On the “Tantawy Technique” and other methods for analyzing the family of fractional Burgers’ equations: Applications to plasma physics

Abstract

This paper explores some innovative and modern techniques for analyzing the family of fractional Burgers-type equations, which are extensively utilized in studying shock waves in plasma physics and marine environments. The most notable novel and effective technique employed in this study for the first time is the “Tantawy Technique,” named after its creator, Prof. Samir El-Tantawy. Two other effective methods, known as, the Aboodh residual power series method (ARPSM) and the Aboodh new iteration method (ANIM), are also applied to analyze the suggested problems and compare them with the Tantawy Technique results. We employ these methods to address the complexities associated with fractional partial differential equations (PDEs), specifically using the Caputo operator for fractional derivatives. The study provides a comprehensive analysis of the convergence and accuracy of the proposed methods through various numerical examples. All three proposed methods can effectively yield precise and rapid solutions for fractional Burgers’ type equations. We assess the precision of all derived approximations by comparing them with the exact solutions for the integer case and calculating the absolute errors for these approximations. This work contributes significantly to fractional calculus and nonlinear differential equations, offering practical tools for scientists and engineers to solve complex fractional PDEs with enhanced accuracy. The results will improve our understanding and investigation of several complex phenomena that occur in nature and different nonlinear media, such as optics fiber, fluid mechanics, and plasma physics. As real-world applications for this study, we will investigate various evolutionary equations, such as the Burgers and KdV–Burgers equations used in plasma physics, to explain the properties of fractional shock waves while considering viscosity.

Keywords

Fractional Burgers’ family equations Aboodh residual power series method Aboodh new iteration method Caputo operator fractional shock waves

Introduction

Fractional calculus (FC) is a branch of mathematics that deals with the fractional order’s derivative. In the middle of the 16th century, researchers explored the concept of FC. They found it to be a more exciting part of mathematics, as it provided comprehensive detail of any physical phenomena. FC is also known as non-Newtonian calculus.^1–4 FC has gained significant utility across a wide range of scientific fields.^5–10 Though not all of them are frequently utilized, fractional derivatives can be defined in several ways. Among fractional derivatives, Riemann–Liouville (R–L), Caputo fractional derivative (CFD), Atangana–Baleanu, and Caputo–Fabrizio are the most popular, and many more.^11–23 In some situations, fractional derivatives may be more useful than integer-order derivatives for modeling complicated systems and advanced non-linear processes. This is because of two primary aspects. Firstly, we are no longer limited to an integer order for the derivative operator; instead, we are free to select an arbitrary order. Second, when the system has a long-term memory, non-integer-order derivatives are advantageous because they depend on both the past and current situations.

Consider the fractional KdV Burgers (KdVB) equation as^24–27

D_{t}^{p} ψ - a \partial_{x}^{2} ψ + b \partial_{x}^{3} ψ + c ψ \partial_{x} ψ = 0,

(1)

where

ψ \equiv ψ (x, t)

and 0 < p ≤ 1. Here, the coefficients

(a, b, c)

are related to the physical/engineering model under consideration.

Burgers’ equation was first proposed by Bateman²⁸ in 1918. In his studies from 1939 until 1965, Burgers provided further detail on this subject.²⁹ Burgers equation is used in several domains, including number theory, gas dynamics, fluid mechanics, elastic waves, and hydrodynamic waves.^30–32 Scholars have searched widely for a numerical or analytical solution to equation (1). For example, Ozis and Ozdes³³ used a simple variational approach to solve Burgers’ equation. To solve Burgers’ equation, Aksan and Ozdes³⁴ developed a variational technique utilizing the discretization in time method. Using finite difference methods, Kutluay et al.³⁵ discovered the numerical solution to Burgers equation. In their work, Varolu and Finn used a weighted residue method.³⁶ Wanless and Caldwell³⁷ had some success with finite elements. Evans and Abdullah used a group explicit approach.³⁸ Consider the following fractional Burgers equation^39–41

D_{t}^{p} ψ + ψ \partial_{x} ψ - \partial_{x}^{2} ψ = 0,

(2)

where

ψ \equiv ψ (x, t)

and 0 < p ≤ 1.

Singhal and Mittal⁴² employed the Galerkin approach to discover numerical solutions to Burgers’ equation. Given this, it’s reasonable to assume that scientists have put in considerable effort to find an analytical and/or numerical solution to the fractional Burgers’ equation. Esen and Tasbozan⁴³ used cubic B-spline finite elements for the Burger equation solution. According to Esen et al.,⁴⁴ the time-fractional Burgers’ equation was essentially solved analytically using HAM. The fractional Riccati expansion approach was also used by Abdel Salam et al.⁴⁵ to solve Burgers’ equation of fractional order.

It was in 2013 that Omar Abu Arqub created RPSM.⁴⁶ We combine the residual error function and the Taylor series to form it. An infinite convergence series gives the solution for DEs.⁴⁷ Various types of DEs, including KdV Burger’s equation, fuzzy DEs, Boussinesq DEs, and many more, have prompted the development of novel RPSM algorithms that aim to produce effective and accurate approximation solutions.^48–54 Researchers combined two effective methods to solve fractional-order differential equations (FODEs) to create a unique methodology.^55–60 In this study, we used a novel combination approach known as the ARPSM to discover both approximate and precise solutions for time-fractional partial differential equations (PDEs) with changing coefficients. This innovative approach is remarkable because it combines the RPSM with the Aboodh transform method.^61,62

One major area for improvement with the approaches outlined earlier is their required amount of computing work and complexity. We came up with a new technique to solve the time-fractional biological population model,⁶³ which is the Aboodh new iterative method (ANIM). Combining the Aboodh transform (AT) with the new iterative method significantly reduces computing effort and complexity. The suggested approach yields a convergent series solution.^64,65

As stated in References^61–65 the two least complicated approaches to solving fractional differential equations are the Aboodh residual power series method (ARPSM) and the ANIM. These methods provide not only analytical solutions that are completely transparent and instantaneous but also numerical solutions to PDEs without discretization. The main aim of this work is to investigate the efficiency of ARPSM and ANIM in resolving the KdV Burgers (KdVB) and Burgers equations. These two techniques have specifically handled numerous fractional linear PDEs and nonlinear PDEs.

Moreover, this investigation will introduce a novel analytical technique that will make the analysis of both linear and nonlinear fractional differential equations easier and help overcome the most challenging problems in analyzing more complicated ones. A vital attribute of the “Tantawy Technique” is its lack of necessity for discretization, Adomian polynomials, linearization, or intricate calculations like other methods to produce higher-order approximations. Thus, owing to its simplicity and rapidity, it is anticipated that it will spread very swiftly among researchers to examine several intricate physical, engineering, and biological challenges that were previously tough to assess using alternative methods or were evaluated with inadequate precision. This new technique is designated as the “Tantawy Technique” in honor of Prof. Samir El-Tantawy to safeguard intellectual property rights. Our analysis will demonstrate through various examples that the “Tantawy Technique” surpasses most others that necessitate intricate calculations, which are time-consuming and may demand high-performance computers. In contrast, the “Tantawy Technique” is the most straightforward, accessible, and faster than all other techniques.

In this paper, we solve and analyze the fractional Burgers equation and fractional KdVB equation using ARPSM, ANIM, and Tantawy Technique. These methods provide numerical solutions that are more accurate and precise than other numerical techniques. The numerical findings undergo a comparative study of the employed methods to ascertain their respective accuracies. We shall demonstrate that the outcomes of the offered methodologies are predominantly consistent, indicating their efficacy and reliability. Nonetheless, the novel method, the “Tantawy Technique,” surpasses the other two approaches, augmenting its worth and efficiency. This work will provide a foundation for researchers to efficiently solve and analyze various partial differential equations due to the effectiveness of the presented approaches.

Definitions

Definition 2.1

⁶⁶ The Aboodh transform (AT) for the exponential-order and piecewise continuous function $ψ \equiv ψ (x, t)$ is defined by

A [ψ] = Ψ (x, s) = \frac{1}{s} \int_{0}^{\infty} ψ e^{- s t} d t, \forall σ_{1} \leq s \leq σ_{2} & t \geq 0,

And the inverse AT (IAT) reads

A^{- 1} [Ψ (x, s)] = ψ = \frac{1}{2 π i} \int_{u - i \infty}^{u + i \infty} s e^{s t} Ψ (x, s) d s,

where $ψ \equiv ψ (x, t)$ and $x = (x_{1}, x_{2}, \dots, x_{m}) \in R^{m}$ & $m \in N$

Lemma 2.1

^67,68 Let’s consider two exponentially order and piecewise continuous functions ψ₁ ≡ ψ₁(x, t) and ψ₂ ≡ ψ₂(x, t) in the interval [0, ∞[. The AT for these functions read A[ψ₁] = Ψ₁(x, s) and A[ψ₂] = Ψ₂(x, s). Consequently, the subsequent characteristics are fulfilled:

1. A [ϱ₁ψ₁ + ϱ₂ψ₂] = ϱ₁Ψ₁ (x, s) + ϱ₂Ψ₂ (x, s),

2. A⁻¹ [ϱ₁Ψ₁ (x, s) + ϱ₂Ψ₂ (x, s)] = ϱ₁ψ₁ (x, t) + ϱ₂ψ₂ (x, t),

3. $A [J_{t}^{p} ψ (x, t)] = Ψ (x, s) / s^{p}$ ,

4. $A [D_{t}^{p} ψ (x, t)] = s^{p} Ψ (x, s) - \sum_{r = 0}^{l - 1} ψ^{(r)} (x, 0) / s^{r - p + 2}$ , ∀ l − 1 < p ≤ l & $l \in N$ ,

where $(ϱ_{1}, ϱ_{2})$ are arbitrary constants

Definition 2.2

⁶⁹ The derivative of the function ψ ≡ ψ(x, t) of fractional-order p in the Caputo sense reads

D_{t}^{p} ψ = J_{t}^{n - p} ψ^{(n)} (x, t), t \geq 0, n - 1 < p \leq n, \forall n, p \in R^{m},

where $x = (x_{1}, x_{2}, \dots, x_{m}) \in R^{m}$ and $J_{t}^{n - p}$ indicates the R-L integral of ψ.

Definition 2.3

⁷⁰ The multiple fractional power series (MFPS) about t = t₀ is given by

\sum_{l = 0}^{\infty} f_{l} (x) {(t - t_{0})}^{l p} = f_{0} {(t - t_{0})}^{0} + f_{1} {(t - t_{0})}^{p} + f_{2} {(t - t_{0})}^{2 p} + \dots,

where

x = (x_{1}, x_{2}, \dots, x_{m}) \in R^{m}

and

m \in N

, t is variable and f_l(x) indicates the coefficients of the series, and the series is centered about t₀.

Lemma 2.2

Suppose that ψ ≡ ψ(x, t) is an exponential order function and its AT reads: A[ψ] = Ψ(x, s). Hence, we get

A [D_{t}^{l p} ψ] = s^{l p} Ψ (x, s) - \sum_{j = 0}^{l - 1} s^{p (l - j) - 2} D_{t}^{j p} ψ (x, 0),

(3)

where 0 < p ≤ 1,

x = (x_{1}, x_{2}, \dots, x_{m}) \in R^{m}

and

m \in N

and

D_{t}^{l p} = D_{t}^{p} . D_{t}^{p} . \dots . D_{t}^{p} (l - t i m e s)

Lemma 2.3

Consider ψ ≡ ψ(x, t) to be the exponential order function and its AT reads: A[ψ] = Ψ(x, s). Accordingly, the power series expansion (PSE) for Ψ(x, s) is given by

Ψ (x, s) = \sum_{l = 0}^{\infty} \frac{f_{l} (x)}{s^{l p + 2}}, \forall s > 0,

(4)

where,

x = (x_{1}, x_{2}, \dots, x_{m}) \in R^{m}, m \in N

Lemma 2.4

Let us assume that the function A[ψ] = Ψ(x, s) has an MFPS representation in the new form of Taylor’s series (4). Thus, we have

f_{0} (x) = \lim_{s \to \infty} s^{2} Ψ (x, s) = ψ (x, 0) .

(5)

Theorem 2.5

Suppose that the MFPS for the function A[ψ] = Ψ(x, s) is defined as

Ψ (x, s) = \sum_{l = 0}^{\infty} \frac{f_{l} (x)}{s^{l p + 2}}, \forall s > 0,

where

x = (x_{1}, x_{2}, \dots, x_{m}) \in R^{m}

and

m \in N

and then we get

f_{l} (x) = D_{t}^{l p} ψ (x, 0),

(6)

where

D_{t}^{l p} = D_{t}^{p} . D_{t}^{p} . \dots . D_{t}^{p} (l - t i m e s)

Theorem 2.6

If $| s^{2} A [D_{t}^{(r + 1) p} ψ (x, t)] | \leq T$ , the residual R_r(x, κ) of innovative form of MFPS validate the inequality:

| R_{r} (x, s) | \leq \frac{T}{s^{(r + 1) p + 2}}, 0 < s \leq κ and 0 < p \leq 1 .

(7)

Methodologies

This section provides a concise overview of both ARPSM, ANIM, followed by their use in fractional KdVB and Burgers equations analysis. The “Tantawy Technique” will be further discussed in the subsequent segments of this section.

ARPSM for analyzing fractional differential equations

Here, we describe some steps for solving PDE using ARPSM:

Step (1) Introduce the following general form to the FPDE

D_{t}^{q p} ψ + N (ψ) + L (ψ) = 0,

(8)

where N(ψ) and

L (ψ)

are, respectively, the nonlinear and linear terms/operators and

ψ \equiv ψ (x, t)

Step (2) Applying AT on equation (8) yields

A [D_{t}^{q p} ψ + N (ψ) + L (ψ)] = 0,

(9)

which equivalent

A [D_{t}^{q p} ψ] = - A [N (ψ)] - A [L (ψ)] .

(10)

Step (3) Using equation (3) (for l = q) in equation (10), we get

Ψ (x, s) = \sum_{j = 0}^{q - 1} \frac{D_{t}^{j} ψ (x, 0)}{s^{j p + 2}} - \frac{Y (s)}{s^{q p}} - \frac{F (s)}{s^{q p}},

(11)

where A [L(ψ)] = F(s) and A [N(ψ)] = Y(s).

Step (4) Based on the RPSM, the solution of equation (11) can be rewritten in the following PSE

Ψ (x, s) = \sum_{l = 0}^{\infty} \frac{f_{l} (x)}{s^{l p + 2}}, s > 0,

(12)

Step (5) According to the properties of AT defined in the first Lemma, we have

f_{0} (x) = \lim_{s \to \infty} s^{2} Ψ (x, s) = ψ (x, 0),

and by using equation (6), we obtain

f_{1} (x) = D_{t}^{p} ψ (x, 0), f_{2} (x) = D_{t}^{2 p} ψ (x, 0), f_{w} (x) = D_{t}^{w p} ψ (x, 0), \dots .

Step (6) The rth truncated series of Ψ(x, s) reads

Ψ_{r} (x, s) = \sum_{l = 0}^{r} \frac{f_{l} (x)}{s^{l p + 2}} = \frac{f_{0} (x)}{s^{2}} + \sum_{l = 1}^{r} \frac{f_{l} (x)}{s^{l p + 2}}, \forall s > 0,

(13)

Step (7) To determine the unknown functions f_l(x) present in equation (13), we formulate the Aboodh residual function (ARF): $A [R e s (x, s)]$ to equation (11) as follows

A [R e s (x, s)] = Ψ (x, s) - \sum_{j = 0}^{q - 1} \frac{D_{t}^{j} ψ (x, 0)}{s^{j p + 2}} + \frac{Y (s)}{s^{j p}} + \frac{F (s)}{s^{j p}},

and the r^th-truncated ARF of equation (11) reads

A [R e s_{r} (x, s)] = Ψ_{r} (x, s) - \sum_{j = 0}^{q - 1} \frac{D_{t}^{j} ψ (x, 0)}{s^{j p + 2}} + \frac{Y (s)}{s^{j p}} + \frac{F (s)}{s^{j p}} .

(14)

Step (8) Substituting the expansion form of Ψ_r (x, s) given in equation (13) into equation (14), we finally get

A [R e s_{r} (x, s)] = (\frac{f_{0} (x)}{s^{2}} + \sum_{l = 1}^{r} \frac{f_{l} (x)}{s^{l p + 2}}) - \sum_{j = 0}^{q - 1} \frac{D_{t}^{j} ψ (x, 0)}{s^{j p + 2}} + \frac{Y (s)}{s^{j p}} + \frac{F (s)}{s^{j p}} .

(15)

Step (9) Multiplying equation (15) by s^rp+2, and then take lim_s→∞ for the obtained result

\lim_{s \to \infty} [s^{r p + 2} A R e s_{r} (x, s)] = \lim_{s \to \infty} s^{r p + 2} [\frac{f_{0} (x)}{s^{2}} + \sum_{l = 1}^{r} \frac{f_{l} (x)}{s^{l p + 2}} - \sum_{j = 0}^{q - 1} \frac{D_{ω}^{j} ψ (x, 0)}{s^{j p + 2}} + \frac{(s)}{s^{j p}} + \frac{F (s)}{s^{j p}}] .

(16)

Step (10) Solving the subsequent equation for f_r(x)

\lim_{s \to \infty} (s^{r p + 2} A [R e s_{r} (x, s)]) = 0 .

Step (11) Inserting the obtained values of f_r(x) into equation (13), we get Ψ_r (x, s). By applying IAT on the obtained results, then the solution ψ_r (x, t) is obtained.

In the following sections, we shall employ this methodology to tackle prevalent physical issues in plasma physics, including fractional KdVB equation and Burgers equation.

ARPSM for analyzing KdVB equation

The fluid theory helps one to explain and characterize numerous nonlinear phenomena occurring in plasma physics and fluid media. Using this theory, the fluid equations (continuity and momentum equations for inertial fluid charged particles, inertialess charged particles that follow (non)-Maxwellian distribution, and Poisson’s equation) controlling the motion of plasma-charged particles can be reduced to an evolution equation by an appropriate approach, such as a reductive perturbation technique. Then, numerous analytical and numerical methods are dedicated to solving the given evolution equation to characterize the feature properties of the nonlinear waves described by this evolutionary equation. For example, El-Tantawy²⁴ reduced the fluid-governed equations of a homogeneous unmagnetized electronegative dusty plasma composed of inertialess thermal electrons and negative ions and fluid inertial positive ions as well as stationary negative charged dust grains using the fluid theory for the integer-order KdVB equation²⁴

\partial_{t} ψ - a \partial_{x}^{2} ψ + b \partial_{x}^{3} ψ + c \partial_{x} ψ = 0,

(17)

where the coefficients

(a, b, c)

contain all the parameters of the physical system under study, and for simplicity, we use

ψ \equiv ψ (x, t)

Now, by using a suitable transformation, we can convert the planar KdVB equation (17) to the following fractional KdVB equation

D_{t}^{p} ψ - a \partial_{x}^{2} ψ + b \partial_{x}^{3} ψ + c ψ \partial_{x} ψ = 0,

(18)

which subject to the initial condition (IC):

ψ_{0} \equiv ψ (x, 0) = \frac{1}{25 b c} [25 b λ + 3 a^{2} - 6 a^{2} \tanh (ζ_{0}) - 3 a^{2} \tanh^{2} (ζ_{0})],

(19)

where ζ₀ = ax/10b, ψ ≡ ψ(x, t), and 0 < p ≤ 1.

The exact solution for equation (18) at p = 1, that is, the exact solution for equation (17) reads

ψ = \frac{1}{25 b c} [25 b λ + 3 a^{2} - 6 a^{2} \tanh (ζ) - 3 a^{2} \tanh^{2} (ζ)],

(20)

with

ζ = a / 10 b (x - λ t)

, where λ indicates the moving frame velocity.

By applying AT on equation (18) and with the help of IC (19), we get

\begin{array}{l} A [D_{t}^{p} ψ] & = A [a \partial_{x}^{2} ψ] - A [b \partial_{x}^{3} ψ] - A [c ψ \partial_{x} ψ], \\ = a \partial_{x}^{2} A [ψ] - b \partial_{x}^{3} A [ψ] - c A [A^{- 1} [ψ] \times \partial_{x} A^{- 1} [ψ]], \end{array}

and by applying the properties of AT as mentioned above, we finally get

Ψ_{s} = \frac{ψ_{0}}{s^{2}} + \frac{a}{s^{p}} [\partial_{x}^{2} Ψ_{s}] - \frac{b}{s^{p}} [\partial_{x}^{3} Ψ_{s}] - \frac{c}{s^{p}} A [A^{- 1} [Ψ_{s}] \times \partial_{x} A^{- 1} [Ψ_{s}]] = 0,

(21)

where

Ψ_{s} \equiv Ψ (x, s)

Based on the RPSM, the solution of equation (21) can be rewritten in the following series form

Ψ_{s} = \frac{ψ_{0}}{s^{2}} + \sum_{l = 1}^{\infty} \frac{f_{l}}{s^{l p + 2}},

(22)

and for the rth truncated series, we get

{Ψ_{s}|}_{r} \equiv ψ_{s (r)} = \frac{ψ_{0}}{s^{2}} + \sum_{l = 1}^{r} \frac{f_{l}}{s^{l p + 2}}, \forall s > 0 & l = 1,2,3, \dots, r,

(23)

where

f_{l} \equiv f_{l} (x)

The ARF of equation (21) reads

A [R e s (x, s)] = (Ψ_{s} - \frac{ψ_{0}}{s^{2}}) - \frac{a}{s^{p}} [\partial_{x}^{2} Ψ_{s}] + \frac{b}{s^{p}} [\partial_{x}^{3} Ψ_{s}] + \frac{c}{s^{p}} A [A^{- 1} [Ψ_{s}] \times \partial_{x} A^{- 1} [Ψ_{s}]],

(24)

and the r^th-RAF of equation (21) is expressed in the following manner

A [R e s_{r} (x, s)] = (Ψ_{s (r)} - \frac{ψ_{0}}{s^{2}}) - \frac{a}{s^{p}} [\partial_{x}^{2} Ψ_{s (r)}] + \frac{b}{s^{p}} [\partial_{x}^{3} Ψ_{s (r)}] + \frac{c}{s^{p}} A [A^{- 1} [Ψ_{s (r)}] \times \partial_{x} A^{- 1} [Ψ_{s (r)}]] .

(25)

Multiply equation (25) by s^lp+2 and taking lim_s→∞ for the both sides of the obtained equation and then solve

\lim_{s \to \infty} (s^{l p + 2} A [R e s_{l} (x, s)]) = 0

, we finally get the unknown functions f_l as follows

\begin{align} f_{1} = \frac{3 a^{3} λ}{125 b^{2} c} {sech}^{2} (ζ_{0}) [1 + \tanh (ζ_{0})], \\ f_{2} = \frac{3 a^{4} λ^{2}}{1250 b^{3} c} {sech}^{4} (ζ_{0}) [- 2 + \cosh (2 ζ_{0}) + \sinh (2 ζ_{0})], \\ ⋮ . \end{align}

(26)

Inserting the values of f_l into equation (23) yields

\begin{align} Ψ_{s} & = \frac{ψ_{0}}{s^{2}} + \frac{f_{1}}{s^{p + 2}} + \frac{f_{2}}{s^{2 p + 2}} + \dots \\ = \frac{ψ_{0}}{s^{2}} + \frac{1}{s^{p + 2}} \{\frac{3 a^{3} λ}{125 b^{2} c} {sech}^{2} (ζ_{0}) [1 + \tanh (ζ_{0})]\} \\ + & \frac{1}{s^{2 p + 2}} \{\frac{3 a^{4} λ^{2}}{1250 b^{3} c} {sech}^{4} (ζ_{0}) [- 2 + \cosh (2 ζ_{0}) + \sinh (2 ζ_{0})]\} \end{align}

(27)

By applying the IAT on equation (27), the following approximate solution until second-order approximation to equation (18) is obtained

\begin{align} ψ & = \frac{1}{25 b c} [25 b λ + 3 a^{2} - 6 a^{2} \tanh (ζ_{0}) - 3 a^{2} \tanh^{2} (ζ_{0})] \\ + \frac{3 a^{3} λ t^{p}}{125 b^{2} c Γ [1 + p]} {sech}^{2} (ζ_{0}) [1 + \tanh (ζ_{0})] \\ + \frac{3 a^{4} λ^{2} t^{2 p}}{1250 b^{3} c Γ [1 + 2 p]} {sech}^{4} (ζ_{0}) [- 2 + \cosh (2 ζ_{0}) + \sinh (2 ζ_{0})] . \end{align}

(28)

In Figure 1, the approximation (28) using ARPSM to equation (18) is examined against the fractional parameter p. Also, for confirming the precision of the approximation (28), we conducted a comparison between the approximation (28) and the exact solution (20) for equation (18) at p = 1, as depicted in Figure 2. The obtained findings clearly demonstrate that the derived approximation (28) exhibits substantial accuracy and stability across the study domain, which enhances the efficiency of the used method.

Figure 1.

The approximation (28) of the KdVB equation (18) using ARPSM is analyzed graphically against p in (a) 3D graphic for p = .1, (b) 3D graphic for p = .7, (c) 3D graphic for p = 1, and (d) 2D graphic for t = 15. Here, a = b = c = 1 and λ = 0.3.

Figure 2.

A comparison between the approximation (28) using ARPSM and the exact solution (20): (a) 3D graphic in the $(x, t)$ -plane and (b) 2D graphic for t = 1.

ARPSM for analyzing Burgers Equation

Tamang and Saha³⁹ studied the phase analysis of dust-acoustic waves (DAWs) in strongly coupled dust plasmas composed of Maxwellian ions and electrons using the Burgers equation. The RPT was applied to reduce the fluid govern equations for this plasma model to the following Burgers equation

D_{t}^{p} ψ + A ψ \partial_{x} ψ - B \partial_{x}^{2} ψ = 0,

(29)

where

(A, B)

indicate the coefficients of the nonlinear and dissipative terms, respectively. For simplicity only we consider

(A, B) = (1,1)

and by considering a suitable transformation, then equation (29) can be converted to the following fractional form

D_{t}^{p} ψ + ψ \partial_{x} ψ - \partial_{x}^{2} ψ = 0,

(30)

with the IC

ψ_{0} \equiv ψ (x, 0) = \frac{1}{2} k (1 - \tanh (ζ_{0})),

(31)

where ζ₀ = kx/4, ψ ≡ ψ(x, t), and 0 < p ≤ 1.

The exact solution to equation (30) at p = 1, reads

ψ = \frac{1}{2} k (1 - \tanh (ζ)),

(32)

with

ζ = 1 / 4 k (x - k / 2 t)

Applying AT to equation (30) and with the help of the IC (31), we get

(Ψ_{s} - \frac{ψ_{0}}{s^{2}}) + \frac{1}{s^{p}} A [A^{- 1} [Ψ_{s}] \times \partial_{x} A^{- 1} [Ψ_{s}]] - \frac{1}{s^{p}} [\partial_{x}^{2} Ψ_{s}] = 0,

(33)

where

Ψ_{s} \equiv Ψ (x, s)

Based on the RPSM, the solution of equation (30) can be rewritten in the following series form

According to the RPSM and for the r^th-term in the solution series, we have

Ψ_{s} = \frac{ψ_{0}}{s^{2}} + \sum_{l = 1}^{\infty} \frac{f_{l}}{s^{l p + 2}},

(34)

and the following rth truncated series is considered

{Ψ_{s}|}_{r} \equiv ψ_{s (r)} = \frac{ψ_{0}}{s^{2}} + \sum_{l = 1}^{r} \frac{f_{l}}{s^{l p + 2}}, \forall s > 0,

(35)

where

Ψ_{s} \equiv Ψ (x, s)

and

f_{l} \equiv f_{l} (x) \forall l = 1,2,3,4 \dots

, r.

The ARF for equation (33) is defined by

A [R e s (x, s)] = Ψ_{s} - \frac{ψ_{0}}{s^{2}} + \frac{1}{s^{p}} A [A^{- 1} [Ψ_{s}] \times \partial_{x} A^{- 1} [Ψ_{s}]] - \frac{1}{s^{p}} [\partial_{x}^{2} Ψ_{s}] = 0,

(36)

and the r^th-RAF for equation (33) is expressed in the following manner

A [R e s_{r} (x, s)] = Ψ_{s (r)} - \frac{ψ_{0}}{s^{2}} + \frac{1}{s^{p}} A [A^{- 1} [Ψ_{s (r)}] \times \partial_{x} A^{- 1} [Ψ_{s (r)}]] - \frac{1}{s^{p}} [\partial_{x}^{2} Ψ_{s (r)}] = 0 .

(37)

Multiply equation (37) by s^lp+2 and taking lim_s→∞ for the both sides of the obtained equation and then solve

\lim_{s \to \infty} (s^{l p + 2} A [R e s_{l} (x, s)]) = 0

, we finally get the unknown functions f_l as follows

f_{1} = \frac{k^{3}}{16} {sech}^{2} (ζ_{0}), f_{2} = \frac{k^{5}}{64} {sech}^{2} (ζ_{0}) \tanh (ζ_{0}), \dots .

(38)

Inserting the values of f_l into equation (34), we have

Ψ_{s} = \frac{k (1 - \tanh (ζ_{0}))}{2 s^{2}} + \frac{k^{3} {sech}^{2} (ζ_{0})}{16 s^{p + 2}} + \frac{k^{5} \tanh (ζ_{0}) {sech}^{2} (ζ_{0})}{64 s^{2 p + 2}} + \dots .

(39)

Now, by applying the IAT on equation (39), the following approximate solution until second-order approximation to equation (30) is obtained

ψ = \frac{1}{2} k (1 - \tanh (ζ_{0})) + \frac{k^{3} {sech}^{2} (ζ_{0})}{16 Γ [1 + p]} t^{p} + \frac{k^{5} \tanh (ζ_{0}) {sech}^{2} (ζ_{0})}{64 Γ [1 + 2 p]} t^{2 p} + \dots .

(40)

Figure 3 displays the numerical analysis of the approximation (40) using ARPSM to equation (30), in relation to the fractional parameter p. Furthermore, at p = 1, we conducted a comparison between the approximation (40) and the exact solution (32) for equation (30) as depicted in Figure 4, in order to confirm the precision of the approximation (40).

Figure 3.

The approximation (40) of Burgers equation (30) using ARPSM is analyzed graphically against p in (a) 3D graphic for p = .1, (b) 3D graphic for p = .3, (c) 3D graphic for p = 1, and (d) 2D graphic for t = 0.1. Here, k = 2.5.

Figure 4.

A comparison between the approximation (40) using ARPSM and the exact solution (32): (a) 3D graphic in the $(x, t)$ -plane and (b) 2D graphic for ω = 0.1 and k = 2.5.

ANIM for analyzing fractional differential equations

Let consider the following general FPDE

D_{t}^{p} ψ = L (ψ) + N (ψ) + g (x, t), \forall n - 1 < p \leq n,

(41)

with the IC

ψ^{(r)} (x, 0) = h_{r} (x), r = 0,1,2, \dots, n - 1,

(42)

where ψ ≡ ψ(x, t) is an undetermined function that will be determined thereafter, whereas

L (ψ)

and

N (ψ)

are, respectively, the linear and nonlinear terms and

g \equiv g (x, t)

indicates a source function.

Applying AT on equation (41) yields

A [ψ] = \frac{1}{s^{p}} [\sum_{r = 0}^{n - 1} \frac{ψ^{(r)} (x, 0)}{s^{2 - p + r}} + A [g]] + \frac{1}{s^{p}} (A [L (ψ)] + A [N (ψ)]) .

(43)

Taking the IAT to equation (43) yields

ψ = A^{- 1} \{\frac{1}{s^{p}} [\sum_{r = 0}^{n - 1} \frac{ψ^{(r)} (x, 0)}{s^{2 - p + r}} + A [g]] + \frac{1}{s^{p}} (A [L (ψ)] + A [N (ψ)])\} .

(44)

The operators

L (ψ)

and

N (ψ)

can be decomposed as follows

\begin{align} L (ψ) + N (ψ) & = L (\sum_{r = 0}^{\infty} ψ_{r}) + N (\sum_{r = 0}^{\infty} ψ_{r}) \\ = L (ψ_{0}) + \sum_{r = 1}^{\infty} \{L (\sum_{i = 0}^{r} ψ_{i})\} \\ + N (ψ_{0}) + \sum_{r = 1}^{\infty} \{N (\sum_{i = 0}^{r} ψ_{i}) - N (\sum_{i = 0}^{r - 1} ψ_{i})\} . \end{align}

(45)

According to ANIM, the m^th-order approximate series can be defined as

ψ = \sum_{k = 0}^{m} ψ_{k} = ψ_{1} + ψ_{2} + ψ_{3} + \dots + ψ_{m}, \forall m \in N .

(46)

Inserting equations (45) and (46) into equation (44) and applying IAT, we get

\begin{aligned} \sum_{r = 0}^{\infty} ψ_{r} = A^{- 1} [\frac{1}{s^{p}} (\sum_{r = 0}^{n - 1} \frac{ψ^{(r)} (x, 0)}{s^{2 - p + r}} + A [g])] \\ + A^{- 1} [\frac{1}{s^{p}} (A [L (ψ_{0}) + N (ψ_{0})])] \\ + A^{- 1} [\frac{1}{s^{p}} \{A [\sum_{r = 1}^{\infty} (L (ψ_{r}) + \{N (\sum_{i = 0}^{r} ψ_{i}) - N (\sum_{i = 0}^{r - 1} ψ_{i})\})]\}] . \end{aligned}

(47)

Accordingly, the following iterations are obtained

\begin{aligned} ψ_{0} = A^{- 1} [\frac{1}{s^{p}} (\sum_{r = 0}^{n - 1} \frac{ψ^{(r)} (x, 0)}{s^{2 - p + r}} + A [g])], \forall n - 1 < p \leq n, \\ ψ_{1} = A^{- 1} [\frac{1}{s^{p}} (A [L (ψ_{0}) + N (ψ_{0})])] \\ ⋮ \\ ψ_{i + 1} = A^{- 1} [\frac{1}{s^{p}} \{A [\sum_{r = 1}^{\infty} (L (ψ_{r}) + \{N (\sum_{i = 0}^{r} ψ_{i}) - N (\sum_{i = 0}^{r - 1} ψ_{i})\})]\}] \end{aligned}

(48)

Ultimately, the approximation to equation (41) using ATIM is typically expressed as

ψ (x, t) = \sum_{r = 0}^{n - 1} ψ_{r} .

(49)

ANIM for analyzing KdVB equation

To apply ANIM for analyzing equation (18), we initially reformulate it in the following manner

D_{t}^{p} ψ = a \partial_{x}^{2} ψ - b \partial_{x}^{3} ψ - c ψ \partial_{x} ψ,

(50)

which subjected to the IC (19) and its exact solution at p = 1, is defined in equation (20). Here, ψ ≡ ψ(x, t) and 0 < p ≤ 1.

According to equation (50), the linear and nonlinear operators and function source are, respectively, read

\{\begin{matrix} L (ψ) = a \partial_{x}^{2} ψ - b \partial_{x}^{3} ψ, \\ N (ψ) = - c ψ \partial_{x} ψ, \\ g = 0 . \end{matrix}

Applying AT to equation (50) yields

A [ψ] = \frac{1}{s^{p}} [\sum_{r = 0}^{n - 1} \frac{ψ^{(r)} (x, 0)}{s^{2 - p + r}} + A [a \partial_{x}^{2} ψ - b \partial_{x}^{3} ψ - c ψ \partial_{x} ψ]],

(51)

Taking IAT to equation (51) yields

ψ = A^{- 1} [\frac{1}{s^{p}} (\sum_{r = 0}^{n - 1} \frac{ψ^{(r)} (x, 0)}{s^{2 - p + r}} + A [a \partial_{x}^{2} ψ - b \partial_{x}^{3} ψ - c ψ \partial_{x} ψ])] .

(52)

According to equation (48), we have

\begin{aligned} ψ_{0} = A^{- 1} [\frac{1}{s^{p}} (\sum_{r = 0}^{n - 1} \frac{ψ^{(r)} (x, 0)}{s^{2 - p + r}})] = A^{- 1} [\frac{ψ (x, 0)}{s^{2}}] = A^{- 1} [\frac{ψ_{0}}{s^{2}}] = ψ_{0} \\ = \frac{1}{25 b c} [25 b λ + 3 a^{2} - 6 a^{2} \tanh (ζ_{0}) - 3 a^{2} \tanh^{2} (ζ_{0})], \\ ψ_{1} = A^{- 1} [\frac{1}{s^{p}} (A [L (ψ_{0}) + N (ψ_{0})])] = A^{- 1} [\frac{1}{s^{p}} (A [a \partial_{x}^{2} ψ_{0} - b \partial_{x}^{3} ψ_{0}] - A [c ψ_{0} \partial_{x} ψ_{0}])] \\ = \frac{3 a^{3} λ {sech}^{2} (ζ_{0}) (1 + \tanh (ζ_{0}))}{125 b^{2} c Γ [1 + p]} t^{p}, \end{aligned}

(53)

and

\begin{align} ψ_{2} & = A^{- 1} [\frac{1}{s^{p}} \{A [L (ψ_{1}) + (N (ψ_{0} + ψ_{1}) - N (ψ_{0}))]\}] \\ = A^{- 1} [\frac{1}{s^{p}} \{A [a \partial_{x}^{2} ψ_{1} - b \partial_{x}^{3} ψ_{1} + \{(- c ψ_{0} \partial_{x} ψ_{0} - c ψ_{1} \partial_{x} ψ_{1}) - (- c ψ_{0} \partial_{x} ψ_{0})\}]\}] \\ = \frac{3 a^{4} λ^{2} {sech}^{2} (ζ_{0}) (1 + \tanh (ζ_{0})) (- 1 + 3 \tanh (ζ_{0}))}{1250 b^{3} c Γ [1 + 2 p]} t^{2 p} \\ + \frac{9 \times 4^{p} a^{7} λ^{2} Γ [\frac{1}{2} + p]}{156250 b^{5} c \sqrt{π} Γ [1 + p] Γ [1 + 3 p]} {sech}^{4} (ζ_{0}) {(1 + \tanh (ζ_{0}))}^{2} (- 1 + 3 \tanh (ζ_{0})) t^{3 p} . \end{align}

(54)

By collecting the values of ψ₀, ψ₁, and ψ₂ given in equations (53) and (54), finally the approximate solution until the second-order approximation to equation (18)/(50) is obtained as follows

\begin{aligned} ψ & = ψ_{0} + ψ_{1} + ψ_{2} + \dots \\ = & \frac{1}{25 b c} [25 b λ + 3 a^{2} - 6 a^{2} \tanh (ζ_{0}) - 3 a^{2} \tanh^{2} (ζ_{0})] \\ + \frac{3 a^{3} λ {sech}^{2} (ζ_{0}) (1 + \tanh (ζ_{0}))}{125 b^{2} c Γ [1 + p]} t^{p} + \frac{3 a^{4} λ^{2} {sech}^{2} (ζ_{0}) (1 + \tanh (ζ_{0})) (- 1 + 3 \tanh (ζ_{0}))}{1250 b^{3} c Γ [1 + 2 p]} t^{2 p} \\ + \frac{9 \times 4^{p} a^{7} λ^{2} Γ [\frac{1}{2} + p]}{156250 b^{5} c \sqrt{π} Γ [1 + p] Γ [1 + 3 p]} {sech}^{4} (ζ_{0}) {(1 + \tanh (ζ_{0}))}^{2} (- 1 + 3 \tanh (ζ_{0})) t^{3 p} . \end{aligned}

(55)

The approximation (55) to equation (18)/(50) using ANIM is examined graphically and numerically against the fractional parameter p as illustrated in Figure 5. Also, the comparison between between the approximation (55) and the exact solution (20) for equation (18)/(50) at p = 1 is investigated as evident in Figure 6, to confirm the accuracy of the approximation (55). The obtained results unequivocally show that the derived approximation (55) demonstrates significant accuracy and stability throughout the study domain, which confirms the high efficiency of the used approach.

Figure 5.

The approximation (55) the KdVB equation (18) using ATIM is analyzed graphically against p in (a) 3D graphic for p = .1, (b) 3D graphic for p = .7, (c) 3D graphic for p = 1, and (d) 2D graphic for t = 15. Here, a = b = c = 1 and λ = 0.3.

Figure 6.

A comparison between the approximation (55) using ARPSM and the exact solution (20): (a) 3D graphic in the $(x, t)$ -plane and (b) 2D graphic for t = 1.

ANIM for analyzing Burgers Equation

Consider the following fractional Burgers equation

D_{t}^{p} ψ = - ψ \partial_{x} ψ + \partial_{x}^{2} ψ,

(56)

which subject to the IC:

ψ_{0} \equiv ψ (x, 0) = \frac{1}{2} k (1 - \tanh (\frac{k x}{4})),

(57)

and its exact solution for p = 1, is given in equation (32).

According to equation (56), the linear and nonlinear operators and function source are, respectively, read

\{\begin{matrix} L (ψ) = \partial_{x}^{2} ψ, \\ N (ψ) = - ψ \partial_{x} ψ, \\ g = 0 . \end{matrix}

Applying AT on equation (56) yields

A [ψ] = \frac{1}{s^{p}} [\sum_{r = 0}^{n - 1} \frac{ψ^{(r)} (x, 0)}{s^{2 - p + r}} + A [- ψ \partial_{x} ψ + \partial_{x}^{2} ψ]] .

(58)

Taking IAT to equation (58) implies

ψ = A^{- 1} [\frac{1}{s^{p}} (\sum_{r = 0}^{n - 1} \frac{ψ^{(r)} (x, 0)}{s^{2 - p + r}} + A [- ψ \partial_{x} ψ + \partial_{x}^{2} ψ])] .

(59)

According to equation (48), we get

\begin{aligned} ψ_{0} = A^{- 1} [\frac{1}{s^{p}} (\sum_{r = 0}^{n - 1} \frac{ψ^{(r)} (x, 0)}{s^{2 - p + r}})] = A^{- 1} [\frac{ψ (x, 0)}{s^{2}}] = A^{- 1} [\frac{ψ_{0}}{s^{2}}] = ψ_{0} \\ = \frac{1}{2} k (1 - \tanh (ζ_{0})), \\ ψ_{1} = A^{- 1} [\frac{1}{s^{p}} (A [L (ψ_{0}) + N (ψ_{0})])] = A^{- 1} [\frac{1}{s^{p}} (A [\partial_{x}^{2} ψ_{0}] - A [ψ_{0} \partial_{x} ψ_{0}])] \\ = \frac{k^{3} {sech}^{2} (ζ_{0})}{16 Γ [1 + p]} t^{p}, \end{aligned}

(60)

and

\begin{align} ψ_{2} & = A^{- 1} [\frac{1}{s^{p}} \{A [\partial_{x}^{2} ψ_{1} + \{(- ψ_{0} \partial_{x} ψ_{0} - ψ_{1} \partial_{x} ψ_{1}) - (- ψ_{0} \partial_{x} ψ_{0})\}]\}] \\ = \frac{k^{5} {sech}^{2} (ζ_{0}) \tanh (ζ_{0})}{64 Γ [1 + 2 p]} t^{2 p} + \frac{k^{7} Γ [1 + 2 p] {sech}^{4} (ζ_{0}) \tanh (ζ_{0})}{512 Γ {[1 + p]}^{2} Γ [1 + 3 p]} t^{3 p} . \end{align}

(61)

By collecting the values of ψ₀, ψ₁, and ψ₂ given in equations (60) and (61), finally the approximate solution until the second-order approximation to equation (29)/(56) is obtained as follows

\begin{aligned} ψ & = ψ_{0} + ψ_{1} + ψ_{2} + \dots = \frac{1}{2} k (1 - \tanh (ζ_{0})) + \frac{k^{3} {sech}^{2} (ζ_{0})}{16 Γ [1 + p]} t^{p} \\ + \frac{k^{5} {sech}^{2} (ζ_{0}) \tanh (ζ_{0})}{64 Γ [1 + 2 p]} t^{2 p} + \frac{k^{7} Γ [1 + 2 p] {sech}^{4} (ζ_{0}) \tanh (ζ_{0})}{512 Γ {[1 + p]}^{2} Γ [1 + 3 p]} t^{3 p} . \end{aligned}

(62)

Here, we examine the approximation (62) graphically and numerically against the fractional parameter p as illustrated in Figure 7. Furthermore, to validate the precision of the obtained approximations and the effectiveness of the used approach, we conducted a comparison between the approximate solution (62) and the exact solution (32), as depicted in Figure 8.

Figure 7.

The approximation (62) of Burgers equation (30) using ANIM is analyzed graphically against p in (a) 3D graphic for p = .1, (b) 3D graphic for p = .3, (c) 3D graphic for p = 1, and (d) 2D graphic for t = 0.1. Here, k = 2.5.

Figure 8.

A comparison between the approximation (62) using ANIM and the exact solution (32): (a) 3D graphic in the $(x, t)$ -plane and (b) 2D graphic for t = 0.1.

The “Tantawy Technique” for analyzing fractional differential equations

The examination of FDEs has garnered considerable interest in recent years owing to its capacity to represent complex physical and engineering problems. While various approaches and techniques have been developed to address these equations, many researchers still face difficulties and challenges while applying the existing methods (RPSM, NIM, ADM, HPM, VIM, etc.) to analyze various fractional evolutionary wave equations associated with several physical phenomena. Consequently, significant work has been devoted to developing a user-friendly method that any researcher can use to analyze all fractional differential equations without requiring costly computations. The efforts resulted in the “Tantawy Technique,” which does not require costly computations, perturbation techniques, Adomian polynomials, linearization, or discretization. Prof. Samir El-Tantawy is the exclusive inventor of this novel technique, so it is called in his honor.

In light of the novelty of the “Tantawy Technique,” we shall now delineate it in detailed aspects as follows

Step (1) Let us introduce the following fractional differential equation

D_{t}^{p} ψ = L (ψ) + N (ψ),

(63)

with the IC

ψ (x, 0) = f,

(64)

where ψ ≡ ψ(x, t) and

f \equiv f (x)

, whereas

L (ψ)

and

N (ψ)

are, respectively, the linear and nonlinear terms.

Step (2) For analyzing problem (63) using the “Tantawy Technique,” the following Ansatz is introduced

ψ = f + \sum_{i = 1}^{\infty} t^{i p} f_{i} = f + t^{p} f_{1} + t^{2 p} f_{2} + t^{3 p} f_{3} + \dots + t^{n p} f_{n} + \dots,

(65)

in equation (63) to get

D_{t}^{p} [f + \sum_{i = 1}^{\infty} t^{i p} f_{i}] = L (f + \sum_{i = 1}^{\infty} t^{i p} f_{i}) + N (f + \sum_{i = 1}^{\infty} t^{i p} f_{i}),

(66)

where $f_{i} \equiv f_{i} (x)$ are undetermined functions.

Step (3) By considering n^th-order in the solution series (65), then equation (66) can be rewritten as

D_{t}^{p} [f + \sum_{i = 1}^{n} t^{i p} f_{i}] = L (f + \sum_{i = 1}^{n} t^{i p} f_{i}) + N (f + \sum_{i = 1}^{n} t^{i p} f_{i}) .

(67)

Step (4) Applying Caputo/Jumarie fractional derivative operator (CFDO) and, collecting all terms corresponding to the same power of t^ip ∀ i = 0, 1, 2, 3, …, we get

T_{0} + T_{1} t^{p} + T_{2} t^{2 p} + T_{3} t^{3 p} + \dots = 0,

(68)

with

\begin{align} Coeff. (t^{0 p}) & : T_{0} \equiv T_{0} (f, f^{'}, f^{''}, f^{(3)}, \dots, f_{1}), \\ Coeff. (t^{p}) & : T_{1} \equiv T_{1} (f, f^{'}, f^{''}, f^{(3)}, \dots, f_{1}, f_{1}^{'}, f_{1}^{''}, \dots, f_{2}), \\ Coeff. (t^{2 p}) & : T_{2} \equiv T_{2} (f, f^{'}, f^{''}, f^{(3)}, \dots, f_{1}, f_{1}^{'}, f_{1}^{''}, \dots, f_{2}, f_{2}^{'}, f_{2}^{''}, \dots, f_{3}), \\ Coeff. (t^{3 p}) & : T_{3} \equiv T_{3} (f, f^{'}, f^{''}, f^{(3)}, \dots, f_{1}, f_{1}^{'}, f_{1}^{''}, \dots, f_{2}, f_{2}^{'}, f_{2}^{''}, \dots, f_{3}, f_{3}^{'}, f_{3}^{''}, \dots, f_{4}), \\ ⋮ . \end{align}

(69)

Step (5) Equating the coefficients T_i ∀ i = 0, 1, 2, 3, …, to zero, we get a system of equations and by solving this system in f₁, f₂, f₃, we finally get implicit values for f_i as functions of f and its derivatives

\begin{align} f_{1} & = (g_{1} (p), f, f^{'}, f^{''}, f^{(3)}, \dots), \\ f_{2} & = (g_{2} (p), f, f^{'}, f^{''}, f^{(3)}, \dots, f_{1}, f_{1}^{'}, f_{1}^{''}, \dots), \\ f_{3} & = (g_{3} (p), f, f^{'}, f^{''}, f^{(3)}, \dots, f_{1}, f_{1}^{'}, f_{1}^{''}, \dots, f_{2}, f_{2}^{'}, f_{2}^{''}, \dots), \\ f_{4} & = (g_{4} (p), f, f^{'}, f^{''}, f^{(3)}, \dots, f_{1}, f_{1}^{'}, f_{1}^{''}, \dots, f_{2}, f_{2}^{'}, f_{2}^{''}, \dots, f_{3}, f_{3}^{'}, f_{3}^{''}, \dots), \\ ⋮, \end{align}

(70)

where $g_{i} (p) \forall i = 1,2,3, \dots$ , are known functions, which appeared after applying the CFDO.

Step (6) Using the explicit value of the IC f in equation (70), we can get the explicit values for f_i. By substituting the values of f_i into equation (65), we ultimately obtain the approximate solution to the problem (63).

The “Tantawy Technique” for analyzing KdVB equation

The subsequent steps can be presented to apply the “Tantawy Technique” for analyzing example (1):

Step (1) Inserting the Ansatz (65) up to nth-order

ψ = f + \sum_{i = 1}^{n} t^{i p} f_{i},

(71)

into the problem (18), yields

\begin{align} D_{t}^{p} (f + \sum_{i = 1}^{n} t^{i p} f_{i}) - a \partial_{x}^{2} (f + \sum_{i = 1}^{n} t^{i p} f_{i}) + b \partial_{x}^{3} (f + \sum_{i = 1}^{n} t^{i p} f_{i}) \\ + c (f + \sum_{i = 1}^{n} t^{i p} f_{i}) \times \partial_{x} (f + \sum_{i = 1}^{n} t^{i p} f_{i}) = 0 . \end{align}

(72)

Step (2) Using the definition of the Caputo/Jumarie derivative operator in equation (72) and collecting all terms corresponding to the same power of t^ip ∀ i = 0, 1, 2, 3, …, we get

T_{0} + T_{1} t^{p} + T_{2} t^{2 p} + T_{3} t^{3 p} + \dots = 0,

(73)

with

\begin{array}{l} T_{0} & = g_{1} (p) f_{1} + c f f^{'} - a f^{''} + b f^{(3)}, \\ T_{1} & = g_{2} (p) f_{2} + c (f_{1} f^{'} + f f_{1}^{'}) - a f_{1}^{''} + b f_{1}^{(3)}, \\ T_{2} & = g_{3} (p) f_{3} + c (f_{2} f^{'} + f_{1} f_{1}^{'} + f f_{2}^{'}) - a f_{2}^{''} + b f_{2}^{(3)}, \\ ⋮, \end{array}

and the values of $g_{i} (p)$ based on the definition of the CFDO read

\begin{align} g_{1} (p) & = Γ [1 + p], \\ g_{2} (p) & = \frac{Γ [1 + 2 p]}{Γ [1 + p]}, \\ g_{3} (p) & = \frac{Γ [1 + 3 p]}{Γ [1 + 2 p]}, \\ ⋮ \\ g_{i} (p) & = \frac{Γ [1 + i p]}{Γ [1 + (i - 1) p]} . \end{align}

(74)

Step (3) By setting the coefficients T₀, T₁, T₁, … to zero and solving them for f₁, f₂, f₃, we ultimately obtain the implicit values of f_i as functions of f

\begin{align} f_{1} & = \frac{1}{Γ [1 + p]} (- c f f^{'} + a f^{''} - b f^{(3)}), \end{align}

(75)

\begin{align} f_{2} & = \frac{1}{Γ [1 + 2 p]} (\begin{matrix} 2 c^{2} f f^{' 2} + c^{2} f^{2} f^{''} - 4 a c f^{'} f^{''} + 3 b c f^{'' 2} - 2 a c f f^{(3)} + 5 b c f^{'} f^{(3)} \\ + a^{2} f^{(4)} + 2 b c f f^{(4)} - 2 a b f^{(5)} + b^{2} f^{(6)} \end{matrix}), \end{align}

(76)

\begin{align} f_{3} & = Large size value . \end{align}

(77)

Step (4) By substituting the value of the IC f

f = \frac{1}{25 b c} [25 b λ + 3 a^{2} - 6 a^{2} \tanh (ζ_{0}) - 3 a^{2} \tanh^{2} (ζ_{0})],

(78)

into the values of f₁, f₂, f₃ given in equation (75) –(77), we get

\begin{align} f_{1} & = \frac{3 a^{3} λ}{125 b^{2} c Γ [1 + p]} {sech}^{2} (ζ_{0}) (1 + \tanh (ζ_{0})), \\ f_{2} & = \frac{24 a^{4} λ^{2}}{625 b^{3} c Γ [1 + 2 p]} \frac{e^{4 ζ_{0}} (e^{2 ζ_{0}} - 2)}{{(e^{2 ζ_{0}} + 1)}^{4}}, \\ f_{3} & = \frac{3 a^{5} λ^{2} (\tanh (ζ_{0}) + 1) {sech}^{6} (ζ_{0})}{1250000 b^{5} c Γ {(ρ + 1)}^{2} Γ (3 ρ + 1)} \\ \times \{\begin{matrix} Γ {(ρ + 1)}^{2} [\begin{matrix} - 6 (8 a^{2} + 25 b λ) \sinh (2 ζ_{0}) \\ - 2 \cosh (2 ζ_{0}) (24 a^{2} + 75 b λ \sinh (2 ζ_{0}) + 50 b λ) \\ + 96 a^{2} + 125 b λ \cosh (4 ζ_{0}) - 225 b λ \end{matrix}] \\ + 24 a^{2} Γ (2 ρ + 1) (e^{2 ζ_{0}} - 2) \end{matrix}\} . \end{align}

(79)

Step (5) By substituting the acquired values of f₁, f₂, f₃, into Ansatz (71), we can get the approximate solution for example (1) until the third-order approximation as follows

\begin{align} ψ & = f + t^{p} f_{1} + t^{2 p} f_{2} + t^{3 p} f_{3} + \dots \\ = \frac{1}{25 b c} [25 b λ + 3 a^{2} - 6 a^{2} \tanh (ζ_{0}) - 3 a^{2} \tanh^{2} (ζ_{0})] \\ + \frac{3 a^{3} λ t^{p}}{125 b^{2} c Γ [1 + p]} {sech}^{2} (ζ_{0}) (1 + \tanh (ζ_{0})) \\ + \frac{24 a^{4} λ^{2} t^{2 p}}{625 b^{3} c Γ [1 + 2 p]} \frac{e^{4 ζ_{0}} (e^{2 ζ_{0}} - 2)}{{(e^{2 ζ_{0}} + 1)}^{4}} \\ + \frac{3 a^{5} λ^{2} t^{3 p} (1 + \tanh (ζ_{0})) {sech}^{6} (ζ_{0})}{1250000 b^{5} c Γ {(ρ + 1)}^{2} Γ (3 ρ + 1)} \\ \times \{\begin{matrix} Γ {(ρ + 1)}^{2} [\begin{matrix} - 6 (8 a^{2} + 25 b λ) \sinh (2 ζ_{0}) \\ - 2 \cosh (2 ζ_{0}) (24 a^{2} + 75 b λ \sinh (2 ζ_{0}) + 50 b λ) \\ + 96 a^{2} + 125 b λ \cosh (4 ζ_{0}) - 225 b λ \end{matrix}] \\ + 24 a^{2} Γ (2 ρ + 1) (e^{2 ζ_{0}} - 2) \end{matrix}\} . \end{align}

(80)

Figure 9 illustrates the approximation (80) employing the Tantawy Technique against p for the problem (1). Additionally, we compared the approximation (80) with the exact solution (20) for the problem (1) at p = 1, as seen in Figure 10. Furthermore, the absolute error R_∞ of the approximations (28), (55), and (80) using ARPSM, ATIM, and the Tantawy Technique are estimated, respectively, as illustrated in Figure 11

\begin{array}{l} {R_{\infty}|}_{ARPSM} & = {|Approx. (28) - Exact (20)|}_{p = 1}, \\ {R_{\infty}|}_{ATIM} & = {|Approx. (55) - Exact (20)|}_{p = 1}, \\ {R_{\infty}|}_{Tantawy} & = {|Approx. (80) - Exact (20)|}_{p = 1} . \end{array}

Additionally, we performed a numerical comparison of the absolute errors of the approximations (28), (55), and (80) utilizing ARPSM, ATIM, and the “Tantawy Technique,” respectively, as elucidated in Table 1. The results indicate that the approximation (80) displays high accuracy and more stability throughout the study domain, which enhances the high efficiency of the “Tantawy Technique.” The comparative results indicated that the approximations (28) and (55) closely resemble the exact solution for the integer case, exhibiting great accuracy and minimal error. However, upon examining the third approximation (80) derived from the “Tantawy Technique,” we discover remarkable findings that exhibit exceptional accuracy, significantly surpassing those produced by alternative methods. The comparison table clearly indicates that the “Tantawy Technique” yields far more accurate findings than alternative methods, which enhances its power in analyzing more complicated physical and engineering problems.

Figure 9.

The approximation (80) of the KdVB equation (18) using the Tantawy Technique is analyzed graphically against p in (a) 3D graphic for p = .1, (b) 3D graphic for p = .7, (c) 3D graphic for p = 1, and (d) 2D graphic for t = 15. Here, a = b = c = 1 and λ = 0.3.

Figure 10.

A comparison between the approximation (80) using the Tantawy Technique and the exact solution (20): (a) 3D graphic in the $(x, t)$ -plane and (b) 2D graphic for t = 1.

Figure 11.

The absolute error R_∞ of the approximations (28): (a) –(b), (55): (c) –(d), and (80): (e) –(f) using ARPSM, ANIM, and the Tantawy Technique is estimated in (a), (c), and (e) $(x, t)$ -plane and (b), (d), and (f) against x for t = 0.01.

Table 1.

A numerical comparison between the absolute error R_∞ of the approximations (28), (55), and (80) utilizing ARPSM, ANIM, and the Tantawy Technique, respectively, for the KdVB equation (18). Here, t = 0.01 and a = b = c = 1 and λ = 0.1.

x	10⁻¹⁷R_∞\|_Tantawy	10⁻¹⁴R_∞\|_ARPSM	10⁻¹⁴×R_∞\|_ANIM
0	0	7.99638	11.2022
0.1	2.77556	8.31557	10.6776
0.2	0	8.62921	10.1363
0.3	2.77556	8.92619	9.57845
0.4	2.77556	9.2204	8.99558
0.5	2.77556	9.50906	8.39606
0.6	5.55112	9.78939	7.77434
0.7	2.77556	10.0531	7.14984
0.8	2.77556	10.314	6.4948
0.9	0	10.561	5.83422
1	0	10.7969	5.15976

The “Tantawy Technique” for analyzing Burgers Equation

Here, we can summarize the main steps of the “Tantawy Technique” for analyzing example (2):

Step (1) Inserting the Ansatz (65) up to nth-order

ψ = f + \sum_{i = 1}^{n} t^{i p} f_{i},

(81)

into the problem (30)

D_{t}^{p} (f + \sum_{i = 1}^{n} t^{i p} f_{i}) + (f + \sum_{i = 1}^{n} t^{i p} f_{i}) \partial_{x} (f + \sum_{i = 1}^{n} t^{i p} f_{i}) - \partial_{x}^{2} (f + \sum_{i = 1}^{n} t^{i p} f_{i}) = 0,

(82)

Step (2) By using the definition of the Caputo/Jumarie derivative operator in equation (82) and collecting all terms of the same power of t^ip ∀ i = 0, 1, 2, 3, …, we get

T_{0} + T_{1} t^{p} + T_{2} t^{2 p} + T_{3} t^{3 p} + \dots = 0,

(83)

with

\begin{array}{l} T_{0} & = g_{1} (p) f_{1} + f f^{'} - f^{''}, \\ T_{1} & = g_{2} (p) f_{2} + f_{1} f^{'} + f f_{1}^{'} - f_{1}^{''}, \\ T_{2} & = g_{3} (p) f_{3} + f_{2} f^{'} + f_{1} f_{1}^{'} + f f_{2}^{'} - f_{2}^{''}, \\ ⋮, \end{array}

where the values of $g_{i} (p) \forall i = 1,2,3, \dots$ , are defined in equation (74)

Step (3) Equating the coefficients T₀, T₁, T₁, … to zero and solving them for f₁, f₂, f₃, the following implicit values of f_i as functions of f are obtained

\begin{align} f_{1} & = \frac{1}{Γ [1 + p]} (- f f^{'} + f^{''}), \end{align}

(84)

\begin{align} f_{2} & = \frac{1}{Γ [1 + 2 p]} (f^{2} f^{''} - 4 f^{'} f^{''} + 2 f {(f^{'})}^{2} - 2 f f^{(3)} + f^{(4)}), \end{align}

(85)

and

f_{3} = \begin{matrix} \frac{\begin{matrix} Γ {[ρ + 1]}^{2} [\begin{matrix} f^{(6)} - f^{(3)} f^{3} - 9 f^{(4)} f^{'} + 2 (8 f^{' 2} - 7 f^{(3)}) f^{''} \\ + f^{2} (3 f^{(4)} - 7 f^{'} f^{''}) + f (- 3 f^{(5)} + 10 f^{'' 2} - 4 f^{' 3} + 16 f^{(3)} f^{'}) \end{matrix}] \\ - Γ [2 ρ + 1] (f f^{'} - f^{''}) (- f^{(3)} + f f^{''} + f^{' 2}) \end{matrix}}{Γ {[ρ + 1]}^{2} Γ [3 ρ + 1]} \end{matrix} .

(86)

Step (4) Inserting the value of the IC f

f = \frac{1}{2} k (1 - \tanh (ζ_{0})),

(87)

into the values of f₁, f₂, f₃ given in equation (84)–(86), we get

\begin{align} f_{1} & = \frac{k^{3}}{16 Γ [1 + p]} {sech}^{2} (ζ_{0}), \\ f_{2} & = \frac{k^{5}}{64 Γ [1 + 2 p]} {sech}^{2} (ζ_{0}) \tanh (ζ_{0}), \\ f_{3} & = \frac{k^{7} {sech}^{4} (ζ_{0})}{512 Γ {[p + 1]}^{2} Γ [1 + 3 p]} \{\begin{matrix} Γ [1 + 2 p] \tanh (ζ_{0}) \\ + Γ {[p + 1]}^{2} [\cosh (2 ζ_{0}) - 2 (1 + \tanh (ζ_{0}))] \end{matrix}\} . \end{align}

(88)

Step (5) By substituting the acquired values of f₁, f₂, f₃, into Ansatz (71), we can get the approximate solution for example (2) until the third-order approximation as follows

\begin{align} ψ & = f + t^{p} f_{1} + t^{2 p} f_{2} + t^{3 p} f_{3} + \dots \\ = \frac{1}{2} k (1 - \tanh (ζ_{0})) + \frac{k^{3} t^{p}}{16 Γ [1 + p]} {sech}^{2} (ζ_{0}) + \frac{k^{5} t^{2 p}}{64 Γ [1 + 2 p]} {sech}^{2} (ζ_{0}) \tanh (ζ_{0}) \\ + \frac{k^{7} t^{3 p} {sech}^{4} (ζ_{0})}{512 Γ {[p + 1]}^{2} Γ [1 + 3 p]} \{\begin{matrix} Γ [1 + 2 p] \tanh (ζ_{0}) \\ + Γ {[p + 1]}^{2} [\cosh (2 ζ_{0}) - 2 (1 + \tanh (ζ_{0}))] \end{matrix}\} . \end{align}

(89)

The approximation (89) of example (2) utilizing the “Tantawy Technique” is examined against the fractional parameter p, as illustrated in Figure 12. Furthermore, Figure 13 visually contrasts the approximation (89) with the exact solution (32) for the problem (2) at p = 1. Also, the absolute error of the approximations (40), (62), and (89) utilizing ARPSM, ATIM, and the Tantawy Technique are evaluated, respectively, as illustrated in Figure 3

\begin{array}{l} {R_{\infty}|}_{ARPSM} & = {|Approx. (40) - Exact (32)|}_{p = 1}, \\ {R_{\infty}|}_{ATIM} & = {|Approx. (62) - Exact (32)|}_{p = 1}, \\ {R_{\infty}|}_{Tantawy} & = {|Approx. (89) - Exact (32)|}_{p = 1} . \end{array}

Additionally, we performed a numerical comparison of the absolute error of the approximations (40), (62), and (89) utilizing ARPSM, ATIM, and the “Tantawy Technique,” respectively, as illustrated in Table 2 and Figure 14. The findings demonstrate that the approximation (89) exhibits high accuracy and stability throughout the study domain, enhancing the Tantawy Technique’s efficiency. Moreover, it is observed that the approximation (89) using the “Tantawy Technique” is more accurate than the other two approximations (40) and (89).

Figure 12.

The approximation (89) of the Burgers equation (30) using the Tantawy Technique is analyzed graphically against p in (a) 3D graphic for p = .1, (b) 3D graphic for p = .3, (c) 3D graphic for p = 1, and (d) 2D graphic for t = 0.1. Here, k = 2.5.

Figure 13.

A comparison between the approximation (89) using the Tantawy Technique and the exact solution (32): (a) 3D graphic in the $(x, t)$ -plane and (b) 2D graphic for t = 0.1.

Table 2.

A numerical comparison between the absolute error R_∞ of the approximations (40), (62), and (89) utilizing ARPSM, ANIM, and the Tantawy Technique for the Burgers equation (30). Here, t = 0.01 and k = 2.5.

x	10⁻¹⁰R_∞\|_Tantawy	10⁻⁷×R_∞\|_ARPSM	10⁻⁷×R_∞\| _ATIM
0	0.0485034	1.98677	1.98677
0.1	1.87193	1.95782	2.20392
0.2	3.67014	1.86902	2.34799
0.3	5.23427	1.72611	2.41284
0.4	6.47759	1.53803	2.39799
0.5	7.34617	1.31598	2.3085
0.6	7.82145	1.07235	2.15408
0.7	7.91811	0.819573	1.94788
0.8	7.67808	0.569163	1.70491
0.9	7.16218	0.330973	1.44057
1	6.44121	0.112725	1.16932

Figure 14.

The absolute error R_∞ of the approximations (40): (a)–(b), (62): (c)–(d), and (89): (e)–(f) using ARPSM, ANIM, and the Tantawy Technique is estimated in (a), (c), and (e) $(x, t)$ -plane and (b), (d), and (f) against x for t = 0.01.

Summary and conclusion

In this study, Aboodh residual power series method (ARPSM), Aboodh new iteration method (ANIM), and the novel “Tantawy Technique” have been applied to analyze and solve the most common evolutionary equations in plasma physics and fluid mechanics, including the time fractional KdV-Burgers and Burgers equations. This class of evolutionary equations is used to describe the shock wave propagation that appears in viscosity plasma systems.²⁴ These approaches, in conjunction with the Caputo operator for managing fractional derivatives, have demonstrated their efficacy in delivering precise and efficient solutions. The two aforementioned evolutionary equations were analyzed using the proposed methods, and several approximate solutions were obtained until the second/or third-order approximation with high accuracy. We also analyzed these approximations graphically and numerically to verify their accuracy and the efficiency of the used methods. Moreover, we calculated and analyzed the absolute error of all derived approximations to also verify the accuracy of the derived approximations. Through the results we obtained, the accuracy and stability of the derived solutions were proven, which enhanced the efficiency of the methods used in analyzing many nonlinear and complicated fractional evolutionary equations. However, it was also found that ARPSM is more accurate than ANIM in analyzing fractional equations. However, the “Tantawy Technique” has shown impressive results despite its application simplicity and performance speed. As real applications for the current study, we used some different plasma models that were reduced from their fluid-governed equations to the fractional Burgers and KdV-Burgers equations. We implemented the two suggested approaches on this set of fractional partial differential equations (PDEs) to get analytical approximations that characterize the fractional shock waves propagating in various plasma systems. Furthermore, we investigated the impact of the fractional parameter on the characteristics of these waves to get insight into the dynamics of their propagation within the plasma medium.

In conclusion, our comprehensive analysis and numerical examples demonstrate that ARPSM and ATIM are robust tools for addressing the complexities of fractional PDEs. Nonetheless, implementing these methods may pose particular difficulties, particularly in complicated and strong nonlinearity problems. The “Tantawy Technique” surpasses all these procedures in terms of application simplicity, analytical speed, and result accuracy, notably superior to other techniques. We have validated the convergence and accuracy of these methods through detailed comparisons with known solutions, highlighting their potential for solving a wide range of fractional differential equations. These methods enhance the precision of the solutions and simplify the computational process, making them valuable for researchers and practitioners in various fields of science and engineering.

Future work:

• In the coming works, the Tantawy technique can be compared with various alternative methods, including the homotopy perturbation method (HPM) with Mohand Transform,⁷¹ HPM in the framework of two-scale transform,⁷² He–Laplace method,⁷³ Laplace HPM,⁷⁴ and many others in analyzing and studying various fractional evolutionary equations.

• The promising results demonstrated by the “Tantawy Technique” in analyzing the suggested problems indicate their applicability to the examination of various strong nonlinearity evolutionary wave equations in their fractional forms that govern the propagation of nonlinear waves across numerous physical and engineering systems, including plasma physics, optical fibers, and the propagation of water and fluid waves. For instance, the Tantawy Technique can be utilized for analyzing strong nonlinearity plasma wave equations in their fractional form, such as the KdV-type equations,^75–80 the Kawahara-type equations^81–83 the modified Kawahara-type equations,^84–87 the Gardner Kawahara-type equations,^88,89 the NLSE-type equations,^90–94 and others various of evolutionary wave equations. Consequently, in forthcoming studies, plasma physics researchers may employ the “Tantawy Technique” to examine the influence of the fractional-order parameter on the dynamics of various nonlinear waves that arise and propagate in diverse plasma systems.

Footnotes

Acknowledgments

The authors extend their appreciation to the Deanship of Scientific Research and Libraries in Princess Nourah bint Abdulrahman University for funding this research work through the Research Group project, Grant No. (RG-1445-0005).

Author’s Note

A sole inventor of the “Tantawy Technique”: Samir A El-Tantawy.

Authors Contributions

All authors contributed equally and approved the final version of the current manuscript.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The authors extend their appreciation to the Deanship of Scientific Research and Libraries in Princess Nourah bint Abdulrahman University for funding this research work through the Research Group project, Grant No. (RG-1445-0005).

Use of artificial intelligence (AI) tools declaration

The authors declare they have not used AI tools in the creation of this article.

ORCID iDs

CGL Tiofack

Samir A El-Tantawy

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