ ) + 13,464 F 2 ( θ ) − 96 a F 3 ( θ ) − 744 F 3 ( θ ) + 4 cosh ( θ 4 + 117,096 ) sech 14 1 2 Δ ,

where

Δ = 1 − ω 2 ( x + x 0 ) , F 1 ( θ ) = cosh ( θ 1 ) , F 2 ( θ ) = cosh ( θ 2 ) , F 3 ( θ ) = cosh ( θ 3 ) , r 1 = 91,776 a 2 ω 8 − 3,840 a 2 , r 2 = ω 8 − 367,104 a 2 ω 6 + 209,552 a ω 6 + 15,360 a 2 , r 3 = θ 3 ω 6 + 550,656 a 2 ω 4 − 622,496 a ω 4 − 23,040 a 2 , r 4 = ω 4 + 130,242 ω 4 − 367,104 a 2 ω 2 + 616,336 a ω 2 + 15,360 a 2 , r 5 = 246,456 ω 2 + 91,776 a 2 − 203,392 a − 4,512 a 2 F 2 ( θ ) ω 2 − 16,912 a F 2 ( θ ) ω 2 − 26,184 F 2 ( θ ) ω 2 + 264 F 3 ( θ ) ω 2 + 1,224 F 3 ( θ ) ω 2 − 4 cosh ( 6 Δ ) ω 2 r 6 = 29,888 a 2 ( ω 2 − 1 ) 4 + 4 a ( 3,169 ω 2 − 3,386 ) ( ω 2 − 1 ) 2 , θ 1 = 3 Δ , θ 2 = 4 Δ , θ 3 = 5 Δ , θ 4 = ( 6 Δ ) .

It is worth noting that similar techniques can be used to calculate further terms to obtain

(30) ψ ( x , t ) = ∑ n = 0 ∞ ψ n ( x , t ) .

5.1 Discussion

From Figure 1(a) and (b), one can see that the wave amplitude decreases as the fractional orders α and fractal dimension β increase. The fractional order α shows how much the system relies on memory or heredity. A smaller α denotes a stronger memory impact, whereas a large value of α shows a weaker memory effect. The fractional derivative approaches a conventional integer-order derivative when α rises to larger values (nearer to 2). In the simulation, extra damping is added to the system as the fractional order α increases. The greater damping caused by higher fractional orders tends to diminish the wave’s energy over time. The damping effect, which reduces the wave amplitude, is increasingly noticeable as α increases. The wave’s amplitude changes in accordance with its own magnitude due to nonlinear influences. The nonlinear effects lose dominance as α increases, which reduces the amplitude modulation. The reduction in wave amplitude is caused by the attenuation of nonlinear effects. The spatial derivative ψ x x ( x , t ) and the fourth-order spatial derivative ψ x x x x ( x , t ) are dispersive terms in the Boussinesq equation. The fractional order α affects the wave’s dispersion characteristics. The dispersion effects become less significant as α rises, which results in less wave spreading and, ultimately, a reduction in wave amplitude. In Figure 2, we compare the exact solution with the approximative solution obtained by MDLDM. It is noticed that the approximation closely matches the exact solution, demonstrating the efficacy of our mathematical approach. The spatial and temporal evolution of the wave profile (wave shape) as it moves through the medium is represented by the 3D surface plots in Figure 3 to the exact and approximate solution. The amplitude of the wave is shown as a function of both space ( x ) and time ( t ) using 3D surface curves. The surface curves increase when the angular frequency ω is increased. The angular frequency ω and the wave’s temporal oscillation rate are connected in the Boussinesq equation. The wave’s phase speed is inversely proportional to ω . The wave’s temporal oscillations happen more frequently as ω increases, which increases the phase speed. There is dispersion in the Boussinesq equation, which means that waves of different frequencies (wavenumbers) move at various rates. The phase speed of the wave increases as the angular frequency, increases. This indicates that as time increases, the wavefronts (crests and troughs) move through space more quickly. As a result, the wavefronts on the surface curves move “ahead.” The parameters of the wave’s dispersion are also impacted by the rise in phase speed brought on by the greater angular frequency ω . Higher-frequency components move more quickly than lower-frequency components in a dispersive medium. The wave’s frequency rises and its dispersion characteristics alter as ω increases, which makes the surface curves appear to “go ahead.”

$Figure 1 Effect of fractional orders on the solitary profiles for Eq. (30) is shown when (a) the fractional order α \alpha is changing but β \beta is constant and (b) the fractional order β \beta is changing but α \alpha is constant.$

Figure 1

Effect of fractional orders on the solitary profiles for Eq. (30) is shown when (a) the fractional order α is changing but β is constant and (b) the fractional order β is changing but α is constant.

Figure 2

Comparison between exact versus approximate solutions.

Figure 3

Effect of the angular frequency on (a) exact solution and (b) approximate solution.

Example 2

Consider

(31) D t α , β 0 F F P ψ − ψ x x − ψ x x x x − ψ x x 2 = 0 ,

with initial condition

(32) ψ ( x , 0 ) = E sech ( ϑ ) + b − 1 2 , ψ t ( x , 0 ) = E 2 ς 3 E 6 sech ( ϑ ) tanh ( ϑ ) ,

where ϑ = E 6 ( x − x 0 ) . The exact solution of Eq. (31) can be obtained in the form [43]:

(33) ψ ( x , t ) = E sech 2 E 6 ( x − ς t − x 0 ) + b − 1 2 .

Using the techniques discussed in Section 3.1, together with Eq. (32), we obtain the series solution of Eq. (31) in the form:

ψ 0 = E sech ( ϑ ) + E 2 ς t 3 E 6 sech ( ϑ ) tanh ( ϑ ) + b − 1 2 , ψ 1 = E 2 Γ ( β + 1 ) t α + β − 1 108 Γ ( α + β ) × [ 3 sech ( ϑ ) ( 12 b + E ) + 72 sech 2 ( ϑ ) ( E + E sech 2 ( ϑ ) − b sech ( ϑ ) ) − 4 E sech 3 ( ϑ ) ( 15 + 26 sech ( ϑ ) ) ] + 6 ς Γ ( β + 2 ) t α + β 108 Γ ( α + β + 1 ) × [ sech ( ϑ ) tanh ( ϑ ) ( 60 E 3 sech 2 ( ϑ ) − 48 E 3 sech ( ϑ ) − 12 b − E 3 − 120 E 3 sech 4 ( ϑ ) + 144 E 3 sech 3 ( ϑ ) ) + 72 b sech 2 ( ϑ ) ] + E 4 ς 2 Γ ( β + 3 ) t α + β + 1 27 Γ ( α + β + 2 ) [ 12 sech 2 ( ϑ ) − 66 sech 4 ( ϑ ) + 60 ] ,

and so on. Finally,

(34) ψ = ∑ n = 0 ∞ ψ n .

5.2 Discussion

In the above example, E is the amplitude of the solitonic solution, b is a fitting parameter, and ς = ± 2 ( b + E ∕ 3 ) . We considered − 30 ≤ x ≤ 30 for numerical computation. The calculations were carried out for various time ( t ) . The outcomes demonstrate that wave propagation has a symmetrical form. The initial standing wave breaks up into two smaller, more symmetric solitary waves that are travelling in opposite directions and have an oscillating tail. The solitary wave’s amplitude diminishes over time, and its tail expands over a long distance. The absolute difference between the solutions Eqs (34) and (33) is shown in the left panel of Figure 4. It can be seen that the approximate and exact solutions are comparatively close to each other, which shows the solitonic behaviour of the given system. The estimated solution is displayed for various values of time (t) in the right panel. It has been noted that the single-wave solution increases over time. The findings are shown in Figure 5, right panel while preserving β = 1 and varying α by 1.0, 1.5, and 2.9 to show the effect of variable, time ( t ) on solution [Eq. (34)]. It is clear that negligible changes occur in the amplitude when the fractional derivative changes for different values of α . Similarly, the system’s solitonic behaviour is extremely close to one another for various values of β for time t = 1 . Figure 6 displays a surface plot for Table 1. Finally, Figure 7 demonstrates the error for the solution of the Boussinesq equation for a range of time steps. It is observed that the inclination shadows of the ordinary form of error reduce as the step size increases.

$Figure 4 Comparing Eq. (33) and Eq. (34) for α = 2 \alpha =2 and β = 1 \beta =1 . The estimated solutions for various times (t) are displayed in the right panel.$

Figure 4

Comparing Eq. (33) and Eq. (34) for α = 2 and β = 1 . The estimated solutions for various times (t) are displayed in the right panel.

$Figure 5 For the parameters used in Figure 4, the solution profiles for Eq. (34) for different values of α = 2 \alpha =2 and β = 1 \beta =1 .$

Figure 5

For the parameters used in Figure 4, the solution profiles for Eq. (34) for different values of α = 2 and β = 1 .

$Figure 6 The left panel shows the surface plot for Eq. (34), while the right panel shows surface plot for Eq. (33) for β = 1 \beta =1 and α = 2 \alpha =2 .$

Figure 6

The left panel shows the surface plot for Eq. (34), while the right panel shows surface plot for Eq. (33) for β = 1 and α = 2 .

Table 1

Error estimation table for the particular values considered in Figures 4–6

( x , t )	Exact	Appro	∣ Exact–Appro ∣	(x,t)	Exact	Appro	∣ Exact–Appro ∣
( ‒ 5 , 0.05 )	0.0956	0.0956	1.6914 × 1 0 ‒ 5	( ‒ 4.5 , 0.05 )	0.1086	0.1086	1.4248 × 1 0 ‒ 5
( ‒ 4 , 0.05 )	0.1223	0.1223	1.0746 × 1 0 ‒ 5	( ‒ 3.5 , 0.05 )	0.1363	0.1363	6.4842 × 1 0 ‒ 6
( ‒ 3 , 0.05 )	0.1502	0.1502	1.6369 × 1 0 ‒ 6	( ‒ 2.5 , 0.05 )	0.1635	0.1635	3.5154 × 1 0 ‒ 6
( ‒ 2 , 0.05 )	0.1756	0.1755	8.6007 × 1 0 ‒ 6	( ‒ 1.5 , 0.05 )	0.1857	0.1857	1.3191 × 1 0 ‒ 5
( ‒ 1 , 0.05 )	0.1935	0.1935	1.6855 × 1 0 ‒ 5	( ‒ 0.5 , 0.05 )	0.1984	0.1983	1.9222 × 1 0 ‒ 5
( 0 , 0.05 )	0.2000	0.2000	2.0041 × 1 0 ‒ 5	( 1 , 0.05 )	0.1935	0.1935	1.6855 × 1 0 ‒ 5
( 1.5 , 0.05 )	0.1857	0.1857	1.3191 × 1 0 ‒ 5	( 2 , 0.05 )	0.1756	0.1755	8.6007 × 1 0 ‒ 6
( 2.5 , 0.05 )	0.1635	0.1635	3.5154 × 1 0 ‒ 6	( 3 , 0.05 )	0.1502	0.1502	1.6369 × 1 0 ‒ 6
( 3.5 , 0.05 )	0.1363	0.1363	6.4842 × 1 0 ‒ 6	( 4 , 0.05 )	0.1223	0.1223	1.0746 × 1 0 ‒ 5
( 4.5 , 0.05 )	0.1086	0.1086	1.4248 × 1 0 ‒ 5	( 5 , 0.05 )	0.0956	0.0956	1.6914 × 1 0 ‒ 5

Figure 7

Surface plot for absolute error approbation shown in Table 1.

6 Conclusion

The nonlinear fractional Boussinesq equation with a Caputo derivative with fractal dimension is investigated. The MDLDM is used to calculate the general approximate solution of the model. One can see that the proposed technique is comparatively a suitable analytical technique for studying the proposed model with good precision in space and time. The findings are accurately supported by numerical investigations. It is demonstrated that the suggested approach gives an insignificant estimate of the solution error and mathematical precision in keeping with all the invariants of the considered model. The numerical results show that when the fractional and fractal orders ( α , β ) increase, the wave amplitude decreases. For α = 2 , the fractional derivative approaches to an integer order. Due to a nonlinear phenomenon, the wave’s amplitude varies with its own magnitude. As α increases, the nonlinear effects become unstable, and the amplitude modulation decreases. The attenuation of nonlinear effects results in a decrease in the wave amplitude. Additionally, it is revealed that the wave’s dispersion characteristics are influenced by the fractional order α . As α increases, the effects of dispersion become less visible, which reduces wave spreading and, eventually, reduces wave amplitude.

Acknowledgments

This research was supported by Researchers Supporting Project number (RSP2023R447), King Saud University, Riyadh, Saudi Arabia.

Author contributions: Author has accepted responsibility for the entire content of this manuscript and approved its submission.
Conflict of interest: It is declared that the author has no conflict of interest regarding this manuscript.
Data availability statement: The data regarding this manuscript is available within the manuscript.

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Received: 2023-08-31

Revised: 2023-09-21

Accepted: 2023-09-24

Published Online: 2023-10-24

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https://doi.org/10.1515/nleng-2022-0338

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fractal-fractional operator; double-Laplace transform; Boussinesq equation; decomposition technique; power law kernel; convergence and stability analysis

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