There has been a growing interest in fractional differential equations (FDEs) in recent years. The fractional approach is a strong modeling paradigm in mechanics and materials, wave propagation, anomalous diffusion, and turbulence. Natural phenomena exhibit anomalous diffusion, in which the underlying stochastic process does not follow Brownian motion. Compared to the Gaussian process, the mean-square variance may rise more quickly for superdiffusion or more slowly for subdiffusion. Due to long-range correlations in dynamics or anomalously large particle jumps, non-Gaussion diffusion models can be constructed utilizing nonlocal-in-time or nonlocal in-space operators, such as Caputo or Riemann–Liouville derivatives. The advantage of the fractional model is that anomalous diffusion is well described [1–11]. The singularity of the kernel poses a difficulty for the authors of Caputo and Riemann derivatives. Considering the fact that the kernel is utilized to clarify the memory impact of the physical system, it is indisputable that this limitation restricts both derivatives from accurately assessing the full effect of the memory. Caputo and Fabrizio (CF) [12] introduced a novel fractional operator with an exponential kernel during the mid-1990s as part of their effort to do so. The utilization of the nonsingular kernel of this derivative produces more logical outcomes when compared to the conventional method. A compilation of CF operator implementations has been expanded around in Ref. [13–15]. The research articles cited encompass a diverse range of topics within the field of control systems, vibration isolation, and neural network approximation. Guo et al. delve into fixed-time safe tracking control and non-singular fixed-time tracking control of uncertain nonlinear systems [16, 17] 3. Lu et al. focus on nonlinear vibration isolation systems with high-static-low-dynamic stiffness [18, 19]. Additionally, Luo et al. explore adaptive optimal control of affine nonlinear systems using identifier-critic neural network approximation [20]. These studies contribute valuable insights and advancements to their respective areas, showcasing the ongoing innovation and research efforts in control theory and engineering applications.

Determining an exact solution to partial differential equations (PDEs) of fractional order is exceedingly challenging. The ability to precisely and numerically solve such equations is critical in applied mathematics. As a result, innovative approaches have been developed to obtain analytical solutions that demonstrate a significant level of accuracy compared to the precise solutions [21–23]. The resolution of differential equations often involves the utilization of integral transformations. Employing integral transformations makes resolving IVPs and BVPs in differential and integral equations possible efficiently. An extensive array of scholars examined the consequences of various integral transforms applied to distinct classes of differential equations [24–26]. The Laplace transform is the integral transform that is most commonly utilized [27]. In 1998, Watugala [28] introduced the Sumudu transform, which proved to be an efficient approach to addressing control engineering and differential equations challenges. In 2011, T. Elzaki and S. Elzaki proposed the “Elzaki Transform” as an innovative integral transform; its utilization in the resolution of partial differential equations has since become widespread [29]. In 2013, Aboodh additionally presented the “Aboodh Transform (AT)” and applied it to the resolution of PDEs [30]. A variety of transformations are documented in the literature.

Omar Abu Arqub created the RPSM in 2013 [31]. The RPSM combines the residual error function with Taylor’s series. After that, this approach was used to find convergence series approximations for both nonlinear and linear differential equations. The RPSM was first introduced in 2013 to solve fuzzy differential equations. More improvements were made to this technique. For instance, Arqub et al. [32] developed a novel collection of RPSM algorithms to promptly find power series solutions for ordinary DEs. Furthermore, Arqub et al. [33] introduced a novel and appealing RPSM method for fractional-order nonlinear boundary value problems. El-Ajou et al. [34] introduced an innovative iterative approach utilizing RPSM to approximate fractional-order solutions to the KdV-burgers equations. A novel approach was introduced by Xu et al. [35], which involved fractional power series solutions for Boussinesq DEs of the second and fourth orders. Zhang et al. [36] synthesized least square methods and RPSM to develop a robust numerical technique. Consult [37–39] for additional readings on RPSM in greater depth.

Scientists utilized two distinct methodologies to solve fractional-order differential equations (FODEs). A sequence of solutions to the new equation form is obtained by mapping the original equation onto the space produced by the AT [40]. The solution to the original equation is obtained by applying the inverse Aboodh transform. Components of the Sumudu transform, and the homotopy perturbation approach are combined in this novel method. As power series expansions, the novel technique, which does not require discretization, linearization, or perturbation, can solve both linear and nonlinear PDEs. The determination of the coefficients can be accomplished through a limited number of calculations, in contrast to RPSM, which necessitates numerous iterations of fractional derivative computations during the solution phases. The proposed methodology has the potential to yield an accurate and closed-form approximation by leveraging a rapid convergence series.

For solving fractional differential equations, the Aboodh transform iteration method (ATIM) [41–43] and the Aboodh residual power series method (ARPSM) [44, 45] are regarded as the most straightforward techniques. These methods generate numerical approximations for solutions to linear and nonlinear differential equations without requiring discretization or linearization and immediately and visibly display the symbolic terms of analytical solutions. Comparing and contrasting the effectiveness of ARPSM and ATIM in solving nonlinear PDEs, specifically damped Burger’s equation, is the primary objective of this study. It is worth mentioning that these two methods have been employed to resolve many fractional differential problems, both linear and nonlinear.

Definition 2.1

[46] It is assumed that the function Θ(ζ, η) is of exponential order and piecewise continuous.

For τ ≥ 0, the AT of Θ(ζ, η) is defined as follows: A Θ ζ , η = Λ ζ , ϵ = 1 ϵ ∫ 0 ∞ Θ ζ , η e − η ϵ d η , r 1 ≤ ϵ ≤ r 2 .

Below is a description of the inverse of AT: A − 1 Λ ζ , ϵ = Θ ζ , η = 1 2 π i ∫ u − i ∞ u + i ∞ Λ ζ , η ϵ e η ϵ d η Where ζ = ( ζ 1 , ζ 2 , … , ζ p ) ∈ R and p ∈ N .

Lemma 2.1

[47, 48] Two functions of exponential order, Θ₁(ζ, η) and Θ₂(ζ, η), are defined. They are piecewise continuous on 0 , ∞ ]. Let us assume that A[Θ₁(ζ, η)] = Λ₁(ζ, η), A[Θ₂(ζ, η)] = Λ₂(ζ, η) and λ ₁, λ ₂ are real constants. Thus, the following features are valid:

1. A [λ ₁Θ₁ (ζ, η) + λ ₂Θ₂ (ζ, η)] = λ ₁Λ₁ (ζ, ϵ) + λ ₂Λ₂ (ζ, η),

Definition 2.2

[49] The Caputo defines the fractional derivative of the function Θ(ζ, η) in terms of order p. D η p Θ ζ , η = J η m − p Θ m ζ , η , r ≥ 0 , m − 1 < p ≤ m , where ζ = ( ζ 1 , ζ 2 , … , ζ p ) ∈ R p and m , p ∈ R , J η m − p is the R-L integral of Θ(ζ, η).

Definition 2.3

[50] The power series has the following form. ∑ r = 0 ∞ ℏ r ζ η − η 0 r p = ℏ 0 η − η 0 0 + ℏ 1 η − η 0 p + ℏ 2 η − η 0 2 p + ⋯ , where ζ = ( ζ 1 , ζ 2 , … , ζ p ) ∈ R p and p ∈ N . This kind of series is called a multiple fractional power series (MFPS) for η ₀, where the variable is η and the series coefficients are ℏ _r (ζ)′s.

Lemma 2.2

Let us assume that Θ(ζ, η) is the exponential order function. In this case, A[Θ(ζ, η)] = Λ(ζ, ϵ) is the definition of the AT. Therefore, A D η r p Θ ζ , η = ϵ r p Λ ζ , ϵ − ∑ j = 0 r − 1 ϵ p r − j − 2 D η j p Θ ζ , 0 , 0 < p ≤ 1 , (1) where ζ = ( ζ 1 , ζ 2 , … , ζ p ) ∈ R p and p ∈ N and D η r p = D η p . D η p . ⋯ . D η p ( r − t i m e s )

Proof. We can demonstrate Eq. 2 via induction. The following outcomes arise from selecting r = 1 in Eq. 2: A D η 2 p Θ ζ , η = ϵ 2 p Λ ζ , ϵ − ϵ 2 p − 2 Θ ζ , 0 − ϵ p − 2 D η p Θ ζ , 0

For r = 1, Lemma 2.1, part (4), asserts that Eq. 2 is valid. By changing r = 2 in Eq. 2, we get A D r 2 p Θ ζ , η = ϵ 2 p Λ ζ , ϵ − ϵ 2 p − 2 Θ ζ , 0 − ϵ p − 2 D η p Θ ζ , 0 . (2)

In light of Eq. 2’s left-hand side, we can conclude L . H . S = A D η 2 p Θ ζ , η . (3)

Eq. 3 may be expressed in the following way: L . H . S = A D η p Θ ζ , η . (4)

Let us assume z ζ , η = D η p Θ ζ , η . (5)

Thus, Eq. 4 becomes as L . H . S = A D η p z ζ , η . (6)

The use of the Caputo type fractional derivative results in a modification of Eq. 6. L . H . S = A J 1 − p z ′ ζ , η . (7)

The R-L integral for the AT is found in Eq. 7, which makes it possible to derive the following: L . H . S = A z ′ ζ , η ϵ 1 − p . (8)

Equation 8 is transformed into the following form by using the differential characteristic of the AT: L . H . S = ϵ p Z ζ , ϵ − z ζ , 0 ϵ 2 − p , (9)

From Eq. 5, we obtain: Z ζ , ϵ = ϵ p Λ ζ , ϵ − Θ ζ , 0 ϵ 2 − p , where A [z (ζ, η)] = Z (ζ, ϵ). Therefore, Eq. 9 is converted to L . H . S = ϵ 2 p Λ ζ , ϵ − Θ ζ , 0 ϵ 2 − 2 p − D η p Θ ζ , 0 ϵ 2 − p , (10)

According to Eq. 2, then Eq. 10 is compatible. Let us assume the validity of Eq. 2 for r = K. This allows us to change r = K in Eq. 2: A D η K p Θ ζ , η = ϵ K p Λ ζ , ϵ − ∑ j = 0 K − 1 ϵ p K − j − 2 D η j p D η j p Θ ζ , 0 , 0 < p ≤ 1 . (11)

Proving Eq. 2 for the value of r = K + 1 is the next step. Based on Eq. 2, we may write A D η K + 1 p Θ ζ , η = ϵ K + 1 p Λ ζ , ϵ − ∑ j = 0 K ϵ p K + 1 − j − 2 D η j p Θ ζ , 0 . (12)

After analysis of the LHS of Eq. 12, we get L . H . S = A D η K p D η K p . (13)

Suppose that D η K p = g ζ , η .

Equation 13 yields L . H . S = A D η p g ζ , η . (14)

By using the R-L integral formula and the Caputo fractional derivative, we may convert Eq. 14 into the following expression. L . H . S = ϵ p A D η K p Θ ζ , η − g ζ , 0 ϵ 2 − p . (15)

Equation 11 is unitized to provide Eq. 15. L . H . S = ϵ r p Λ ζ , ϵ − ∑ j = 0 r − 1 ϵ p r − j − 2 D η j p Θ ζ , 0 , (16)

Moreover, Eq. 16 yields the following result. L . H . S = A D η r p Θ ζ , 0 .

Therefore, Eq. 2 holds for r = K + 1. Thus, we used the mathematical induction approach and shows that Eq. 2 holds true for all positive integers.

Extending the concept of multiple fractional A lemma demonstrating Taylor’s formula is shown below. The ARPSM, which will be covered in more detail later on, will benefit from this formula.

Lemma 2.3

Assume that the function Θ(ζ, η) behaves exponentially order. The statement A[Θ(ζ, η)] = Λ(ζ, ϵ) represents the AT of Θ(ζ, η), and it is multiple fractional Taylor’s series expressed as: Λ ζ , ϵ = ∑ r = 0 ∞ ℏ r ζ ϵ r p + 2 , ϵ > 0 , (17) where, ζ = ( s 1 , ζ 2 , … , ζ p ) ∈ R p , p ∈ N .

Proof. Now we examine the fractional order of Taylor’s series as Θ ζ , η = ℏ 0 ζ + ℏ 1 ζ η p Γ p + 1 + + ℏ 2 ζ η 2 p Γ 2 p + 1 + ⋯ . (18)

Equation 18 may be transformed using the AT to get the following equality: A Θ ζ , η = A ℏ 0 ζ + A ℏ 1 ζ η p Γ p + 1 + A ℏ 1 ζ η 2 p Γ 2 p + 1 + ⋯

For this, we use the AT’s characteristics. A Θ ζ , η = ℏ 0 ζ 1 ϵ 2 + ℏ 1 ζ Γ p + 1 Γ p + 1 1 ϵ p + 2 + ℏ 2 ζ Γ 2 p + 1 Γ 2 p + 1 1 ϵ 2 p + 2 ⋯

Hence, in the AT, we obtains (17), a new version of Taylor’s series.

Lemma 2.4

Define the MFPS representation of the function expressed in the new form of Taylor’s series (17) as A[Θ(ζ, η)] = Λ(ζ, ϵ). Next, we have ℏ 0 ζ = lim ϵ → ∞ ϵ 2 Λ ζ , ϵ = Θ ζ , 0 . (19)

Proof. The subsequent is derived from the new form of Taylor’s series: ℏ 0 ζ = ϵ 2 Λ ζ , ϵ − ℏ 1 ζ ϵ p − ℏ 2 ζ ϵ 2 p − ⋯ (20)

The required result, denoted by Eq. 20, is obtained by applying lim _ϵ→∞ to Eq. 19 and performing a brief computation.

Theorem 2.5

Let us suppose that the function A[Θ(ζ, η)] = Λ(ζ, ϵ) has MFPS form given by Λ ζ , ϵ = ∑ 0 ∞ ℏ r ζ ϵ r p + 2 , ϵ > 0 , where ζ = ( ζ 1 , ζ 2 , … , ζ p ) ∈ R p and p ∈ N . Then we have ℏ r ζ = D r r p Θ ζ , 0 , where, D η r p = D η p . D η p . ⋯ . D η p ( r − t i m e s ) .

Proof. This is the revised version of the Taylor’s series that we have. ℏ 1 ζ = ϵ p + 2 Λ ζ , ϵ − ϵ p ℏ 0 ζ − ℏ 2 ζ ϵ p − ℏ 3 ζ ϵ 2 p − ⋯ (21)

Using Eq. 21 and the lim _ϵ→∞ , we are able to get ℏ 1 ζ = lim ϵ → ∞ ϵ p + 2 Λ ζ , ϵ − ϵ p ℏ 0 ζ − lim ϵ → ∞ ℏ 2 ζ ϵ p − lim ϵ → ∞ ℏ 3 ζ ϵ 2 p − ⋯

Taking limit, we arrive at the equality that follows: ℏ 1 ζ = lim ϵ → ∞ ϵ p + 2 Λ ζ , ϵ − ϵ p ℏ 0 ζ . (22)

Following is the result that is obtained by applying Lemma (2.2) to Eq. 22: ℏ 1 ζ = lim ϵ → ∞ ϵ 2 A D η p Θ ζ , η ϵ . (23)

Through the use of Lemma (2.3) to Eq. 23, the equation is changed into ℏ 1 ζ = D η p Θ ζ , 0 .

Once again, by taking into consideration the new implementation of Taylor’s series and assuming limit ϵ → ∞, we have arrived at the result that ℏ 2 ζ = ϵ 2 p + 2 Λ ζ , ϵ − ϵ 2 p ℏ 0 ζ − ϵ p ℏ 1 ζ − ℏ 3 ζ ϵ p − ⋯

Lemma (2.3) leads us to get the following: ℏ 2 ζ = lim ϵ → ∞ ϵ 2 ϵ 2 p Λ ζ , ϵ − ϵ 2 p − 2 ℏ 0 ζ − ϵ p − 2 ℏ 1 ζ . (24)

With the help of Lemmas (2.2) and (2.4), Eq. 24 is transformed into ℏ 2 ζ = D η 2 p Θ ζ , 0 .

When we apply the same method to the subsequent Taylor’s series, we obtain the following results: ℏ 3 ζ = lim ϵ → ∞ ϵ 2 A D η 2 p Θ ζ , p ϵ .

The final equation can be found by applying Lemma (2.4). ℏ 3 ζ = D η 3 p Θ ζ , 0 .

So, in general ℏ r ζ = D η r p Θ ζ , 0 .

Thus, the proof comes to an end.

In the succeeding theorem, the conditions that determine the convergence of the new version of Taylor’s formula are established and detailed in further depth.

Theorem 2.6

The revised formula for multiple fractional Taylor’s, given in Lemma (2.3), is denoted by the expression A[Θ(ζ, η)] = Λ(ζ, ϵ). The new version of multiple fractional Taylor’s formula’s residual R _K (ζ, ϵ) satisfies the following inequality if | ϵ a A [ D η ( K + 1 ) p Θ ( ζ , η ) ] | ≤ T , on 0 < ϵ ≤ s is associated with 0 < p ≤ 1: | R K ζ , ϵ | ≤ T ϵ K = 1 p + 2 , 0 < ϵ ≤ s .

Proof. To start the proof, Let assume: For r = 0, 1, 2, … , K + 1, A [ D η r p Θ ( ζ , η ) ] ( ϵ ) is defined on 0 < ϵ ≤ s. Let, | ϵ 2 A [ D η K + 1 Θ ( ζ , t a u ) ] | ≤ T , o n 0 < ϵ ≤ s . Based on the revised version of Taylor’s series, determine the following relationship: R K ζ , ϵ = Λ ζ , ϵ − ∑ r = 0 K ℏ r ζ ϵ r p + 2 . (25)

Applying Theorem (2.5) allows for the transformation of Eq. 25. R K ζ , ϵ = Λ ζ , ϵ − ∑ r = 0 K D η r p Θ ζ , 0 ϵ r p + 2 . (26)

It is necessary to multiply ϵ ^(K+1)a+2 on both sides of Eq. 26 which leads to ϵ K + 1 p + 2 R K ζ , ϵ = ϵ 2 ϵ K + 1 p Λ ζ , ϵ − ∑ r = 0 K ϵ K + 1 − r p − 2 D η r p Θ ζ , 0 . (27)

The use of Lemma (2.2) to Eq. 27 results in ϵ K + 1 p + 2 R K ζ , ϵ = ϵ 2 A D η K + 1 p Θ ζ , η . (28)

Equation 28 is obtained by applying the absolute sign to the equation. | ϵ K + 1 p + 2 R K ζ , ϵ | = | ϵ 2 A D η K + 1 p Θ ζ , η | . (29)

By applied the condition given in Eq. 29, we can arrive at the result as will be given below. − T ϵ K + 1 p + 2 ≤ R K ζ , ϵ ≤ T ϵ K + 1 p + 2 . (30)

Equation 30 yields the required result. | R K ζ , ϵ | ≤ T ϵ K + 1 p + 2 .

Hence, a novel criterion for series convergence is established.