EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 1, Article Number 5807 ISSN 1307-5543 – ejpam.com Published by New York Business Global The Double ARA-Sawi Transform Raed R. Abu Awwad1, Monther Al-Momani2, Ali Jaradat2, Baha’ Abughazaleh3,∗, Ahmad Al-Natoor3 1 Department of Mathematics, University of Petra, Amman, Jordan 2 Department of Mathematics, Amman Arab University, Amman, Jordan 3 Department of Mathematics, Isra University, Amman, Jordan Abstract. This study introduces a novel integral transform derived by integrating the ARA and Sawi transforms. The paper explores the foundational properties and establishes the existence of this new transform. It presents advanced results for partial differential equations in higher dimensions and extends the double convolution theorem to two dimensions. These developments are applied to solve specific types of differential equations, demonstrating practical applications in physics and related scientific fields. 2020 Mathematics Subject Classifications: 44A05 Key Words and Phrases: The ARA transform, The Sawi transform, The double integral trans- form, The ARA-Sawi transform. 1. Introduction Implementing Integral transform which is one of the powerful mathematical techniques, which transforms a function to another domain. By applying the inverse of the integral transform, we return to the original space after transforming the function. Such transforms play a crucial role in engineering (dealing with signals), economics (input-output relationships), physics (quantum mechanics), and chemistry (reaction ki- netics), where they are valuable in elucidating complex real systems. As a result, mathe- maticians are constantly coming up with new techniques to solve an ever-growing class of differential equations. Among the innovative integral transforms emerging in recent years are the ARA and Sawi transforms. The ARA transform, introduced in 2020 by [9], and the Sawi transform, ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i1.5807 Email addresses: rabuawwad@uop.edu.jo (R. Abu Awwad) montheralmomani72@gmail.com (M. Al-Momani), a.jaradat@aau.edu.jo (A. Jaradat), baha.abughazaleh@iu.edu.jo (B. Abughazaleh), ahmad.alnatoor@iu.edu.jo (A. Al-Natoor) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) R. Abu Awwad et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5807 2 of 16 introduced in 2021 by [7], have gained attention for their unique properties and applications in various fields. Additionally, there exist several double transforms designed to handle multi-variable differential equations. In the broad spectrum of double transforms, we encounter new methods to help solve differential equations in higher dimensions. Examples of such double transforms include the Double Laplace transform [2], the Double Laplace ARA Transform [10], Double Laplace-Sawi Transform [3], the Double Laplace-Shehu transform [5], the Double Sawi transform [6], and the Double Mellin-ARA Transform [1]. In the present work, we propose a new double transform called the Double ARA-Sawi Transform (DA-SWT) aimed at globalizing differential equation analysis. We explore its fundamental properties, characterize the necessary conditions for its existence, and demonstrate its power in convolution theory and derivative operations. By applying this novel transform method, we present new strategies for dealing with partial differential equations and integral equations. The innovation in this work lies in the combination of the ARA and Sawi transforms, creating a new approach that combines the strengths of both. This combination enhances the simplicity and applicability of addressing complex mathematical problems. 2. The ARA and Sawi transforms In this section, we provide an overview and highlight key properties of the single transforms, namely the ARA and Sawi transforms. 2.1. The ARA transform Definition 1. The ARA transform of order k of a continuous function s(ν) on the interval (0,∞) is expressed as follows: Ak(s(ν))(ρ) = S(k, ρ) = ρ ∞∫ 0 νk−1e−ρνs(ν)dν, ρ > 0, for k = 1, 2, 3, .... In particular, if k = 1, the ARA transform of order 1 is expressed as A1(s(ν))(ρ) = S(ρ) = ρ ∞∫ 0 e−ρνs(ν) dν, ρ > 0. In the rest of the study, we denote A1(s(ν))(ρ) by A(s(ν))(ρ). Some basic properties of the ARA transform are now given. Let S(ρ) = A(s(ν)), then for nonzero constants γ and δ, we have A(γs1(ν) + δs2(ν)) = γA(s1(ν)) + δA(s2(ν)), (1) where s1(ν) and s2(ν) are continuous functions on (0,∞). R. Abu Awwad et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5807 3 of 16 A(νγ) = Γ(γ + 1) ργ , (2) A(eγν) = ρ ρ− γ , γ ∈ R, (3) A(s′(ν)) = ρS(ρ)− ρs(0), (4) A(s′′(ν)) = ρ2S(ρ)− ρ2s(0)− ρs′(0). (5) 2.2. The Sawi transform Definition 2. The Sawi transform of a continuous function r(σ) on (0,∞) expressed as follows R(ϖ) = W (r(σ)) = 1 ϖ2 ∞∫ 0 e− σ ϖ r(σ)dσ, ϖ > 0. Let us now explore the core properties that define the Sawi transform. Suppose that R1(ϖ) = W (r1(σ)) and R2(ϖ) = W (r2(σ)),with γ and δ as nonzero real numbers, the following properties hold W (γr1(σ) + δr2(σ)) = γW (r1(σ)) + δW (r2(σ)), (6) W (σγ) = Γ(γ + 1)ϖγ−1, (7) W (eδσ) = 1 ϖ (1− δϖ) , (8) W (r′(σ)) = 1 ϖ R(ϖ)− 1 ϖ2 r(0), (9) W (r′′(σ)) = 1 ϖ2 R(ϖ)− 1 ϖ3 r(0)− 1 ϖ2 r′(0). (10) 3. Double ARA-Sawi transform This section announces the Double ARA-Sawi Transformation (DA-SWT). We start by stating the basic properties of the DA-SWT, such as linearity. Then we state a new result regarding the partial derivatives and another new result regarding the convolution R. Abu Awwad et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5807 4 of 16 theorem. We also state how we use these results to compute the DA-SWT of some basic functions. The definition of the DA-SWT is: G(λ,ϖ) = AρWσ(g(ρ, σ)) = λ ϖ2 ∞∫ 0 ∞∫ 0 e−λρ− σ ϖ g(ρ, σ) dρdσ, (11) where g(ρ, σ) is a continuous function on (0,∞)× (0,∞). Clearly, AρWσ(g(ρ, σ)) is linear transformation. In fact, for nonzero constants γ and δ, we have AρWσ(γg1(ρ, σ)+δg2(ρ, σ)) = λ ϖ2 ∞∫ 0 ∞∫ 0 e−λρ− σ ϖ (γg1(ρ, σ) + δg2(ρ, σ)) dρdσ = γ λ ϖ2 ∞∫ 0 ∞∫ 0 e−λρ− σ ϖ g1(ρ, σ) dρdσ + δ λ ϖ2 ∞∫ 0 ∞∫ 0 e−λρ− σ ϖ g2(ρ, σ) dρdσ = γAρWσ(g1(ρ, σ)) + δAρWσ(g2(ρ, σ)). If g(ρ, σ) can be written as g(ρ, σ) = s(ρ)r(σ) for some continuous functions s and r, then AρWσ(g(ρ, σ)) = A(s(ρ))W (r(σ)). In fact AρWσ(g(ρ, σ)) = AρWσ(s(ρ)r(σ)) = λ ϖ2 ∞∫ 0 ∞∫ 0 e−λρ− σ ϖ s(ρ)r(σ)dρdσ = λ ∞∫ 0 e−λρs(ρ)dρ  1 ϖ2 ∞∫ 0 e− σ ϖ r(σ)dσ  = A(s(ρ))W (r(σ)). 3.1. Existence condition for Double ARA-Sawi transform Definition 3. A function g(ρ, σ) is said to be of exponential orders γ and δ on 0 ≤ ρ < ∞ and 0 ≤ σ < ∞ if there exist K,X, Y > 0 such that |g(ρ, σ)| ≤ Keγρ+δσ, for all ρ > X, σ > Y. Theorem 1. Let g(ρ, σ) be a continuous function on the region (0,∞) × (0,∞) of ex- ponential orders γ and δ. Then G(λ,ϖ) exists for λ,ϖ and γ whenever Re (λ) > γ and Re ( 1 ϖ ) > δ. R. Abu Awwad et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5807 5 of 16 Proof. We have |G(λ,ϖ)| = ∣∣∣∣∣∣ λ ϖ2 ∞∫ 0 ∞∫ 0 e−λρ− σ ϖ g(ρ, σ) dρdσ ∣∣∣∣∣∣ ≤ λ ϖ2 ∞∫ 0 ∞∫ 0 e−λρ− σ ϖ |g(ρ, σ)| dρdσ ≤ K λ ϖ2 ∞∫ 0 ∞∫ 0 e−λρ− σ ϖ eγρ+δσdρdσ = K ∞∫ 0 ∞∫ 0 ( λe−(λ−γ)ρ )( 1 ϖ2 e−( 1 ϖ −δ)σ ) dρdσ = K λ ∞∫ 0 e−(λ−γ)ρdρ  1 ϖ2 ∞∫ 0 e−( 1 ϖ −δ)σdσ  = Kλ ϖ (λ− γ) (1− δϖ) , where Re (λ) > γ and Re ( 1 ϖ ) > δ. Double ARA-Sawi transform for some basic functions (i) AρWσ(1) = λ ϖ2 ∞∫ 0 ∞∫ 0 e−λρ− σ ϖ dρdσ = λ ∞∫ 0 e−λρdρ  1 ϖ2 ∞∫ 0 e− σ ϖ dσ  = 1× 1 ϖ = 1 ϖ , Re(λ) > 0. (ii) AρWσ(e γρ+δσ) = λ ϖ2 ∞∫ 0 ∞∫ 0 e−λρ− σ ϖ eγρ+δσdρdσ =  ∞ λ ∫ 0 eγρ−λρdρ  1 ϖ2 ∞∫ 0 eδσ− σ ϖ dσ  = λ λ− γ × 1 ϖ (1− δϖ) = λ ϖ (λ− γ) (1− δϖ) , Re(λ) > Re(γ). (iii) AρWσ(ρ γσδ) = λ ϖ2 ∞∫ 0 ∞∫ 0 e−λρ− σ ϖ ργσδdρdσ = λ ∞∫ 0 ργe−λρdρ  1 ϖ2 ∞∫ 0 σδe− σ ϖ dσ  R. Abu Awwad et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5807 6 of 16 = Γ(γ + 1) λγ × Γ(δ + 1)ϖδ−1 = ϖδ−1 λγ Γ(γ + 1)Γ(δ + 1), Re(λ) > 0 and Re(γ) > −1. 3.2. Derivatives properties Now, we present some basic properties of the DA-SWT Let G(λ,ϖ) = AρWσ(g(ρ, σ)) where g(ρ, σ) is a continuous function on (0,∞)×(0,∞). Then (i) AρWσ ( ∂g(ρ, σ) ∂ρ ) = λG(λ,ϖ)− λW (g(0, σ)), (12) (ii) AρWσ ( ∂2g(ρ, σ) ∂ρ2 ) = λ2G(λ,ϖ)− λ2W (g(0, σ))− λW (gρ(0, σ)), (iii) AρWσ ( ∂g(ρ, σ) ∂σ ) = 1 ϖ G(λ,ϖ)− 1 ϖ2 A(g(ρ, 0)), (13) (iv) AρWσ ( ∂2g(ρ, σ) ∂σ2 ) = 1 ϖ2 G(λ,ϖ)− 1 ϖ3 A(g(ρ, 0))− 1 ϖ2 A(gσ(ρ, 0)), (14) (v) AρWσ ( ∂2g(ρ, σ) ∂ρ∂σ ) = λ ϖ G(λ,ϖ)− λ ϖ2 A(g(ρ, 0))− λ ϖ W (g(0, σ))+ λ ϖ2 g(0, 0). (15) Proof. (1) AρWσ ( ∂g(ρ,σ) ∂ρ ) = λ ϖ2 ∞∫ 0 ∞∫ 0 e−λρ− σ ϖ ∂g(ρ,σ) ∂ρ dρdσ = λ ϖ2 ∞∫ 0 e− σ ϖ ∞∫ 0 e−λρ ∂g(ρ,σ) ∂ρ dρdσ. By integrating by parts, we get AρWσ ( ∂g(ρ,σ) ∂ρ ) = λ ϖ2 ∞∫ 0 e− σ ϖ ( −g(0, σ) + λ ∞∫ 0 e−λρg(ρ, σ) dρ ) dσ = − λ ϖ2 ∞∫ 0 e− σ ϖ g(0, σ)dσ + λ2 ϖ2 ∞∫ 0 ∞∫ 0 e−λρ− σ ϖ g(ρ, σ) dρdσ R. Abu Awwad et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5807 7 of 16 = λG(λ,ϖ)− λW (g(0, σ)). (2) AρWσ ( ∂2g(ρ,σ) ∂ρ2 ) = λ ϖ2 ∞∫ 0 ∞∫ 0 e−λρ− σ ϖ ∂2g(ρ,σ) ∂ρ2 dρdσ = λ ϖ2 ∞∫ 0 e− σ ϖ ∞∫ 0 e−λρ ∂2g(ρ,σ) ∂ρ2 dρdσ. By integrating by parts, we get AρWσ ( ∂2g(ρ,σ) ∂ρ2 ) = λ ϖ2 ∞∫ 0 e− σ ϖ ( −gρ(0, σ)− λg(0, σ) + λ2 ∞∫ 0 e−λρg(ρ, σ)dρ ) dσ = − λ ϖ2 ∞∫ 0 e− σ ϖ gρ(0, σ)dσ − λ2 ϖ2 ∞∫ 0 e− σ ϖ g(0, σ)dσ + λ3 ϖ2 ∞∫ 0 ∞∫ 0 e−λρ− σ ϖ g(ρ, σ)dρdσ = λ2G(λ,ϖ)− λ2W (g(0, σ))− λW (gρ(0, σ)). (3) AρWσ ( ∂g(ρ,σ) ∂σ ) = λ ϖ2 ∞∫ 0 ∞∫ 0 e−λρ− σ ϖ ∂g(ρ,σ) ∂σ dρdσ = λ ϖ2 ∞∫ 0 e−λρ ∞∫ 0 e− σ ϖ ∂g(ρ,σ) ∂σ dσdρ. By integrating by parts, we get AρWσ ( ∂g(ρ,σ) ∂σ ) = λ ϖ2 ∞∫ 0 e−λρ ( −g(ρ, 0) + 1 ϖ ∞∫ 0 e− σ ϖ g(ρ, σ)dσ ) dρ = − λ ϖ2 ∞∫ 0 e−λρg(ρ, 0)dρ+ λ ϖ3 ∞∫ 0 ∞∫ 0 e−λρ− σ ϖ g(ρ, σ) dσdρ = 1 ϖG(λ,ϖ)− 1 ϖ2A(g(ρ, 0)). (4) AρWσ ( ∂2g(ρ,σ) ∂σ2 ) = λ ϖ2 ∞∫ 0 ∞∫ 0 e−λρ− σ ϖ ∂2g(ρ,σ) ∂σ2 dρdσ = λ ϖ2 ∞∫ 0 e−λρ ∞∫ 0 e− σ ϖ ∂2g(ρ,σ) ∂σ2 dσdρ. By integrating by parts, we get AρWσ ( ∂2g(ρ,σ) ∂σ2 ) = λ ϖ2 ∞∫ 0 e−λρ ( −gσ(ρ, 0)− 1 ϖg(ρ, 0) + 1 ϖ2 ∞∫ 0 e− σ ϖ g(ρ, σ)dσ ) dρ = − λ ϖ2 ∞∫ 0 e−λρgσ(ρ, 0)dρ− λ ϖ3 ∞∫ 0 e−λρg(ρ, 0)dρ+ λ ϖ4 ∞∫ 0 ∞∫ 0 e−λρ− σ ϖ g(ρ, σ)dσdρ = 1 ϖ2G(λ,ϖ)− 1 ϖ3A(g(ρ, 0))− 1 ϖ2A(gσ(ρ, 0)). (5) AρWσ ( ∂2g(ρ,σ) ∂ρ∂σ ) = λ ϖ2 ∞∫ 0 ∞∫ 0 e−λρ− σ ϖ ∂2g(ρ,σ) ∂ρ∂σ dρdσ = λ ϖ2 ∞∫ 0 e− σ ϖ ∞∫ 0 e−λρ ∂2g(ρ,σ) ∂ρ∂σ dρdσ By integrating by parts, we get AρWσ ( ∂2g(ρ,σ) ∂ρ∂σ ) = λ ϖ2 ∞∫ 0 e− σ ϖ ( −gσ(0, σ) + λ ∞∫ 0 e−λρgσ(ρ, σ) dρ ) dσ = − λ ϖ2 ∞∫ 0 e− σ ϖ gσ(0, σ)dσ + λ2 ϖ2 ∞∫ 0 ∞∫ 0 e−λρ− σ ϖ gσ(ρ, σ)dρdσ R. Abu Awwad et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5807 8 of 16 = −λW (gσ(0, σ)) + λAρWσ (gσ(ρ, σ)) Using Equations 9 and 13 we get AρWσ ( ∂2g(ρ,σ) ∂ρ∂σ ) = λ ϖG(λ,ϖ)− λ ϖ2A(g(ρ, 0))− λ ϖW (g(0, σ)) + λ ϖ2 g(0, 0). 3.3. Convolution Theorem of Double ARA-Sawi transform Let H(ρ, σ) represent the Heaviside unit step function, which is defined as follows: H(ρ− γ, σ − δ) = { 1, ρ > γ and σ > δ 0, otherwise Then we have the following lemma Lemma 1. Let g(ρ, σ) be a continuous function on (0,∞)×(0,∞) and H(ρ, σ) be the Heav- iside unit step function. Then AρWσ(g(ρ−γ, σ−δ)H(ρ−γ, σ−δ)) = e−λγ− δ ϖAρWσ(g(ρ, σ). Proof. We have AρWσ(g(ρ− γ, σ − δ)H(ρ− γ, σ − δ)) (16) = λ ϖ2 ∞∫ 0 ∞∫ 0 e−λρ− σ ϖ g(ρ− γ, σ − δ)H(ρ− γ, σ − δ)dρdσ = λ ϖ2 ∞∫ γ ∞∫ δ e−λρ− σ ϖ g(ρ− γ, σ − δ)dρdσ. Now, by making the substitution z = ρ− γ and w = σ − δ, equation 3.3 becomes: AρWσ(g(ρ− γ, σ − δ)H(ρ− γ, σ − δ)) = λ ϖ2 ∞∫ 0 ∞∫ 0 e−λ(z+γ)− (w+δ) ϖ g(z, w)dzdw = e−λγ− δ ϖAρWσ(g(ρ, σ)). Definition 4. Let g(ρ, σ) and k(ρ, σ) be continuous functions. We define the convolution in the DA-SWT as (g ∗ ∗k)(ρ, σ) = ρ∫ 0 σ∫ 0 g(ρ− γ, σ − δ)k(γ, δ)dγdδ. R. Abu Awwad et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5807 9 of 16 In the following theorem, we compute DA-SWT of the convolution of two functions Theorem 2. Let G(λ,ϖ) = AρWσ(g(ρ, σ)) and K(λ,ϖ) = AρWσ(k(ρ, σ)). Then AρWσ((g ∗ ∗k)(ρ, σ)) = ϖ2 λ G(λ,ϖ)K(λ,ϖ). Proof. AρWσ((g∗∗k)(ρ, σ)) = λ ϖ2 ∞∫ 0 ∞∫ 0 e−λρ− σ ϖ (g ∗ ∗k)(ρ, σ)dρdσ = λ ϖ2 ∞∫ 0 ∞∫ 0 e−λρ− σ ϖ  ρ∫ 0 σ∫ 0 g(ρ− γ, σ − δ)k(γ, δ)dγdδ  dρdσ. (17) Using the Heaviside unit step function, We can write equation 17 as AρWσ((g∗∗g)(ρ, σ)) = λ ϖ2 ∞∫ 0 ∞∫ 0 e−λρ− σ ϖ ∞∫ 0 ∞∫ 0 g(ρ− γ, σ − δ)H(ρ− γ, σ − δ)k(γ, δ))dγdδ  dρdσ = ∞∫ 0 ∞∫ 0 k(γ, δ)  λ ϖ2 ∞∫ 0 ∞∫ 0 e−λρ− σ ϖ g(ρ− γ, σ − δ)H(ρ− γ, σ − δ)dρdσ  dγdδ. So by Lemma 1, We have AρWσ((g ∗ ∗k)(ρ, σ)) = G(λ,ϖ) ∞∫ 0 ∞∫ 0 k(γ, δ)e−λγ− δ ϖ dγdδ = ϖ2 λ G(λ,ϖ)K(λ,ϖ). In Table 1, we have the DAHT of some basic functions. R. Abu Awwad et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5807 10 of 16 Table 1: Table of DAHT g(ρ, σ) AρWσ(g(ρ, σ)) 1 1 ϖ , Re(λ) > 0 ργσδ ϖδ−1 λγ Γ(γ + 1)Γ(δ + 1), Re(λ) > 0 and Re(γ) > −1 eγρ+δσ λ ϖ(λ−γ)(1−δϖ) , Re(λ) > Re(γ) ei(γρ+δσ) iλ ϖ(λ−iγ)(i+δϖ) , Im(γ) + Re(λ) > 0 sin (γρ+ δσ) λ(γ+λϖδ) ϖ(λ2+γ2)(1+δ2ϖ2) , |Im(γ)| < Re(λ) cos (γρ+ δσ) λ(λ−ϖγδ) ϖ(λ2+γ2)(1+δ2ϖ2) , |Im(γ)| < Re(λ) sinh (γρ+ δσ) λ(γ+λϖδ) ϖ(λ2−γ2)(1−δ2ϖ2) , Re(λ) > Re(γ) and Re(λ) + Re(γ) > 0 cosh (γρ+ δσ) λ(λ+ϖγδ) ϖ(λ2−γ2)(1−δ2ϖ2) , Re(λ) > Re(γ) and Re(λ) + Re(γ) > 0 s(ρ)r(σ) A(s(ρ))W (r(σ)) g(ρ− γ, σ − δ)H(ρ− γ, σ − δ) e−λγ− δ ϖAρWσ(g(ρ, σ) (g ∗ ∗f)(ρ, σ) ϖ2 λ AρWσ(g(ρ, σ))AρWσ(f(ρ, σ)) J0 ( c √ ρσ ) 4λ ϖ(4λ+c2ϖ) , Re ( λ+ c2ϖ 4 ) > 0 4. Applications In this section, we use the DA-SWT for solving PDEs and Integro PDEs Example 1. Consider the heat equation gσ − gρρ = 2g + 6σ − 3, where ρ, σ > 0, With ICs g(ρ, 0) = sin ρ, and BCs g (0, σ) = −3σ, gρ (0, σ) = eσ. Solution 1. By applying the single ARA transform to the ICs and the single Sawi trans- form to the BCs, we get A (g(ρ, 0)) = λ 1+λ2 , W (g (0, σ)) = −3, W (gρ (0, σ)) = 1 ϖ(1−ϖ) . Apply the DA-SWT to Equation 1, we get 1 ϖ G(λ,ϖ)− 1 ϖ2 A(g(ρ, 0))− λ2G(λ,ϖ) +λ2W (g(0, σ)) + λW (gρ(0, σ)) = 2G+ 6− 3 ϖ . So, 1− λ2ϖ − 2ϖ ϖ ×G(λ,ϖ) = R. Abu Awwad et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5807 11 of 16 1 ϖ2 × λ 1 + λ2 − λ2 ×−2− λ× 1 ϖ (1−ϖ) + 6− 3 ϖ . By simplifying, we get, G(λ,ϖ) = λ ϖ (1 + λ2) (1−ϖ) − 3. Therefore, g(ρ, σ) = A−1 ρ W−1 σ ( λ ϖ (1 + λ2) (1−ϖ) − 3 ) = eσ sin ρ− 3σ. Its graph is Figure 1: The solution of Example 1 Example 2. Consider the telegraph equation gρρ + gρ − 2gσσ = 2g(ρ, σ), where ρ, σ > 0, With ICs g(ρ, 0) = eρ + 1, gσ(ρ, 0) = 0, and BCs g (0, σ) = 1 + cosσ, gρ (0, σ) = 1. R. Abu Awwad et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5807 12 of 16 Solution 2. By applying the single ARA transform and the single Sawi transform to the ICs, we get A (g(ρ, 0)) = λ λ−1 + 1, A (gσ(ρ, 0)) = 0, W (g (0, σ)) = 1 ϖ + 1 ϖ(1+ϖ2) , W (gρ (0, σ)) = 1 ϖ . Apply the DA-SWT to Equation 2, we get λ2G(λ,ϖ)− λ2W (g(0, σ))− λW (gρ(0, σ)) + λG(λ,ϖ) −λW (g(0, σ))− 2 1 ϖ2 G(λ,ϖ) + 2 1 ϖ3 A(g(ρ, 0)) +2 1 ϖ2 A(gσ(ρ, 0)) = 2G. So, λ2ϖ2 + λϖ2 − 2ϖ2 − 2 ϖ2 ×G(λ,ϖ) = λ2 × ( 1 ϖ + 1 ϖ (1 +ϖ2) ) + λ× 1 ϖ +λ× ( 1 ϖ + 1 ϖ (1 +ϖ2) ) − 2 ϖ3 × ( λ λ− 1 + 1 ) . By simplifying, we get, G(λ,ϖ) = λ ϖ (λ− 1) + 1 1 +ϖ2 . Therefore, g(ρ, σ) = A−1 ρ W−1 σ ( λ ϖ (λ− 1) + 1 1 +ϖ2 ) = eρ + cosσ. Its graph is R. Abu Awwad et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5807 13 of 16 Figure 2: The solution of Example 2 Example 3. Consider the equation of Volterra Integro PDE. gρ + gσ − coshσ − ρ sinhσ − ρ2 sinhσ = 2 ρ∫ 0 σ∫ 0 g(γ, δ))dγdδ, where ρ, σ > 0, (18) With ICs g(ρ, 0) = ρ, g(0, σ) = 0. Solution 3. By applying the single ARA transform and the single Sawi transform to the ICs, we get A (g(ρ, 0)) = 1 λ , W (g (0, σ)) = 0. By Definition 4 and Theorem 2, we have ρ∫ 0 σ∫ 0 g(γ, δ))dγdδ = (1 ∗ ∗g) (ρ, σ) . (19) Apply the DA-SWT to Equation 18, we get λG(λ,ϖ)− λW (g(0, σ)) + 1 ϖ G(λ,ϖ)− 1 ϖ2 A(g(ρ, 0)) R. Abu Awwad et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5807 14 of 16 − 1 ϖ (1−ϖ2) − 1 λ (1−ϖ2) − 2 λ2 (1−ϖ2) = 2ϖ λ G(λ,ϖ). So, λ2ϖ + λ− 2ϖ2 λϖ ×G(λ,ϖ) = 1 ϖ2 × 1 λ + 1 ϖ (1−ϖ2) + 1 λ (1−ϖ2) + 2 λ2 (1−ϖ2) . By simplifying, we get, G(λ,ϖ) = 1 λϖ (1−ϖ2) . Therefore, g(ρ, σ) = A−1 ρ W−1 σ ( 1 λϖ (1−ϖ2) ) = ρ coshσ. Its graph is Figure 3: The solution of Example 3 5. Conclusion In this paper, we introduced the Double ARA-Sawi Transform (DA-SWT) and thor- oughly explored its foundational properties, rigorously characterizing the necessary con- R. Abu Awwad et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5807 15 of 16 ditions for its existence. Through this investigation, we demonstrated the transformative potential of these properties in convolution theory and derivative operations. By establishing a robust theoretical framework and validating its applicability, we high- lighted the practical advantages of the DA-SWT in problem-solving. Where relevant, we connected earlier numerical procedures that benefited from our previous research, show- casing how the DA-SWT builds upon and enhances existing methodologies. We foresee significant potential for the DA-SWT in addressing fractional and con- formable partial differential equations (PDEs) and integro-PDEs with variable coefficients. This innovative transform method paves the way for future advancements in solving com- plex mathematical and scientific problems. Additional results related to conformable PDEs and Integro PDEs are available in references [4, 8]. 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