Noninteger calculus is a prominent branch of mathematics that employs fractional order operators to mimic and clarify physical processes. Another facet of this topic is the application of noninteger order derivatives to both integration and differentiation difficulties. For zero-order the ordinary derivative is recovered. A fractional derivative has a noninteger order and meets specific conditions [1]. A few of the benefits of fractional derivatives include the memory effect and preserved demonstrative physical qualities. Using these operators, more recent and accurate research has been discovered. Thus, the theory and practice of fractional calculus are seeing a surge in popularity. The memory effect allows fractional order models to absorb all prior information, which improves their ability to anticipate and evaluate dynamical models. The efficient characteristics of fractional order calculus make it applicable to numerous fields, including biology and physics [2–6] economics and finance [7, 8], mechanical modeling and mathematical modeling, and [9–11] science and engineering. The singularity of the kernel in Caputo and Riemann derivatives presents a challenge for the authors. Given that the kernel is employed to elucidate the memory impact of the physical system, it is evident that this constraint prevents both derivatives from precisely interpreting the memory’s complete effect [12–14]. In an endeavor to develop a novel fractional operator featuring an exponential kernel, Caputo and Fabrizio (CF) [15] proposed one in the mid-nineties. The nonsingular kernel of this derivative yields more rational results than the classical approach. A selection of CF operator applications is detailed in [16–18].

There are two categories of partial differential equations: linear and nonlinear. Solving a fractional-order partial differential equation accurately is a difficult task. However, developing numerical and exact solutions to these equations is crucial in applied mathematics and theoretical physics [19–21]. Consequently, innovative methods have been developed for analytical solutions that closely approximate the exact solutions [22, 23]. Differential equations were often solved using integral transforms. Integral transformations are helpful in solving initial value problems (IVPs) and boundary value problems (BVPs) in differential and integral equations. A wide range of authors investigated the effects of diverse types of integral transforms implemented on various categories of differential equations. The most frequently employed integral transform is the Laplace transform [24]. Watugala [25] introduced the Sumudu transform in 1998 as a powerful technique for solving differential equations and addressing engineering problems. T. Elzaki and S. Elzaki [26] introduced the “Elzaki Transform” as a new integral transform in 2011, which is extensively used in solving partial differential equations. Aboodh introduced the “Aboodh Transform” in 2013 and applied it to address partial differential equations (27). Numerous transformations are present in the literature.

In 2013, Omar Abu Arqub developed the RPSM [28]. The RPSM is a semi-analytical approach that combines Taylor’s series with the residual error function. Convergence series for nonlinear and linear DEs are both solved using it. In 2013, RPSM was first used to resolve fuzzy differential equations. To quickly get power series solutions for common DEs, Arqub et al. [29] created a novel RPSM method. Arqub et al. [30] developed a unique and appealing RPSM technique for fractional DEs issues. El-Ajou et al. [31] proposed a special iterative technique using RPSM to estimate fractional KdV-burger equations. Xu et al. [32] introduced a novel approach for solving Boussinesq DEs using fractional power series. Zhang et al. devised a robust numerical approach [33]. For further information on RPSM, see [34–36].

Differential equations have been crucial in several aspects of applied mathematics and theoretical physics for an extended period, and their importance has increased with the advent of computers [37, 38]. Examining and analyzing differential equations used in applications reveal numerous intricate mathematical challenges, resulting in various methods for solving them. Various integral transforms, such as Laplace, Fourier, Mellin, Hankel, and Sumudu, were commonly used for solving differential equations. Khalid Aboodh introduced a new integral transform called the Aboodh transform and applied it to solve both ordinary and partial differential equations (39)–(41).

The Aboodh residual power series method (ARPSM) [42, 43] and the Aboodh transform iteration method (ATIM) [39–41] are regarded as the most straightforward methods to solve fractional differential equations. These approaches give numeric results for partial differential equations that do not need discretization or linearization and make the symbol elements of analytic solutions more visible and accessible. The fundamental goal of this research is to contrast and evaluate the efficacy of ARPSM and ATIM in solving linear and nonlinear PDE problems. It is worth noting that many linear and nonlinear fractional differential equations have been solved using these two approaches.

Definition 2.1

[44] The function ϕ(α, β) is assumed to be piecewise continuous and of exponential order.

For ϕ(α, β) and τ ≥ 0 Aboodh transform (AT) is given as: A ϕ α , β = Ψ α , ξ = 1 ξ ∫ 0 ∞ ϕ α , β e − β ξ d β , r 1 ≤ ξ ≤ r 2 .

The Aboodh inverse transform (AIT) is described as: A − 1 Ψ α , ξ = ϕ α , β = 1 2 π i ∫ u − i ∞ u + i ∞ Ψ α , β ξ e β ξ d β Where α = ( α 1 , α 2 , … , α p ) ∈ R and p ∈ N

Lemma 2.1

[45, 46] Define two functions ϕ ₁(α, ϕ ₂ β) that are piecewise continuous on [0, ∞[ and of exponential order. Suppose that A[ϕ ₁(α, β)] = Ψ₁(α, β), A[ϕ ₂(α, β)] = Ψ₂(α, β) and λ ₁, λ ₂ are constants. Consequently, the following characteristics hold:

1. A [λ ₁ ϕ ₁ (α, β) + λ ₂ ϕ ₂ (α, β)] = λ ₁Ψ₁ (α, ξ) + λ ₂Ψ₂ (α, β),

Definition 2.2

[47] The fractional derivative of the function ϕ(α, β) is defined in terms of order p according to the Caputo. D β p ϕ α , β = J β m − p ϕ m α , β , r ≥ 0 , m − 1 < p ≤ m , where α = ( α 1 , α 2 , … , α p ) ∈ R p and m , p ∈ R , J β m − p is the R-L integral of ϕ(α, β).

Definition 2.3

[48] The following is the form of the power series representation. ∑ r = 0 ∞ ℏ r α β − β 0 r p = ℏ 0 β − β 0 0 + ℏ 1 β − β 0 p + ℏ 2 β − β 0 2 p + ⋯ , where α = ( α 1 , α 2 , … , α p ) ∈ R p and p ∈ N . This series is called multiple fractional power series (MFPS) about β ₀, where the series coefficients are ℏ _r (α)′s and β is variable.

Lemma 2.2

Let us suppose that the exponential order function is ϕ(α, β). In such case, the AT is defined as A[ϕ(α, β)] = Ψ(α, ξ). Hence,

Proof. Using induction, we can prove Eq. 2. Choosing r = 1 in Eq. 2 leads to the following results: A D β 2 p ϕ α , β = ξ 2 p Ψ α , ξ − ξ 2 p − 2 ϕ α , 0 − ξ p − 2 D β p ϕ α , 0 Lemma 2.1, part (4), states that Eq. 2 is true for r = 1. By substituting r = 2 into Eq. 2, we have A D r 2 p ϕ α , β = ξ 2 p Ψ α , ξ − ξ 2 p − 2 ϕ α , 0 − ξ p − 2 D β p ϕ α , 0 . (2)

Based on Eq. 2 L.H.S, we may deduce L . H . S = A D β 2 p ϕ α , β . (3)

One possible way to express Eq. 3 is as: L . H . S = A D β p ϕ α , β . (4)

Suppose z α , β = D β p ϕ α , β . (5)

Equation 4 therefore becomes as L . H . S = A D β p z α , β . (6)

Due to the utilization of the derivative of Caputo, Eq. 6 modified as. L . H . S = A J 1 − p z ′ α , β . (7)

Equation 7 contains the R-L integral for AT, which enables the following to be obtained: L . H . S = A z ′ α , β ξ 1 − p . (8)

Using the differential property of the AT, Eq. 8 is converted to the following form: L . H . S = ξ p Z α , ξ − z α , 0 ξ 2 − p , (9)

Eq. 5 gives us the following: Z α , ξ = ξ p Ψ α , ξ − ϕ α , 0 ξ 2 − p , where A [z (α, β)] = Z (α, ξ). Therefore, Eq. 9 is converted to L . H . S = ξ 2 p Ψ α , ξ − ϕ α , 0 ξ 2 − 2 p − D β p ϕ α , 0 ξ 2 − p , (10) when r = K. Eq. 10 is compatible with Eq. 2. Suppose that for r = K, Eq. 2 holds true. As a result, we may substitute r = K into Eq. 2: A D β K p ϕ α , β = ξ K p Ψ α , ξ − ∑ j = 0 K − 1 ξ p K − j − 2 D β j p D β j p ϕ α , 0 , 0 < p ≤ 1 . (11)

The next step is to illustrate Eq. 2 for the value of r = K + 1. Using Eq. 2, we can write A D β K + 1 p ϕ α , β = ξ K + 1 p Ψ α , ξ − ∑ j = 0 K ξ p K + 1 − j − 2 D β j p ϕ α , 0 . (12)

When the LHS of Eq. 12 is taken into consideration, we get L . H . S = A D β K p D β K p . (13)

Let D β K p = g α , β .

By Eq. 13, we get L . H . S = A D β p g α , β . (14)

Equation 14 is transformed into the following using the R-L integral and derivative of Caputo. L . H . S = ξ p A D β K p ϕ α , β − g α , 0 ξ 2 − p . (15)

By ulatizing Eq. 11 and Eq. 15 becomes L . H . S = ξ r p Ψ α , ξ − ∑ j = 0 r − 1 ξ p r − j − 2 D β j p ϕ α , 0 , (16) further, the following result is derived from Eq. 16. L . H . S = A D β r p ϕ α , 0 .

Consequently, for r = K + 1, Eq. 2 is true. In light of this, we showed that Eq. 2 holds for every positive integer by applying the mathematical induction method.

A novel version of multiple fractional Taylor’s series (MFTS) is shown in the following lemma. This formula will be helpful for the ARPSM, which will be discussed further below.

Lemma 2.3

Let’s assume ϕ(α, β) be the exponential order function. The AT of ϕ(α, β), which is represented by the expression A[ϕ(α, β)] = Ψ(α, ξ), is characterized by a MFTS notation as:

Proof. Let us analyze Taylor’s series expressed in fractional order as ϕ α , β = ℏ 0 α + ℏ 1 α β p Γ p + 1 + + ℏ 2 α β 2 p Γ 2 p + 1 + ⋯ . (18)

Applying the AT to Eq. 18 yields the subsequent equality: A ϕ α , β = A ℏ 0 α + A ℏ 1 α β p Γ p + 1 + A ℏ 1 α β 2 p Γ 2 p + 1 + ⋯

This is achieved by employing the properties of the AT. A ϕ α , β = ℏ 0 α 1 ξ 2 + ℏ 1 α Γ p + 1 Γ p + 1 1 ξ p + 2 + ℏ 2 α Γ 2 p + 1 Γ 2 p + 1 1 ξ 2 p + 2 ⋯

As a result, 17, which is a novel variant of Taylor’s series in the AT, is acquired.

Lemma 2.4

Let A[ϕ(α, β)] = Ψ(α, ξ) be the MFPS stated in the new form of Taylor’s series 17. Then we have

Proof. From the Taylor’s series new form, the preceding is derived: ℏ 0 α = ξ 2 Ψ α , ξ − ℏ 1 α ξ p − ℏ 2 α ξ 2 p − ⋯ (20)

After applying lim _ξ→∞ to Eq. 19 and doing a short computation, we get the necessary result, which is shown by 20.

Theorem 2.5

Let the MFPS notation of the function A[ϕ(α, β)] = Ψ(α, ξ) is provided by Ψ α , ξ = ∑ 0 ∞ ℏ r α ξ r p + 2 , ξ > 0 , where α = ( α 1 , α 2 , … , α p ) ∈ R p and p ∈ N . Then we have ℏ r α = D r r p ϕ α , 0 , where, D β r p = D β p . D β p . ⋯ . D β p ( r − t i m e s ) .

Proof. We have the Taylor’s series in its new form ℏ 1 α = ξ p + 2 Ψ α , ξ − ξ p ℏ 0 α − ℏ 2 α ξ p − ℏ 3 α ξ 2 p − ⋯ (21)

Using Eq. 21 and lim _ξ→∞ , we get ℏ 1 α = lim ξ → ∞ ξ p + 2 Ψ α , ξ − ξ p ℏ 0 α − lim ξ → ∞ ℏ 2 α ξ p − lim ξ → ∞ ℏ 3 α ξ 2 p − ⋯

Taking the linit, we arrive to the following equality: ℏ 1 α = lim ξ → ∞ ξ p + 2 Ψ α , ξ − ξ p ℏ 0 α . (22)

The application of Lemma 2.2 to Eq. 22 results in the following: ℏ 1 α = lim ξ → ∞ ξ 2 A D β p ϕ α , β ξ . (23)

Moreover, by using Lemma 2.3 to Eq. 23, it is transformed into ℏ 1 α = D β p ϕ α , 0 .

Once more, by taking into account the new form of Taylor’s series and taking limit ξ → ∞, we conclude that ℏ 2 α = ξ 2 p + 2 Ψ α , ξ − ξ 2 p ℏ 0 α − ξ p ℏ 1 α − ℏ 3 α ξ p − ⋯

As a result of Lemma 2.3, we get ℏ 2 α = lim ξ → ∞ ξ 2 ξ 2 p Ψ α , ξ − ξ 2 p − 2 ℏ 0 α − ξ p − 2 ℏ 1 α . (24)

Using Lemmas 2.2 and 2.4, Eq. 24 becomes ℏ 2 α = D β 2 p ϕ α , 0 .

Applying the same process to the new Taylor’s series yields the following results: ℏ 3 α = lim ξ → ∞ ξ 2 A D β 2 p ϕ α , p ξ .

Following the use of Lemma 2.4, the final equation is found. ℏ 3 α = D β 3 p ϕ α , 0 .

In general ℏ r α = D β r p ϕ α , 0 .

Thus, the proof concludes.

The subsequent theorem details and establishes the conditions that govern the convergence of the new form of Taylor’s series.

Theorem 2.6

The expression A[ϕ(α, β)] = Ψ(α, ξ) represents the new form of the formula for multiple fractional Taylor’s, which is presented in Lemma 2.3. If | ξ a A [ D β ( K + 1 ) p ϕ ( α , β ) ] | ≤ T , on 0 < ξ ≤ s with 0 < p ≤ 1, then the following inequality is satisfied by the residual R _K (α, ξ) of the new version of MFTS: | R K α , ξ | ≤ T ξ K = 1 p + 2 , 0 < ξ ≤ s .

Proof. Let A [ D β r p ϕ ( α , β ) ] ( ξ ) for r = 0, 1, 2, … , K + 1, is defined on 0 < ξ ≤ s. Assume, as given, that | ξ 2 A [ D β K + 1 ϕ ( α , t a u ) ] | ≤ T , o n 0 < ξ ≤ s . Determine the following relationship based on the revised version of Taylor’s series: R K α , ξ = Ψ α , ξ − ∑ r = 0 K ℏ r α ξ r p + 2 . (25)

Equation 25 is transformed by using Theorem 2.5. R K α , ξ = Ψ α , ξ − ∑ r = 0 K D β r p ϕ α , 0 ξ r p + 2 . (26)

On both sides of Eq. 26, multiply ξ ^(K+1)a+2. ξ K + 1 p + 2 R K α , ξ = ξ 2 ξ K + 1 p Ψ α , ξ − ∑ r = 0 K ξ K + 1 − r p − 2 D β r p ϕ α , 0 . (27)

Lemma 2.2 applied to Eq. 27 gives ξ K + 1 p + 2 R K α , ξ = ξ 2 A D β K + 1 p ϕ α , β . (28)

Taking absolute of Eq. 28, we obtain | ξ K + 1 p + 2 R K α , ξ | = | ξ 2 A D β K + 1 p ϕ α , β | . (29)

After applying the specified condition in Eq. 29, we arrive to the following result. − T ξ K + 1 p + 2 ≤ R K α , ξ ≤ T ξ K + 1 p + 2 . (30)

Equation 30 provides the necessary outcome. | R K α , ξ | ≤ T ξ K + 1 p + 2 .

In consequence, the new condition for series convergence is developed.