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Two Novel Computational Techniques for Solving Nonlinear Time-Fractional Lax’s Korteweg-de Vries Equation

Basic Science Department, College of Science and Theoretical Studies, Saudi Electronic University, Riyadh 11673, Saudi Arabia

Department of Mathematics, College of Science and Humanities, Prince Sattam bin Abdulaziz University, Al Kharj 11942, Saudi Arabia

Department of Mathematics, Abdul Wali Khan University Mardan, Mardan 23200, Pakistan

Author to whom correspondence should be addressed.

Axioms 2023, 12(4), 400; https://doi.org/10.3390/axioms12040400

Submission received: 13 March 2023 / Revised: 6 April 2023 / Accepted: 18 April 2023 / Published: 20 April 2023

(This article belongs to the Special Issue Fractional Calculus and the Applied Analysis)

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Abstract

This article investigates the seventh-order Lax’s Korteweg–de Vries equation using the Yang transform decomposition method (YTDM) and the homotopy perturbation transform method (HPTM). The physical phenomena that emerge in physics, engineering and chemistry are mathematically expressed by this equation. For instance, the KdV equation was constructed to represent a wide range of physical processes involving the evolution and interaction of nonlinear waves. In the Caputo sense, the fractional derivative is considered. We employed the Yang transform, the Adomian decomposition method and the homotopy perturbation method to obtain the solution to the time-fractional Lax’s Korteweg–de Vries problem. We examined and compared a particular example with the actual result to verify the approaches. By utilizing these methods, we can construct recurrence relations that represent the solution to the problem that is being proposed, and we are then able to present graphical representations that enable us to visually examine all of the results in the proposed case for different fractional order values. Furthermore, the results of the current approach exhibit a good correlation with the precise solution to the problem being studied. Furthermore, the present study offers an example of error analysis. The numerical outcomes obtained by applying the provided approaches demonstrate that the techniques are easy to use and have superior computational performance.

Keywords:

Yang transform; Caputo operator; time-fractional Lax’s Korteweg–de Vries equation; Adomian decomposition method; homotopy perturbation method

MSC:

34A25; 26A33; 35Q53; 35A20

1. Introduction

Researchers have become more interested in fractional calculus (FC) due to the fractional modeling it offers for various natural processes. FC is helpful in illustrating the memory and hereditary properties of numerous events. Fractional differentiation is the expansion of the integer to the non-integer order of differentiation. FC is related to practical efforts and is commonly employed in various fields. This is because fractional calculus is a valuable tool for explaining the dynamic behavior of numerous physical systems. Fractional differential equations (FDEs) are notable for providing memory and transmission qualities for numerous mathematical models [1,2]. The integer order differential operator is commonly considered a local operator, whereas the fractional order differential operator is non-local. This explains how the past and present states of a system influence its future state. This enables fractional calculus to be more useful, which is one of the reasons it is becoming more popular [3,4,5,6]. Fractional differential equations have grown in importance and popularity as a result of their various engineering and scientific applications. For instance, these equations are more frequently employed to explain issues in a variety of physical systems, including quantum physics [7], diffusion processes [8], viscoelasticity and fluid mechanics [9], propagation of complex acoustic oscillations [10], human diseases [11], optics [12], biomathematics [13], chaos theory [14] and many more. In these and other applications, the major benefit of fractional differential equations is their non-locality.

Nonlinear equations are essential for explaining a wide range of events, not just in physics but also in other fields of science and engineering. In order to adequately represent physical processes, a mathematical model involving nonlinear partial differential equations (PDEs) was constructed, because relatively few issues in physics, or in fact in any field of natural science, can be resolved by direct solutions. An overview of the related phenomena is given by the PDE solution’s features. Due to their importance in a wide range of fields, numerous technologies have been implemented to examine the precise and computational solutions of fractional differential equations. Along with modeling, the solutions’ divergence and convergence are equally important. In some cases, it can be very difficult to find analytical solutions to fractional differential equations. This has increased the significance of developing numerical solutions to these issues. There are numerous effective methods in the literature for constructing semi-analytical and computational solutions of fractional differential equations, including the first integral method [15], the extended direct algebraic method [16], the modified Kudryashov method [17], the finite difference method [18], the optimal homotopy asymptotic method (OHAM) [19], the Adomian decomposition method [20], the standard reductive perturbation method [21], the homotopy perturbation transform technique [22,23], the Elzaki transform decomposition method [24], the Haar wavelet method [25], the fractional sub-equation method [26], the differential transform method [27], the Khater method [28] and the variational iteration procedure with transformation [29].

An example of a partial differential equation is the Korteweg–de Vries (KdV) equation, which has been used to explain a variety of physical phenomena as a model for the evolution and interaction of nonlinear waves. It was derived as an evolution equation that governed the propagation of one-dimensional, long, low-amplitude surface gravity waves in a shallow water channel [30]. The KdV equation has now been used in many different branches of physics, including collision-free hydromagnetic waves, stratified internal waves, ion-acoustic waves, plasma physics, and lattice dynamics [31]. Using a KdV model, some hypothetical physical events in the context of quantum mechanics have been expressed. It is used as a model for shock wave propagation, turbulence, solitons, mass transport in fluid dynamics, boundary layer behavior, continuum mechanics and aerodynamics [32]. Hence, in the literature, various finite-difference- [33] and finite-element-based (continuous [34,35,36] and discontinuous [37,38] Galerkin/Petrov–Galerkin) numerical approaches are found to address transport, dispersion, and convection–diffusion problems. The general form of the seventh-order time-fractional Lax’s Korteweg–de Vries equation is given as

\begin{matrix} D_{ψ}^{σ} T (ζ, ψ) + μ_{1} T^{3} (ζ, ψ) T_{ζ} (ζ, ψ) + μ_{2} T_{ζ}^{3} (ζ, ψ) + μ_{3} T (ζ, ψ) T_{ζ} (ζ, ψ) T_{ζ ζ} (ζ, ψ) + μ_{4} T^{2} (ζ, ψ) T_{ζ ζ ζ} (ζ, ψ) \\ + μ_{5} T_{ζ ζ} (ζ, ψ) T_{ζ ζ ζ} (ζ, ψ) + μ_{6} T_{ζ} (ζ, ψ) T_{ζ ζ ζ ζ} (ζ, ψ) + μ_{7} T (ζ, ψ) T_{ζ ζ ζ ζ ζ} (ζ, ψ) + T_{ζ ζ ζ ζ ζ ζ ζ} (ζ, ψ) = 0, \\ 0 < σ \leq 1, \end{matrix}

(1)

where the fractional derivative’s order is indicated by the parameter

σ

and the constant parameters

μ_{i}, i = 1, 2, \dots, 7

are finite and cannot be zero.

T (ζ, ψ)

is a function of the dynamic wave profile of the spatiotemporal patterns that will be eventually derived. The complicated physical processes that arise in physics, biology, engineering and chemistry are mathematically modeled by these equations. Examples include quantum mechanics, plasma physics, the propagation of long waves in shallow water under gravity, fluid mechanics and nonlinear optics. The pseudospectral method [39], variational iteration method [40] and modified Cole–Hopf transformation method [41] are a few of the analytical techniques utilized by numerous scholars to solve the seventh-order Lax’s Korteweg–de Vries equation. The residual power series approach and perturbation iteration algorithm were used in [42] to analyze numerical solutions of the time-fractional Rosenau–Hyman equation, a model that is similar to the KdV model.

The adomian decomposition approach, the homotopy perturbation method (HPM) and the Yang transform (YT) are presented in this paper. The homotopy perturbation method (HPM), developed by Ji-Huan He of Shanghai University, was first used to address nonlinear problems in science and technology in 1998 [43,44]. Many mathematicians used the homotopy perturbation method to solve nonlinear equations that came up in engineering and scientific studies [45,46,47,48]. It is simpler to estimate the series terms using the proposed method than it is using the usual Adomian approach since it does not compute the fractional derivative or fractional integrals in the recursive mechanism. Since YTDM does not require prescribed assumptions, discretization, perturbation and round-off errors are avoided. In the literature, YTDM has been used to solve a wide range of differential equations, including the time-fractional Belousov–Zhabotinskii reaction [49], fractional-order diffusion equations [50] and the time-fractional Fisher equation [51].

The current article is structured as follows: We start with the fundamental idea of fractional calculus in Section 2. We discuss the fundamental concepts of the suggested methods in Section 3 and Section 4. The time-fractional Lax’s Korteweg–de Vries problem is solved using these techniques in Section 5 with the provided initial conditions. Lastly, Section 6 provides the conclusion.

2. Preliminaries

The FC and the Yang transform (YT) are discussed here which will be employed in this framework.

Definition 1.

The fractional derivative Caputo operator is given by [4]

\begin{matrix} D_{ψ}^{σ} T (ζ, ψ) = \frac{1}{Γ (k - σ)} \int_{0}^{ψ} {(ψ - σ)}^{k - σ - 1} T^{(k)} (ζ, σ) d σ, k - 1 < σ \leq k, k \in N . \end{matrix}

(2)

Definition 2.

The YT of the function is given by [52]

Y {T (ψ)} = M (u) = \int_{0}^{\infty} e^{\frac{- ψ}{u}} T (ψ) d ψ, ψ > 0,

(3)

where u is the transform variable.

The inverse YT is

Y^{- 1} {M (u)} = T (ψ) .

(4)

Definition 3.

The inverse Yang transform

Y^{- 1}

is defined by

Y^{- 1} [Y (u)] = T (ψ) = \frac{1}{2 π ι} \int_{σ - ι \infty}^{σ + ι \infty} T (\frac{1}{u}) e^{u ψ} u d u = Σ r e s i d u e s o f T (\frac{1}{u}) e^{u ψ} u .

(5)

Definition 4.

The YT of the fractional derivative function is given by [52]

Y {T^{σ} (ψ)} = \frac{M (u)}{u^{σ}} - \sum_{k = 0}^{n - 1} \frac{T^{k} (0)}{u^{σ - (k + 1)}}, 0 < σ \leq n .

(6)

3. General Implementation of HPTM

Let us assume the general fractional PDE of the form:

D_{ψ}^{σ} T (ζ, ψ) = P_{1} [ζ] T (ζ, ψ) + Q_{1} [ζ] T (ζ, ψ), 0 < σ \leq 1,

(7)

with initial guess

T (ζ, 0) = ξ (ζ),

where

D_{ψ}^{σ} = \frac{\partial^{σ}}{\partial ψ^{σ}}

illustrates the Caputo operator,

P_{1} [ζ]

is a linear function and

Q_{1} [ζ]

is a nonlinear function.

On taking YT, we obtain

Y [D_{ψ}^{σ} T (ζ, ψ)] = Y [P_{1} [ζ] T (ζ, ψ) + Q_{1} [ζ] T (ζ, ψ)],

(8)

\frac{1}{u^{σ}} {M (u) - u T (ζ, 0)} = Y [P_{1} [ζ] T (ζ, ψ) + Q_{1} [ζ] T (ζ, ψ)] .

(9)

By simplification of the above equation, we have

M (u) = u T (ζ, 0) + u^{σ} Y [P_{1} [ζ] T (ζ, ψ) + Q_{1} [ζ] T (ζ, ψ)] .

(10)

On taking inverse YT, we have

T (ζ, ψ) = T (ζ, 0) + Y^{- 1} [u^{σ} Y [P_{1} [ζ] T (ζ, ψ) + Q_{1} [ζ] T (ζ, ψ)]] .

(11)

Now, in terms of HPM

T (ζ, ψ) = \sum_{k = 0}^{\infty} ϵ^{k} T_{k} (ζ, ψ) .

(12)

with parameter

ϵ \in [0, 1]

The nonlinear term is taken as

Q_{1} [ζ] T (ζ, ψ) = \sum_{k = 0}^{\infty} ϵ^{k} H_{n} (T) .

(13)

In addition,

H_{k} (T)

illustrates He’s polynomial and is written as

H_{n} (T_{0}, T_{1}, \dots, T_{n}) = \frac{1}{Γ (n + 1)} D_{ϵ}^{k} {[Q_{1} (\sum_{k = 0}^{\infty} ϵ^{i} T_{i})]}_{ϵ = 0},

(14)

where

D_{ϵ}^{k} = \frac{\partial^{k}}{\partial ϵ^{k}} .

By inserting (14) and (15) into (12), we obtain

\sum_{k = 0}^{\infty} ϵ^{k} T_{k} (ζ, ψ) = T (ζ, 0) + ϵ \times (Y^{- 1} [u^{σ} Y {P_{1} \sum_{k = 0}^{\infty} ϵ^{k} T_{k} (ζ, ψ) + \sum_{k = 0}^{\infty} ϵ^{k} H_{k} (T)}]) .

(15)

Comparing the coefficients of like powers of

ϵ

yields

\begin{matrix} ϵ^{0} : T_{0} (ζ, ψ) = T (ζ, 0), \\ ϵ^{1} : T_{1} (ζ, ψ) = Y^{- 1} [u^{σ} Y (P_{1} [ζ] T_{0} (ζ, ψ) + H_{0} (T))], \\ ϵ^{2} : T_{2} (ζ, ψ) = Y^{- 1} [u^{σ} Y (P_{1} [ζ] T_{1} (ζ, ψ) + H_{1} (T))], \\ . \\ . \\ . \\ ϵ^{k} : T_{k} (ζ, ψ) = Y^{- 1} [u^{σ} Y (P_{1} [ζ] T_{k - 1} (ζ, ψ) + H_{k - 1} (T))], \\ k > 0, k \in N . \end{matrix}

(16)

Lastly, the HPTM solution in series form is taken as

T (ζ, ψ) = lim_{M \to \infty} \sum_{k = 1}^{M} T_{k} (ζ, ψ) .

(17)

4. General Implementation of the YTDM

Let us assume the general fractional PDE of the form:

D_{ψ}^{σ} T (ζ, ψ) = P_{1} (ζ, ψ) + Q_{1} (ζ, ψ), 0 < σ \leq 1,

(18)

with initial guess

T (ζ, 0) = ξ (ζ),

where

D_{ψ}^{σ} = \frac{\partial^{σ}}{\partial ψ^{σ}}

illustrates the Caputo operator,

P_{1}

is a linear function and

Q_{1}

is a non-linear function.

On taking YT, we obtain

\begin{matrix} Y [D_{ψ}^{σ} T (ζ, ψ)] = Y [P_{1} (ζ, ψ) + Q_{1} (ζ, ψ)], \\ \frac{1}{u^{σ}} {M (u) - u T (ζ, 0)} = Y [P_{1} (ζ, ψ) + Q_{1} (ζ, ψ)] . \end{matrix}

(19)

By simplification of the above equation, we have

M (u) = u T (ζ, 0) + u^{σ} Y [P_{1} (ζ, ψ) + Q_{1} (ζ, ψ)] .

(20)

On taking inverse YT, we have

T (ζ, ψ) = T (ζ, 0) + Y^{- 1} [u^{σ} Y [P_{1} (ζ, ψ) + Q_{1} (ζ, ψ)]] .

(21)

Now, in terms of the YTDM,

T (ζ, ψ) = \sum_{m = 0}^{\infty} T_{m} (ζ, ψ) .

(22)

The nonlinear term is taken as

Q_{1} (ζ, ψ) = \sum_{m = 0}^{\infty} A_{m},

(23)

with

A_{m} = \frac{1}{m!} {[\frac{\partial^{m}}{\partial ℓ^{m}} \{Q_{1} (\sum_{k = 0}^{\infty} ℓ^{k} T_{k} (ζ, ψ))\}]}_{ℓ = 0} .

(24)

By inserting (24) and (26) into (23), we obtain

\sum_{m = 0}^{\infty} T_{m} (ζ, ψ) = T (ζ, 0) + Y^{- 1} u^{σ} [Y \{P_{1} (\sum_{m = 0}^{\infty} T_{m} (ζ, ψ)) + \sum_{m = 0}^{\infty} A_{m}\}] .

(25)

Comparing both sides of the above equation yields

T_{0} (ζ, ψ) = T (ζ, 0),

(26)

T_{1} (ζ, ψ) = Y^{- 1} [u^{σ} Y {P_{1} (T_{0} (ζ, ψ)) + A_{0}}] .

Lastly, the YTDM general solution for

m \geq 1

T_{m + 1} (ζ, ψ) = Y^{- 1} [u^{σ} Y {P_{1} (T_{m} (ζ, ψ)) + A_{m}}] .

5. Application

Example 1.

Let us assume the seventh order TFLK–dV equation with

μ_{1} = 140, μ_{2} = 70, μ_{3} = 280, μ_{4} = 70, μ_{5} = 70, μ_{6} = 42, μ_{7} = 14

as:

\begin{matrix} D_{ψ}^{σ} T (ζ, ψ) = - 140 T^{3} (ζ, ψ) T_{ζ} (ζ, ψ) - 70 T_{ζ}^{3} (ζ, ψ) - 280 T (ζ, ψ) T_{ζ} (ζ, ψ) T_{ζ ζ} (ζ, ψ) - 70 T^{2} (ζ, ψ) T_{ζ ζ ζ} (ζ, ψ) \\ - 70 T_{ζ ζ} (ζ, ψ) T_{ζ ζ ζ} (ζ, ψ) - 42 T_{ζ} (ζ, ψ) T_{ζ ζ ζ ζ} (ζ, ψ) - 14 T (ζ, ψ) T_{ζ ζ ζ ζ ζ} (ζ, ψ) - T_{ζ ζ ζ ζ ζ ζ ζ} (ζ, ψ), 0 < σ \leq 1, \end{matrix}

(27)

with initial guess

T (ζ, 0) = 2 ρ^{2} {sech}^{2} (ρ ζ)),

where ρ is an arbitrary constant.

On taking YT, we obtain

\begin{matrix} Y [\frac{\partial^{σ} T}{\partial ψ^{σ}}] = Y [- 140 T^{3} (ζ, ψ) T_{ζ} (ζ, ψ) - 70 T_{ζ}^{3} (ζ, ψ) - 280 T (ζ, ψ) T_{ζ} (ζ, ψ) T_{ζ ζ} (ζ, ψ) - 70 T^{2} (ζ, ψ) T_{ζ ζ ζ} (ζ, ψ) \\ - 70 T_{ζ ζ} (ζ, ψ) T_{ζ ζ ζ} (ζ, ψ) - 42 T_{ζ} (ζ, ψ) T_{ζ ζ ζ ζ} (ζ, ψ) - 14 T (ζ, ψ) T_{ζ ζ ζ ζ ζ} (ζ, ψ) - T_{ζ ζ ζ ζ ζ ζ ζ} (ζ, ψ)] . \end{matrix}

(28)

By simplification of the above equation, we have

\begin{matrix} \frac{1}{u^{σ}} {M (u) - u T (ζ, 0)} = Y [- 140 T^{3} (ζ, ψ) T_{ζ} (ζ, ψ) - 70 T_{ζ}^{3} (ζ, ψ) - 280 T (ζ, ψ) T_{ζ} (ζ, ψ) T_{ζ ζ} (ζ, ψ) - 70 T^{2} (ζ, ψ) \\ T_{ζ ζ ζ} (ζ, ψ) - 70 T_{ζ ζ} (ζ, ψ) T_{ζ ζ ζ} (ζ, ψ) - 42 T_{ζ} (ζ, ψ) T_{ζ ζ ζ ζ} (ζ, ψ) - 14 T (ζ, ψ) T_{ζ ζ ζ ζ ζ} (ζ, ψ) - T_{ζ ζ ζ ζ ζ ζ ζ} (ζ, ψ)], \end{matrix}

(29)

\begin{matrix} M (u) = u T (ζ, 0) + u^{σ} [- 140 T^{3} (ζ, ψ) T_{ζ} (ζ, ψ) - 70 T_{ζ}^{3} (ζ, ψ) - 280 T (ζ, ψ) T_{ζ} (ζ, ψ) T_{ζ ζ} (ζ, ψ) - 70 T^{2} (ζ, ψ) \\ T_{ζ ζ ζ} (ζ, ψ) - 70 T_{ζ ζ} (ζ, ψ) T_{ζ ζ ζ} (ζ, ψ) - 42 T_{ζ} (ζ, ψ) T_{ζ ζ ζ ζ} (ζ, ψ) - 14 T (ζ, ψ) T_{ζ ζ ζ ζ ζ} (ζ, ψ) - T_{ζ ζ ζ ζ ζ ζ ζ} (ζ, ψ)] . \end{matrix}

(30)

On taking inverse YT, we have

\begin{matrix} T (ζ, ψ) = T (ζ, 0) + Y^{- 1} [u^{σ} {Y (- 140 T^{3} (ζ, ψ) T_{ζ} (ζ, ψ) - 70 T_{ζ}^{3} (ζ, ψ) - 280 T (ζ, ψ) T_{ζ} (ζ, ψ) T_{ζ ζ} (ζ, ψ) - 70 \\ T^{2} (ζ, ψ) T_{ζ ζ ζ} (ζ, ψ) - 70 T_{ζ ζ} (ζ, ψ) T_{ζ ζ ζ} (ζ, ψ) - 42 T_{ζ} (ζ, ψ) T_{ζ ζ ζ ζ} (ζ, ψ) - 14 T (ζ, ψ) T_{ζ ζ ζ ζ ζ} (ζ, ψ) - T_{ζ ζ ζ ζ ζ ζ ζ} (ζ, ψ))}], \\ T (ζ, ψ) = 2 ρ^{2} {sech}^{2} (ρ ζ)) + Y^{- 1} [u^{σ} {Y (- 140 T^{3} (ζ, ψ) T_{ζ} (ζ, ψ) - 70 T_{ζ}^{3} (ζ, ψ) - 280 T (ζ, ψ) T_{ζ} (ζ, ψ) T_{ζ ζ} (ζ, ψ) - \\ 70 T^{2} (ζ, ψ) T_{ζ ζ ζ} (ζ, ψ) - 70 T_{ζ ζ} (ζ, ψ) T_{ζ ζ ζ} (ζ, ψ) - 42 T_{ζ} (ζ, ψ) T_{ζ ζ ζ ζ} (ζ, ψ) - 14 T (ζ, ψ) T_{ζ ζ ζ ζ ζ} (ζ, ψ) - T_{ζ ζ ζ ζ ζ ζ ζ} (ζ, ψ))}] . \end{matrix}

(31)

Now, by means of the HPM, the non-linear term is represented by

H_{k} (T)

\begin{matrix} \sum_{k = 0}^{\infty} ϵ^{k} T_{k} (ζ, ψ) = 2 ρ^{2} {sech}^{2} (ρ ζ)) + ϵ (Y^{- 1} [u^{σ} Y [- 140 (\sum_{k = 0}^{\infty} ϵ^{k} H_{k} (T)) - 70 (\sum_{k = 0}^{\infty} ϵ^{k} H_{k} (T)) - \\ 280 (\sum_{k = 0}^{\infty} ϵ^{k} H_{k} (T)) - 70 (\sum_{k = 0}^{\infty} ϵ^{k} H_{k} (T)) - 70 (\sum_{k = 0}^{\infty} ϵ^{k} H_{k} (T)) - 42 (\sum_{k = 0}^{\infty} ϵ^{k} H_{k} (T)) - 14 (\sum_{k = 0}^{\infty} ϵ^{k} H_{k} (T)) - \\ {(\sum_{k = 0}^{\infty} ϵ^{k} T_{k} (ζ, ψ))}_{ζ ζ ζ ζ ζ ζ ζ}]]) . \end{matrix}

(32)

Comparing the coefficients of like powers of ϵ yields

\begin{matrix} ϵ^{0} : T_{0} (ζ, ψ) = 2 ρ^{2} {sech}^{2} (ρ ζ)), \\ ϵ^{1} : T_{1} (ζ, ψ) = \frac{256 ρ^{9} ψ^{σ} tanh (ρ ζ) {sech}^{2} (ρ ζ)}{Γ (σ + 1)}, \\ ϵ^{2} : T_{2} (ζ, ψ) = \frac{16384 ψ^{2 σ} ρ^{16} (cosh (2 ρ ζ) - 2) {sech}^{4} (ρ ζ)}{Γ (2 σ + 1)}, \\ ⋮ \end{matrix}

Lastly, the HPTM solution in series form is taken as

\begin{matrix} T (ζ, ψ) = T_{0} (ζ, ψ) + T_{1} (ζ, ψ) + T_{2} (ζ, ψ) + \dots \\ T (ζ, ψ) = 2 ρ^{2} {sech}^{2} (ρ ζ)) + \frac{256 ρ^{9} ψ^{σ} tanh (ρ ζ) {sech}^{2} (ρ ζ)}{Γ (σ + 1)} + \frac{16384 ψ^{2 σ} ρ^{16} (cosh (2 ρ ζ) - 2) {sech}^{4} (ρ ζ)}{Γ (2 σ + 1)} + \dots \end{matrix}

The YTDM for solving the TFLK–dV equation

On taking YT, we obtain

\begin{matrix} Y \{\frac{\partial^{σ} T}{\partial ψ^{σ}}\} = Y [- 140 T^{3} (ζ, ψ) T_{ζ} (ζ, ψ) - 70 T_{ζ}^{3} (ζ, ψ) - 280 T (ζ, ψ) T_{ζ} (ζ, ψ) T_{ζ ζ} (ζ, ψ) - 70 T^{2} (ζ, ψ) T_{ζ ζ ζ} (ζ, ψ) \\ - 70 T_{ζ ζ} (ζ, ψ) T_{ζ ζ ζ} (ζ, ψ) - 42 T_{ζ} (ζ, ψ) T_{ζ ζ ζ ζ} (ζ, ψ) - 14 T (ζ, ψ) T_{ζ ζ ζ ζ ζ} (ζ, ψ) - T_{ζ ζ ζ ζ ζ ζ ζ} (ζ, ψ)] . \end{matrix}

(33)

By simplification of the above equation, we have

\begin{matrix} \frac{1}{u^{σ}} {M (u) - u T (ζ, 0)} = Y [- 140 T^{3} (ζ, ψ) T_{ζ} (ζ, ψ) - 70 T_{ζ}^{3} (ζ, ψ) - 280 T (ζ, ψ) T_{ζ} (ζ, ψ) T_{ζ ζ} (ζ, ψ) - 70 \\ T^{2} (ζ, ψ) T_{ζ ζ ζ} (ζ, ψ) - 70 T_{ζ ζ} (ζ, ψ) T_{ζ ζ ζ} (ζ, ψ) - 42 T_{ζ} (ζ, ψ) T_{ζ ζ ζ ζ} (ζ, ψ) - 14 T (ζ, ψ) T_{ζ ζ ζ ζ ζ} (ζ, ψ) - T_{ζ ζ ζ ζ ζ ζ ζ} (ζ, ψ)], \end{matrix}

(34)

\begin{matrix} M (u) = u T (ζ, 0) + u^{σ} Y [- 140 T^{3} (ζ, ψ) T_{ζ} (ζ, ψ) - 70 T_{ζ}^{3} (ζ, ψ) - 280 T (ζ, ψ) T_{ζ} (ζ, ψ) T_{ζ ζ} (ζ, ψ) - 70 \\ T^{2} (ζ, ψ) T_{ζ ζ ζ} (ζ, ψ) - 70 T_{ζ ζ} (ζ, ψ) T_{ζ ζ ζ} (ζ, ψ) - 42 T_{ζ} (ζ, ψ) T_{ζ ζ ζ ζ} (ζ, ψ) - 14 T (ζ, ψ) T_{ζ ζ ζ ζ ζ} (ζ, ψ) - T_{ζ ζ ζ ζ ζ ζ ζ} (ζ, ψ)] . \end{matrix}

(35)

On taking inverse YT, we have

\begin{matrix} T (ζ, ψ) = T (ζ, 0) + Y^{- 1} [u^{σ} {Y [- 140 T^{3} (ζ, ψ) T_{ζ} (ζ, ψ) - 70 T_{ζ}^{3} (ζ, ψ) - 280 T (ζ, ψ) T_{ζ} (ζ, ψ) T_{ζ ζ} (ζ, ψ) - 70 \\ T^{2} (ζ, ψ) T_{ζ ζ ζ} (ζ, ψ) - 70 T_{ζ ζ} (ζ, ψ) T_{ζ ζ ζ} (ζ, ψ) - 42 T_{ζ} (ζ, ψ) T_{ζ ζ ζ ζ} (ζ, ψ) - 14 T (ζ, ψ) T_{ζ ζ ζ ζ ζ} (ζ, ψ) - T_{ζ ζ ζ ζ ζ ζ ζ} (ζ, ψ)]}], \\ T (ζ, ψ) = 2 ρ^{2} {sech}^{2} (ρ ζ)) + Y^{- 1} [u^{σ} {Y [- 140 T^{3} (ζ, ψ) T_{ζ} (ζ, ψ) - 70 T_{ζ}^{3} (ζ, ψ) - 280 T (ζ, ψ) T_{ζ} (ζ, ψ) \\ T_{ζ ζ} (ζ, ψ) - 70 T^{2} (ζ, ψ) T_{ζ ζ ζ} (ζ, ψ) - 70 T_{ζ ζ} (ζ, ψ) T_{ζ ζ ζ} (ζ, ψ) - 42 T_{ζ} (ζ, ψ) T_{ζ ζ ζ ζ} (ζ, ψ) - 14 T (ζ, ψ) T_{ζ ζ ζ ζ ζ} (ζ, ψ) - \\ T_{ζ ζ ζ ζ ζ ζ ζ} (ζ, ψ)]}] . \end{matrix}

(36)

Now, in terms of the ADM,

T (ζ, ψ) = \sum_{m = 0}^{\infty} T_{m} (ζ, ψ) .

(37)

The nonlinear term is taken as

T^{3} (ζ, ψ) T_{ζ} (ζ, ψ) = \sum_{m = 0}^{\infty} A_{m}, T_{ζ}^{3} (ζ, ψ) = \sum_{m = 0}^{\infty} B_{m},

T (ζ, ψ) T_{ζ} (ζ, ψ) T_{ζ ζ} (ζ, ψ) = \sum_{m = 0}^{\infty} C_{m}, T^{2} (ζ, ψ) T_{ζ ζ ζ} (ζ, ψ) = \sum_{m = 0}^{\infty} D_{m}, T_{ζ ζ} (ζ, ψ) T_{ζ ζ ζ} (ζ, ψ) = \sum_{m = 0}^{\infty} E_{m},

T_{ζ} (ζ, ψ) T_{ζ ζ ζ ζ} (ζ, ψ) = \sum_{m = 0}^{\infty} T_{m}, T (ζ, ψ) T_{ζ ζ ζ ζ ζ} (ζ, ψ) = \sum_{m = 0}^{\infty} G_{m} .

Thus, we obtain

\begin{matrix} \sum_{m = 0}^{\infty} T_{m} (ζ, ψ) = T (ζ, 0) + Y^{- 1} [u^{σ} Y [- 140 \sum_{m = 0}^{\infty} A_{m} - 70 \sum_{m = 0}^{\infty} B_{m} - 280 \sum_{m = 0}^{\infty} C_{m} - 70 \sum_{m = 0}^{\infty} D_{m} - \\ 70 \sum_{m = 0}^{\infty} E_{m} - 42 \sum_{m = 0}^{\infty} T_{m} - 14 \sum_{m = 0}^{\infty} G_{m} - T_{ζ ζ ζ ζ ζ ζ ζ} (ζ, ψ)]], \\ \sum_{m = 0}^{\infty} T_{m} (ζ, ψ) = 2 ρ^{2} {sech}^{2} (ρ ζ)) + Y^{- 1} [u^{σ} Y [- 140 \sum_{m = 0}^{\infty} A_{m} - 70 \sum_{m = 0}^{\infty} B_{m} - 280 \sum_{m = 0}^{\infty} C_{m} - \\ 70 \sum_{m = 0}^{\infty} D_{m} - 70 \sum_{m = 0}^{\infty} E_{m} - 42 \sum_{m = 0}^{\infty} T_{m} - 14 \sum_{m = 0}^{\infty} G_{m} - T_{ζ ζ ζ ζ ζ ζ ζ} (ζ, ψ)]] . \end{matrix}

(38)

Comparing both sides of the above equation yields

T_{0} (ζ, ψ) = 2 ρ^{2} {sech}^{2} (ρ ζ)) .

m = 0

T_{1} (ζ, ψ) = \frac{256 ρ^{9} ψ^{σ} tanh (ρ ζ) {sech}^{2} (ρ ζ)}{Γ (σ + 1)} .

m = 1

T_{2} (ζ, ψ) = \frac{16384 ψ^{2 σ} ρ^{16} (cosh (2 ρ ζ) - 2) {sech}^{4} (ρ ζ)}{Γ (2 σ + 1)} .

Lastly, the YTDM solution for

(m \geq 3)

is easy to obtain as

T (ζ, ψ) = \sum_{m = 0}^{\infty} T_{m} (ζ, ψ) = T_{0} (ζ, ψ) + T_{1} (ζ, ψ) + T_{2} (ζ, ψ) + \dots

\begin{matrix} T (ζ, ψ) = 2 ρ^{2} {sech}^{2} (ρ ζ)) + \frac{256 ρ^{9} ψ^{σ} tanh (ρ ζ) {sech}^{2} (ρ ζ)}{Γ (σ + 1)} + \frac{16384 ψ^{2 σ} ρ^{16} (cosh (2 ρ ζ) - 2) {sech}^{4} (ρ ζ)}{Γ (2 σ + 1)} + \dots \end{matrix}

By inserting

σ = 1

we have

T (ζ, ψ) = 2 ρ^{2} {sech}^{2} (ρ (ζ - 64 ρ^{6} ψ)) .

(39)

6. Numerical Simulation Studies

This section offers the approximate analytical solution to the mathematical equation

T (ζ, ψ)

. The numerical results show the method’s applicability, and its accuracy is evaluated in the context of exact results. The implementation of our approach gives outcomes with good performance and simplicity. The actual solution plot is depicted in Figure 1a, whereas the suggested approaches solution plot of

T (ζ, ψ)

is depicted in Figure 1b. The graphical representations of

T (ζ, ψ)

for

σ = 0.8

and

0.6

are shown in Figure 2a,b. Similarly, Figure 3a,b shows the plots of

T (ζ, ψ)

for

σ = 0.25, 0.50, 0.75

and 1, while Figure 4 shows the behavior of absolute error for the same equation generated using both techniques. The approximate solution to the equation

T (ζ, ψ)

is provided in Table 1 for different values of

ζ

and

ψ

, while the absolute error comparison is shown in Table 2 for different values of

ζ

and

ψ

. It should be noted that we used third-order approximate solutions throughout the calculations and that we obtained a good approximation of the exact solutions for the addressed problem. Better approximation solutions would have been found if we increased the order of the approximation, which would increase the number of terms in the solution.

7. Conclusions

This article uses Lax’s Korteweg–de Vries equation using the Yang transform and the Caputo derivative. The YTDM and the HPTM are methods that combine the Yang transform, decomposition and perturbation. A numerical example shows the efficacy and precision of the presented approaches. To explain the theoretical perspective and visualize dynamic behavior, two-dimensional and three-dimensional graphical representations of particular solutions are given. We gave the error estimate in terms of absolute error, shown in Table 2, to demonstrate the applicability and consistency of the two approaches utilized in this current work. The approaches utilized are very effectively explained by the tables and plots, which are comparable with the exact solution in the standard situation when

σ = 1

. Finally, multidimensional problems, variable-order nonlinear fractional differential equations and many other problems can be successfully resolved by combining the methodologies that have been considered with the fractional operator.

Author Contributions

Conceptualization, N.K.M., M.M.A., A.K. and A.H.G.; methodology, N.K.M., M.M.A., A.K. and A.H.G.; software, A.K.; validation, N.K.M., M.M.A., A.K. and A.H.G.; formal analysis, A.K.; investigation, A.K.; resources, N.K.M., M.M.A., A.K. and A.H.G.; data curation, A.K. and A.H.G.; writing—original draft preparation, A.K.; writing—review and editing, A.K.; visualization, A.K. and A.H.G.; supervision, A.K. and A.H.G. All authors have read and agreed to the published version of the manuscript.

Funding

This research received no external funding.

Data Availability Statement

The numerical data used to support the findings of this study are included within the article.

Acknowledgments

The authors are pleased to the reviewers for their meticulous reading and suggestions which improved the presentation of the paper.

Conflicts of Interest

The authors declare no conflict of interest.

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Figure 1. Graphical depiction of our techniques and the accurate solution.

Figure 2. Graphical depiction of our techniques solution at

σ = 0.8

and

0.6

Figure 2. Graphical depiction of our techniques solution at

σ = 0.8

and

0.6

Figure 3. Graphical depiction of our techniques solution for various values of

σ

Figure 3. Graphical depiction of our techniques solution for various values of

σ

Figure 4. Graphical depiction our techniques solution in terms of error.

Table 1. Exact and approximate solutions of the TFLK–dV equation at different values of

σ

Table 1. Exact and approximate solutions of the TFLK–dV equation at different values of

σ

$ψ$	$ζ$	$σ = 0.4$	$σ = 0.6$	$σ = 0.8$	$σ = 1 (approx)$	$σ = 1 (exact)$
	0.2	0.31959022	0.31904353	0.31850296	0.31796602	0.31796602
	0.4	0.31512294	0.31405704	0.31300306	0.31195618	0.31195618
0.01	0.6	0.30682292	0.30528988	0.30377397	0.30226828	0.30226828
	0.8	0.29509495	0.29316615	0.29125890	0.28936453	0.28936453
	1	0.28048302	0.27824242	0.27602684	0.27382622	0.27382622
	0.2	0.31961056	0.31905702	0.31851156	0.31797133	0.31797133
	0.4	0.31516260	0.31408335	0.31301984	0.31196655	0.31196655
0.02	0.6	0.30687997	0.30532771	0.30379811	0.30228320	0.30228320
	0.8	0.29516673	0.29321375	0.29128927	0.28938330	0.28938330
	1	0.28056640	0.27829770	0.27606212	0.27384803	0.27384803
	0.2	0.31962793	0.31906917	0.31851973	0.31797664	0.31797664
	0.4	0.31519647	0.31410702	0.31303577	0.31197691	0.31197691
0.03	0.6	0.30692867	0.30536176	0.30382101	0.30229811	0.30229811
	0.8	0.29522800	0.29325659	0.29131809	0.28940207	0.28940207
	1	0.28063758	0.27834747	0.27609560	0.27386984	0.27386984
	0.2	0.31964361	0.31908051	0.31852763	0.31798194	0.31798194
	0.4	0.31522704	0.31412914	0.31305116	0.31198727	0.31198727
0.04	0.6	0.30697264	0.30539357	0.30384315	0.30231302	0.30231302
	0.8	0.29528332	0.29329662	0.29134595	0.28942083	0.28942083
	1	0.28070184	0.27839397	0.27612796	0.27389165	0.27389165
	0.2	0.31965814	0.31909129	0.31853532	0.31798723	0.31798723
	0.4	0.31525537	0.31415017	0.31306617	0.31199762	0.31199762
0.05	0.6	0.30701339	0.30542381	0.30386474	0.30232792	0.30232792
	0.8	0.29533460	0.29333466	0.29137311	0.28943959	0.28943959
	1	0.28076141	0.27843817	0.27615951	0.27391345	0.27391345

Table 2. Absolute error comparison of our techniques at different values of

σ

Table 2. Absolute error comparison of our techniques at different values of

σ

$ψ$	$ζ$	$σ = 0.4$	$σ = 0.6$	$σ = 0.8$	$σ = 1 (HPTM)$	$σ = 1 (YTDM)$
	0.2	1.6241980000 $\times 10^{- 4}$	1.0775117000 $\times 10^{- 3}$	5.3693380000 $\times 10^{- 4}$	3.5000000000 $\times 10^{- 9}$	3.5000000000 $\times 10^{- 9}$
	0.4	3.1667607000 $\times 10^{- 3}$	2.1008643000 $\times 10^{- 3}$	1.0468777000 $\times 10^{- 3}$	3.0000000000 $\times 10^{- 9}$	3.0000000000 $\times 10^{- 9}$
0.01	0.6	4.5546412000 $\times 10^{- 3}$	3.0215991000 $\times 10^{- 3}$	1.5056865000 $\times 10^{- 3}$	2.8000000000 $\times 10^{- 9}$	2.8000000000 $\times 10^{- 9}$
	0.8	5.7304317000 $\times 10^{- 3}$	3.8016311000 $\times 10^{- 3}$	1.8943820000 $\times 10^{- 3}$	2.3000000000 $\times 10^{- 9}$	2.3000000000 $\times 10^{- 9}$
	1	6.6568000000 $\times 10^{- 3}$	4.4161937000 $\times 10^{- 3}$	2.2006227000 $\times 10^{- 3}$	1.7000000000 $\times 10^{- 9}$	1.7000000000 $\times 10^{- 9}$
	0.2	1.6392279000 $\times 10^{- 4}$	1.0856886000 $\times 10^{- 3}$	5.4022870000 $\times 10^{- 4}$	1.3800000000 $\times 10^{- 8}$	1.3800000000 $\times 10^{- 8}$
	0.4	3.1960548000 $\times 10^{- 3}$	2.1167969000 $\times 10^{- 3}$	1.0532916000 $\times 10^{- 3}$	1.2700000000 $\times 10^{- 8}$	1.2700000000 $\times 10^{- 8}$
0.02	0.6	4.5967682000 $\times 10^{- 3}$	3.0445088000 $\times 10^{- 3}$	1.5149057000 $\times 10^{- 3}$	1.1000000000 $\times 10^{- 8}$	1.1000000000 $\times 10^{- 8}$
	0.8	5.7834303000 $\times 10^{- 3}$	3.8304514000 $\times 10^{- 3}$	1.9059776000 $\times 10^{- 3}$	9.1000000000 $\times 10^{- 9}$	9.100000000 $\times 10^{- 9}$
	1	6.7183635000 $\times 10^{- 3}$	4.4496702000 $\times 10^{- 3}$	2.2140901000 $\times 10^{- 3}$	6.9000000000 $\times 10^{- 9}$	6.9000000000 $\times 10^{- 9}$
	0.2	1.6512893000 $\times 10^{- 4}$	1.0925259000 $\times 10^{- 3}$	5.4309060000 $\times 10^{- 4}$	3.0900000000 $\times 10^{- 8}$	3.0900000000 $\times 10^{- 8}$
	0.4	3.2195538000 $\times 10^{- 3}$	2.1301101000 $\times 10^{- 3}$	1.0588540000 $\times 10^{- 3}$	2.8400000000 $\times 10^{- 8}$	2.8400000000 $\times 10^{- 8}$
0.03	0.6	4.6305573000 $\times 10^{- 3}$	3.0636480000 $\times 10^{- 3}$	1.5228971000 $\times 10^{- 3}$	2.4900000000 $\times 10^{- 8}$	2.4900000000 $\times 10^{- 8}$
	0.8	5.8259361000 $\times 10^{- 3}$	3.8545254000 $\times 10^{- 3}$	1.9160259000 $\times 10^{- 3}$	2.0500000000 $\times 10^{- 8}$	2.0500000000 $\times 10^{- 8}$
	1	6.7677359000 $\times 10^{- 3}$	4.4776312000 $\times 10^{- 3}$	2.2257581000 $\times 10^{- 3}$	1.5400000000 $\times 10^{- 8}$	1.5400000000 $\times 10^{- 8}$
	0.2	1.6616697000 $\times 10^{- 4}$	1.0985706000 $\times 10^{- 3}$	5.4568620000 $\times 10^{- 4}$	5.4900000000 $\times 10^{- 8}$	5.4900000000 $\times 10^{- 8}$
	0.4	3.2397683000 $\times 10^{- 3}$	2.1418713000 $\times 10^{- 3}$	1.0638902000 $\times 10^{- 3}$	5.0700000000 $\times 10^{- 8}$	5.0700000000 $\times 10^{- 8}$
0.04	0.6	4.6596185000 $\times 10^{- 3}$	3.0805511000 $\times 10^{- 3}$	1.5301280000 $\times 10^{- 3}$	4.4300000000 $\times 10^{- 8}$	4.4300000000 $\times 10^{- 8}$
	0.8	5.8624910000 $\times 10^{- 3}$	3.8757835000 $\times 10^{- 3}$	1.9251149000 $\times 10^{- 3}$	3.6400000000 $\times 10^{- 8}$	3.6400000000 $\times 10^{- 8}$
	1	6.8101937000 $\times 10^{- 3}$	4.5023194000 $\times 10^{- 3}$	2.2363100000 $\times 10^{- 3}$	2.7300000000 $\times 10^{- 8}$	2.7300000000 $\times 10^{- 8}$
	0.2	1.6709106000 $\times 10^{- 4}$	1.1040627000 $\times 10^{- 3}$	5.4809170000 $\times 10^{- 4}$	8.5800000000 $\times 10^{- 8}$	8.5800000000 $\times 10^{- 8}$
	0.4	3.2577540000 $\times 10^{- 3}$	2.1525477000 $\times 10^{- 3}$	1.0685486000 $\times 10^{- 4}$	7.9200000000 $\times 10^{- 8}$	7.9200000000 $\times 10^{- 8}$
0.05	0.6	4.6854706000 $\times 10^{- 3}$	3.0958905000 $\times 10^{- 3}$	1.5368119000 $\times 10^{- 4}$	6.9300000000 $\times 10^{- 8}$	6.9300000000 $\times 10^{- 8}$
	0.8	5.8950058000 $\times 10^{- 3}$	3.8950718000 $\times 10^{- 3}$	1.9335132000 $\times 10^{- 4}$	5.6800000000 $\times 10^{- 8}$	5.6800000000 $\times 10^{- 8}$
	1	6.8479565000 $\times 10^{- 3}$	4.5247175000 $\times 10^{- 3}$	2.2460576000 $\times 10^{- 4}$	4.2700000000 $\times 10^{- 8}$	4.2700000000 $\times 10^{- 8}$

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Mishra, N.K.; AlBaidani, M.M.; Khan, A.; Ganie, A.H. Two Novel Computational Techniques for Solving Nonlinear Time-Fractional Lax’s Korteweg-de Vries Equation. Axioms 2023, 12, 400. https://doi.org/10.3390/axioms12040400

AMA Style

Mishra NK, AlBaidani MM, Khan A, Ganie AH. Two Novel Computational Techniques for Solving Nonlinear Time-Fractional Lax’s Korteweg-de Vries Equation. Axioms. 2023; 12(4):400. https://doi.org/10.3390/axioms12040400

Chicago/Turabian Style

Mishra, Nidhish Kumar, Mashael M. AlBaidani, Adnan Khan, and Abdul Hamid Ganie. 2023. "Two Novel Computational Techniques for Solving Nonlinear Time-Fractional Lax’s Korteweg-de Vries Equation" Axioms 12, no. 4: 400. https://doi.org/10.3390/axioms12040400

APA Style

Mishra, N. K., AlBaidani, M. M., Khan, A., & Ganie, A. H. (2023). Two Novel Computational Techniques for Solving Nonlinear Time-Fractional Lax’s Korteweg-de Vries Equation. Axioms, 12(4), 400. https://doi.org/10.3390/axioms12040400

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Two Novel Computational Techniques for Solving Nonlinear Time-Fractional Lax’s Korteweg-de Vries Equation

Abstract

1. Introduction

2. Preliminaries

3. General Implementation of HPTM

4. General Implementation of the YTDM

5. Application

6. Numerical Simulation Studies

7. Conclusions

Author Contributions

Funding

Data Availability Statement

Acknowledgments

Conflicts of Interest

References

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