EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 2, Article Number 6099 ISSN 1307-5543 – ejpam.com Published by New York Business Global Solving Partial Differential Equations via the Conformable Double ARA-Sawi Transform Monther Al-Momani1,∗, Ali Jaradat2, Baha’ Abughazaleh3, Abdulkarim Farah3 1 Department of Basic Sciences, Al-Ahliyya Amman University, Amman, Jordan 2 Department of Mathematics, Amman Arab University, Amman, Jordan 3 Department of Mathematics, Isra University, Amman, Jordan Abstract. This paper presents a new method the conformable double ARA-Sawi transform to solve fractional partial differential equations that arise in physics and engineering. These equations often involve derivatives based on conformable calculus, which generalizes classical derivatives to fractional orders. We explore the core properties of the transform, such as its linearity and inter- action with fractional derivatives, and establish conditions for its applicability. To demonstrate its practical use, we apply the method to solve two key equations: the conformable Klein-Gordon equation, which models wave propagation in quantum fields, and the conformable telegraph equa- tion, governing signal transmission in dissipative systems. The results highlight the transform’s ability to simplify complex fractional equations into manageable algebraic forms, offering a system- atic tool for researchers. This work bridges theoretical advancements with real-world applications, paving the way for future studies in areas like nonlinear dynamics and material science. 2020 Mathematics Subject Classifications: 44A05 Key Words and Phrases: ARA transform, Sawi transform, the double ARA-Sawi transform, the conformable double ARA-Sawi transform. 1. Introduction Fractional partial differential equations play a crucial role in modeling complex systems in physics, electrical circuits, fluid dynamics, optics, and mathematical biology. One important development in this field is the conformable fractional derivative, introduced in [1], which retains many key properties of standard derivatives. To solve conformable fractional partial differential equations, researchers have explored several techniques. Among these are the conformable double Laplace transform [2, 3], the Conformable Double Laplace-Sawi Transform [4] and the conformable double Sumudu transform [5]. ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i2.6099 Email addresses: montheralmomani72@gmail.com (M. Al-Momani), a.jaradat@aau.edu.jo (A. Jaradat), baha.abughazaleh@iu.edu.jo (B. Abughazaleh), karim.farah@iu.edu.jo (A. Farah) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) M. Al-Momani et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6099 2 of 15 The double ARA-Sawi transform [6] was proposed, offering a new perspective in han- dling boundary value problems for integer-order partial differential equations. For further insights into integral transforms, see [7–13]. However, when dealing with fractional derivatives, especially of conformable type, there remained a need for a structured and efficient method. The conformable double ARA-Sawi transform (CA-SW) was introduced to fill this gap. Inspired by the success of the double Sawi transform in solving boundary value problems [6], the CA-SW transform adapts and extends the method to the conformable framework, allowing researchers to manage more complex fractional models. The motivation behind this study stems from the increasing demand for robust analyt- ical tools that can systematically address fractional differential equations without relying on cumbersome or purely numerical techniques. Traditional methods often struggle when fractional derivatives appear with variable coefficients or non-standard boundary condi- tions. By developing the CA-SW, we aim to offer a reliable and systematic analytical tool that simplifies these equations into algebraic forms that are easier to solve. This paper focuses on establishing the theoretical foundation of the CA-SW transform. We define its existence conditions, explore its linearity, and derive its interaction with conformable partial derivatives. Several basic examples demonstrate how to compute the transform explicitly for elementary functions. Applications are a key component of this work. We apply the CA-SW transform to solve two important conformable partial differential equations: the conformable Klein- Gordon equation and the conformable telegraph equation. These applications showcase how the CA-SW transform can simplify complex equations, reduce the number of steps needed for their solution, and provide explicit forms of the solutions. By extending the classical ARA-Sawi transform to the conformable setting and demon- strating its practical utility, this study bridges theoretical advancements with real-world applications, paving the way for further developments in fractional dynamics, nonlinear systems, signal transmission, and material science. The CA-SW systematically reduces complex conformable fractional equations into solvable algebraic forms while preserving key properties like linearity and scaling. It extends classical transforms to the conformable setting, offering broader applicability to equations with variable coefficients and non-standard conditions. This makes it a powerful tool for solving advanced problems in physics and engineering. 2. Fundamental Definitions and Theorems This section covers essential definitions and theorems related to conformable fractional derivatives. It defines the conformable fractional derivative, explores its key properties, and presents fundamental theorems needed for applying the CA-SW. M. Al-Momani et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6099 3 of 15 Definition 1. [1] Let 0 < β ≤ 1 and s : (0,∞) → R. The conformable fractional derivative of order β is defined as: dβ dξβ s(ξ) = lim n→0 s(ξ + nξ1−β)− s(ξ) n , where ξ > 0, and ∂β ∂ξβ is referred to as the fractional derivative of order β. Definition 2. [14] Let 0 < β1, β2 ≤ 1 and s(ξ, ρ) : (0,∞)× (0,∞) → R. The conformable partial derivatives of orders β1 and β2 of the function s(ξ, ρ) are defined as: ∂β1 ∂ξβ1 s(ξ, ρ) = lim n→0 s(ξ + nξ1−β1 , ρ)− s(ξ, ρ) n , ∂β2 ∂ρβ2 s(ξ, ρ) = lim n→0 s(ξ, ρ+ nρ1−β2)− s(ξ, ρ) n , where ξ, ρ > 0, ∂β1 ∂ξβ1 and ∂β2 ∂ρβ2 are referred to as fractional derivatives of orders β1 and β2, respectively. Theorem 1. [15]Suppose that s(ξ, ρ) be differentiable at a point ξ, ρ > 0, 0 < β1, β2 ≤ 1, then: ∂β1s ∂ξβ1 = ξ1−β1 ∂s ∂ξ , ∂β2s ∂ρβ2 = ρ1−β2 ∂s ∂ρ . 3. Conformable Double ARA-Sawi transform This section serves to introduce the CA-SW. We commence by delineating its funda- mental properties, encompassing aspects like linearity. Subsequently, we reveal a novel result associated with partial derivatives. Ultimately, we illustrate how these insights enable us to compute the CA-SW for various essential functions. Definition 3. Let s(ξ, ρ) be a continuous function on (0,∞)× (0,∞). Then 1- The Conformable ARA transformation (CA) of s(ξ, ρ), denoted by Aβξ [s(ξ, ρ)], is defined as: J (ψ) = Aβξ (s(ξ, ρ)) = ψ ∞∫ 0 e −ψ ξβ β s(ξ, ρ)ξβ−1dξ, ψ ∈ C. 2- The Conformable Sawi transformation (CSW) of s(ξ, ρ), denoted by Aβρ [s(ξ, ρ)], is defined as: L (κ) =W β ρ (s(ξ, ρ)) = 1 κ2 ∞∫ 0 e − ρβ κβ s(ξ, ρ)ρβ−1dρ, κ ∈ C. M. Al-Momani et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6099 4 of 15 3- CA-SW of s(ξ, ρ), denoted by Aβ1ξ W β2 ρ [s(ξ, ρ)], is defined as: S(ψ,κ) = Aβ1ξ W β2 ρ [s(ξ, ρ)] = ψ κ2 ∫ ∞ 0 ∫ ∞ 0 e − ( ψ ξβ1 β1 + ρβ2 κβ2 ) s(ξ, ρ)ξβ1−1ρβ2−1dξdρ. Theorem 2. Assume that s : (0,∞)×(0,∞) → R such that S(ψ,κ) = Aβ1ξ W β2 ρ [s( ξ β1 β1 , ρ β2 β2 )] exist, then Aβ1ξ W β2 ρ [s( ξβ1 β1 , ρβ2 β2 )] = AξWρ[s(ξ, ρ)], where AξWρ[s(ξ, ρ)] = ψ κ2 ∫ ∞ 0 ∫ ∞ 0 e−(ψξ+ ρ κ )s(ξ, ρ) dξ dρ. Lemma 1. Aβ1ξ W β2 ρ (s(ξ, ρ)) is a linear transformation. Proof. for nonzero constants λ and ν, we have Aβ1ξ W β2 ρ (λs1(ξ, ρ)+νs2(ξ, ρ)) = ψ κ2 ∞∫ 0 ∞∫ 0 e − ( ψ ξβ1 β1 + ρβ2 κβ2 ) (λs1(ξ, ρ) + νs2(ξ, ρ))ξ β1−1ρβ2−1dξdρ, = λ ψ κ2 ∞∫ 0 ∞∫ 0 e − ( ψ ξβ1 β1 + ρβ2 κβ2 ) s1(ξ, ρ)ξ β1−1ρβ2−1dξdρ+ ν ψ κ2 ∞∫ 0 ∞∫ 0 e − ( ψ ξβ1 β1 + ρβ2 κβ2 ) s2(ξ, ρ)ξ β1−1ρβ2−1dξdρ = λAβ1ξ W β2 ρ (s1(ξ, ρ)) + νAβ1ξ W β2 ρ (s2(ξ, ρ)). If s(ξ, ρ) can be written as s(ξ, ρ) = p(ξ)q(ρ) for some continuous functions p and q, then Aβ1ξ W β2 ρ (s(ξ, ρ)) = Aβ1ξ (p(ξ))W β2 ρ (q(ρ)). In fact Aβ1ξ W β2 ρ (s(ξ, ρ)) = Aβ1ξ W β2 ρ (p(ξ)q(ρ)) = ψ κ2 ∞∫ 0 ∞∫ 0 e − ( ψ ξβ1 β1 + ρβ2 κβ2 ) p(ξ)q(ρ)ξβ1−1ρβ2−1dξdρ = ψ∞∫ 0 e −ψ ξβ1 β1 p(ξ)ξβ1−1dξ  1 κ2 ∞∫ 0 e − ρβ2 κβ2 q(ρ)ρβ2−1dρ  = Aβ1ξ (p(ξ))W β2 ρ (q(ρ)). M. Al-Momani et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6099 5 of 15 3.1. CA-SW for some basic functions (i) Aβ1ξ W β2 ρ [c] = AξWρ[c] = c κ , c ∈ R, (ii) Aβ1ξ W β2 ρ [( ξβ1 β1 )λ( ρβ2 β2 )ν] = AξWρ[ξ λρν ] = κν−1 ψλ Γ(λ+ 1)Γ(ν + 1), Re(ψ) > 0 and Re(λ) > −1, (iii) Aβ1ξ W β2 ρ [ e λ ξβ1 β1 +ν ρβ2 β2 ] = AξWρ[e λξ+νρ] = ψ κ (ψ − λ) (1− νκ) ,Re(ψ) > Re(λ). 3.2. Existence condition for CA-SW Definition 4. Let 0 < β1, β2 ≤ 1. Then a function s(ξ, ρ) is said to be of conformable exponential orders λ and ν on 0 < ξ <∞ and 0 < ρ <∞. If there exist K,X, Y > 0 such that |s(ξ, ρ)| ≤ Ke λ ξβ1 β1 +ν ρβ2 β2 , for all ξ β1 β1 > X, ρβ2 β2 > Y. Theorem 3. Let 0 < β1, β2 ≤ 1 and s(ξ, ρ) be a continuous function on the region (0,∞)× (0,∞) of conformable exponential orders λ and ν. Then S(ψ,κ) = Aβ1ξ W β2 ρ [s(ξ, ρ)] exists for ψ,κ whenever Re (ψ) > λ and Re ( 1 κ ) > ν. Proof. We have |S(ψ,κ)| = ∣∣∣∣∣∣ ψκ2 ∞∫ 0 ∞∫ 0 e − ( ψ ξβ1 β1 + ρβ2 κβ2 ) s(ξ, ρ)ξβ1−1ρβ2−1 dξdρ ∣∣∣∣∣∣ ≤ ψ κ2 ∞∫ 0 ∞∫ 0 e − ( ψ ξβ1 β1 + ρβ2 κβ2 ) |s(ξ, ρ)| ξβ1−1ρβ2−1dξdρ ≤ K ψ κ2 ∞∫ 0 ∞∫ 0 e − ( ψ ξβ1 β1 + ρβ2 κβ2 ) e λ ξβ1 β1 +ν ρβ2 β2 ξβ1−1ρβ2−1dξdρ M. Al-Momani et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6099 6 of 15 = K ψ∞∫ 0 e −(ψ−λ) ξ β1 β1 ξβ1−1dξ  1 κ2 ∞∫ 0 e −( 1 κ−ν) ρ β2 β2 ρβ2−1dρ  = Kψ κ (ψ − λ) (1− νκ) , where Re (ψ) > λ and Re ( 1 κ ) > ν. 3.3. Derivatives properties Now, we present some basic properties of the CA-SW Let S(ψ,κ) = Aβ1ξ W β2 ρ (s(ξ, ρ)) where s(ξ, ρ) is a continuous function on (0,∞) × (0,∞). Then (i) Aβ1ξ W β2 ρ ( ∂β1s(ξ, ρ) ∂ξβ1 ) = ψS(ψ,κ)− ψW β2 ρ (s(0, ρ)), (1) (ii) Aβ1ξ W β2 ρ ( ∂2β1s(ξ, ρ) ∂ξ2β1 ) = ψ2S(ψ,κ)− ψ2W β2 ρ (s(0, ρ))− ψW β2 ρ ( ∂β1s(0, ρ) ∂ξβ1 ), (2) (iii) Aβ1ξ W β2 ρ ( ∂β2s(ξ, ρ) ∂ρβ2 ) = 1 κ S(ψ,κ)− 1 κ2 Aβ1ξ (s(ξ, 0)), (3) (iv) Aβ1ξ W β2 ρ ( ∂2β2s(ξ, ρ) ∂ρ2β2 ) = 1 κ2 S(ψ,κ)− 1 κ3 Aβ1ξ (s(ξ, 0))− 1 κ2 Aβ1ξ ( ∂β2s(ξ, 0) ∂ρβ2 ). (4) Proof. Proof of Equation 1Aβ1ξ W β2 ρ ( ∂β1s(ξ,ρ) ∂ξβ1 ) = ψ κ2 ∞∫ 0 ∞∫ 0 e − ( ψ ξβ1 β1 + ρβ2 κβ2 ) ∂β1s(ξ,ρ) ∂ξβ1 ξβ1−1ρβ2−1dξdρ. By Theorem 1, we have ∂β1s(ξ,ρ) ∂ξβ1 = ξ1−β1 ∂s(ξ,ρ)∂ξ . So, Aβ1ξ W β2 ρ ( ∂β1s(ξ,ρ) ∂ξβ1 ) = ψ κ2 ∞∫ 0 e − ρβ2 κβ2 ρβ2−1 ∞∫ 0 e −ψ ξβ1 β1 ∂s(ξ,ρ) ∂ξ dξdρ. By integrating by parts, we get Aβ1ξ W β2 ρ ( ∂β1s(ξ,ρ) ∂ξβ1 ) = ψ κ2 ∞∫ 0 e − ρβ2 κβ2 ρβ2−1 ( −s(0, ρ) + ψ ∞∫ 0 e −ψ ξβ1 β1 s(ξ, ρ)ξβ1−1 dξ ) dρ M. Al-Momani et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6099 7 of 15 = − ψ κ2 ∞∫ 0 e − ρβ2 κβ2 s(0, ρ)ρβ2−1dρ+ ψ2 κ2 ∞∫ 0 ∞∫ 0 e − ( ψ ξβ1 β1 + ρβ2 κβ2 ) s(ξ, ρ)ξβ1−1ρβ2−1dξdρ = ψS(ψ,κ)− ψW β2 ρ (s(0, ρ)). The proof of Equations 2, 3 and 4 can be obtained in the same manner. In Table 1, we have the CA-SW of some basic functions. Table 1: Table of CA-SW s(ξ, ρ) Aβ1ξ W β2 ρ (s(ξ, ρ)) c c κ , Re(ψ) > 0( ξβ1 β1 )λ ( ρβ2 β2 )ν κν−1 ψλ Γ(λ+ 1)Γ(ν + 1), Re(ψ) > 0 and Re(λ) > −1 e λ ξβ1 β1 +ν ρβ2 β2 ψ κ(ψ−λ)(1−νκ) , Re(ψ) > Re(λ) e i ( λ ξβ1 β1 +ν ρβ2 β2 ) iψ κ(ψ−iλ)(i+νκ) , Im(λ) + Re(ψ) > 0 sin ( λ ξ β1 β1 + ν ρ β2 β2 ) ψ(λ+ψκν) κ(ψ2+λ2)(1+ν2κ2) , |Im(λ)| < Re(ψ) cos ( λ ξ β1 β1 + ν ρ β2 β2 ) ψ(ψ−κλν) κ(ψ2+λ2)(1+ν2κ2) , |Im(λ)| < Re(ψ) sinh ( λ ξ β1 β1 + ν ρ β2 β2 ) ψ(λ+ψκν) κ(ψ2−λ2)(1−ν2κ2) , Re(ψ) > Re(λ) and Re(ψ) + Re(λ) > 0 cosh ( λ ξ β1 β1 + ν ρ β2 β2 ) ψ(ψ+κλν) κ(ψ2−λ2)(1−ν2κ2) , Re(ψ) > Re(λ) and Re(ψ) + Re(λ) > 0 p(ξ)q(ρ) Aβ1ξ (p(ξ))W β2 ρ (q(ρ)) 4. Applications This section applies CA-SW to solving conformable partial differential equations. Example 1. Consider the conformable Klein-Gordon equation 3 ∂2β2s(ξ, ρ) ∂ρ2β2 + ∂2β1s(ξ, ρ) ∂ξ2β1 + s(ξ, ρ) = 0, where ξ, ρ > 0, (5) With ICs s(ξ, 0) = sin ( 2ξβ1 β1 ) , ∂ β2s(ξ,0) ∂ρβ2 = − sin ( 2ξβ1 β1 ) , and BCs s (0, ρ) = 0, ∂β1s(0,ρ) ∂ξβ1 = 2e − ρβ2 β2 . Solution 1. By applying the CA to the ICs and the CSW to the BCs, we get Aβ1ξ ( sin ( 2ξβ1 β1 )) = 2ψ ψ2+4 , Aβ1ξ ( − sin ( 2ξβ1 β1 )) = −2ψ ψ2+4 , W β2 ρ (0) = 0, W β2 ρ (2e−ρ) = 2 κ(1+κ) . Apply the CA-SW to Equation 5, we get 3 κ2 S − 6ψ κ3 (ψ2 + 4) + 6ψ κ2 (ψ2 + 4) + ψ2S − 2ψ κ (1 + κ) + S = 0. M. Al-Momani et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6099 8 of 15 So, S(ψ,κ) = 6ψ κ3(ψ2+4) − 6ψ κ2(ψ2+4) + 2ψ κ(1+κ) 3 κ2 + ψ2 + 1 . By simplify, S(ψ,κ) = 2ψ κ (1 + κ) (ψ2 + 4) . So, s(ξ, ρ) = ( Aβ1ξ )−1 ( W β2 ρ )−1 ( 2ψ κ (1 + κ) (ψ2 + 4) ) = e − ρβ2 β2 sin ( 2ξβ1 β1 ) . The following figures show the 3D representation of the solution at β1 = β2 = 0.5, 1. Figure 1: The 3D representation of the solution at β1 = β2 = 0.5, 1. Figure 1 presents the three-dimensional plot of the solution to the conformable Klein- Gordon equation at different fractional orders (β1 = β2 = 0.5 and β1 = β2 = 1). The plot clearly shows that at β1 = β2 = 1, the solution behaves according to the classical structure with sharp transitions. When the fractional orders are reduced to β1 = β2 = 0.5, M. Al-Momani et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6099 9 of 15 the surface becomes smoother and the amplitude variations are more gradual, reflecting the memory and nonlocal properties introduced by the fractional operators. The following two figures illustrate the 2D graph of the solution with respect to ξ and ρ at β1 = β2 = 0.6, 0.8, 1. Figure 2: The 2D graph of the solution with respect to ξ at β1 = β2 = 0.6, 0.8, 1. M. Al-Momani et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6099 10 of 15 Figure 3: The 2D graph of the solution with respect to ρ at β1 = β2 = 0.6, 0.8, 1. Figures 2 and 3 display the two-dimensional behavior of the solution with respect to ξ and ρ, respectively, for different fractional orders (β1 = β2 = 0.6, 0.8, 1). Figure 2 shows the solution as a function of ξ, where lower fractional orders lead to broader and less steep transitions. Similarly, Figure 3 shows the solution as a function of ρ, highlighting that fractional orders lower than 1 result in slower decay and wider spreading of the solution profile. Example 2. Consider the conformable telegraph equation ∂2β1s(ξ, ρ) ∂ξ2β1 + 3 ∂2β2s(ξ, ρ) ∂ρ2β2 + 2 ∂β2s(ξ, ρ) ∂ρβ2 = 9s(ξ, ρ), where ξ, ρ > 0, (6) With initial conditions (ICs) s(ξ, 0) = e −2 ξβ1 β1 , ∂ β2s(ξ,0) ∂ρβ2 = e −2 ξβ1 β1 , and boundary conditions (BCs) s (0, ρ) = e ρβ1 β1 , ∂β1s(0,ρ) ∂ξβ1 = −2e ρβ1 β1 . Solution 2. By applying the CA to the ICs and the CSW to the BCs, we get Aβ1ξ ( e −2 ξβ1 β1 ) = ψ ψ+2 , A β1 ξ ( e −2 ξβ1 β1 ) = ψ ψ+2 , W β2 ρ ( e ρβ1 β1 ) = 1 κ(1−κ) , W β2 ρ ( −2e ρβ1 β1 ) = −2 κ(1−κ) . Apply the CA-SW to Equation 6, we get M. Al-Momani et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6099 11 of 15 ψ2S − ψ2 κ (1− κ) + 2ψ κ (1− κ) + 3 κ2 S − 3ψ κ3 (ψ + 2) − 3ψ κ2 (ψ + 2) + 2 κ S − 2ψ κ2 (ψ + 2) = 9S. So, S(ψ,κ) = ψ2−2ψ κ(1−κ) + 3ψ κ3(ψ+2) + 2ψ κ2(ψ+2) ψ2 + 3 κ2 + 2 κ − 9 = κ2(ψ−2)(ψ+2)+3(1−κ)+5κ(1−κ) κ3(ψ+2)(1−κ) ψ2κ2−9κ2+2κ+3 κ2 . By simplify, S(ψ,κ) = ψ κ (ψ + 2) (1− κ) . So, s(ξ, ρ) = ( Aβ1ξ )−1 ( W β2 ρ )−1 ( ψ κ (ψ + 2) (1− κ) ) = e −2 ξβ1 β1 + ρβ2 β2 . The following figures show the 3D representation of the solution at β1 = β2 = 0.4, 1. Figure 4: The 3D representation of the solution at β1 = β2 = 0.4, 1. M. Al-Momani et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6099 12 of 15 Figure 4 illustrates the three-dimensional plot of the solution to the conformable tele- graph equation at fractional orders β1 = β2 = 0.4 and β1 = β2 = 1. The figure demon- strates that for β1 = β2 = 1, the surface is steeper and more sharply defined, corresponding to the classical case. When β1 = β2 = 0.4, the surface becomes noticeably smoother and the solution exhibits a delayed and slower variation, indicating the influence of fractional derivatives. The following two figures illustrate the 2D graph of the solution with respect to ξ and ρ at β1 = β2 = 0.6, 0.8, 1. Figure 5: The 2D graph of the solution with respect to ξ at β1 = β2 = 0.6, 0.8, 1. M. Al-Momani et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6099 13 of 15 Figure 6: The 2D graph of the solution with respect to ρ at β1 = β2 = 0.6, 0.8, 1. Figures 5 and 6 depict two-dimensional profiles of the solution of the telegraph equation with respect to ξ and ρ, respectively, for fractional orders (β1 = β2 = 0.6, 0.8, 1). Figure 5 shows that as ξ increases, the solution’s behavior changes more gradually for lower frac- tional orders. Figure 6 reveals a similar effect along the ρ direction, where lower fractional orders result in a slower and smoother evolution of the solution, confirming the theoretical expectations based on the properties of conformable derivatives. 5. Conclusion This study introduced the CA-SW and examined its application to conformable frac- tional partial differential equations. We demonstrated its effectiveness in solving these equations, highlighting its potential as a useful mathematical tool. As a newly developed approach, CA-SW presents several open problems and opportunities for further research. 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