EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 1, Article Number 5619 ISSN 1307-5543 – ejpam.com Published by New York Business Global Double Laplace-Sawi Transform Monther Al-Momani1, Ali Jaradat1, Baha’ Abughazaleh2,∗ 1 Department of Mathematics, Amman Arab University, Amman, Jordan 2 Department of Mathematics, Isra University, Amman, Jordan Abstract. The primary objective of this study is to develop a new integral transform by combining the Laplace and Sawi transforms, and to investigate its key properties, existence, and the inver- sion theorem. Furthermore, we introduce new results related to partial differential equations in higher dimensions and extend the double convolution theorem to two dimensions. Using these new properties and theorems, we solve special type differential equations with some real applications in physics and related sciences. 2020 Mathematics Subject Classifications: 44A05, 44A10 Key Words and Phrases: Laplace transform, Sawi transform, Double integral transform, Laplace-Sawi transform 1. Introduction Integral transforms are powerful mathematical tools that convert functions into new domains. After transforming the function can be returned to its original space by applying the inverse of the integral transform. By applying an integral transform, we generate a new function G(δ) through the integration of the product of g (η) and K(η, δ) across the interval [a, b] represented by: b∫ a g(η)K(η, δ)dη They are pivotal in engineering, economics, physics, and chemistry, serving as essential tools for understanding complex real-world phenomena. Thus, mathematicians relentlessly innovate and develop new techniques to tackle ever-broader classes of differential equa- tions, and one of the most celebrated integral transforms is the Laplace transform, first introduced in 1780. Among the innovative integral transforms emerging in recent years is the Sawi transform introduced in 2021 by [1]. These transforms offer powerful new ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i1.5619 Email addresses: monther ok@yahoo.com M. Al-Momani), a.jaradat@aau.edu.jo (A. Jaradat), baha.abughazaleh@iu.edu.jo (B. Abughazaleh) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) M. Al-Momani, A. Jaradat , B. Abughazaleh / Eur. J. Pure Appl. Math, 18 (1) (2025), 5619 2 of 19 tools for tackling both ordinary and fractional differential equations, for more information about the Sawi transform, refer to [2], and for more details on other single transforms, one may refer to [3–5]. Additionally, some Double transforms exist to handle many-variable differential equa- tions. In the wide range of double transforms, we notice fresh methods to help solve differential equations in more than one dimension. The double Laplace transform [6], the Double Laplace-Shehu transform [8], the Double Laplace ARA Transform [7], the Double Sawi transform [9] and Double Mellin-ARA Transform [10]. In the present work, we propose a double transform called the Double Laplace-Sawi Transform (DLSWT) aimed at globalizing differential equation analysis. We go down to its bedrock properties characterizing what is needed for it to exist and demonstrating their power in convolution theory and derivative operation. Applying this novel transform method, we identify new ways of dealing with partial differential equations and integral equations. The novelty of this work lies in the innovative combinations of the Laplace and Sawi transforms, creating a new approach that harnesses the strengths of both trans- forms. This combination enhances the simplicity and applicability in addressing complex mathematical problems. 2. Laplace and Sawi transforms In this section, we provide an overview and highlight key properties of the single transforms, namely the Laplace and Sawi transforms. 2.1. Laplace transform Definition 1. The Laplace transform of a continuous function p(η) on (0,∞) is defined as follows P (δ) = L(p(η)) = ∞∫ 0 e−δηp(η)dη, δ ∈ C. Some basic properties of the Laplace transform are now given. Let P (δ) = L(p(η)), then for nonzero constants u and v, we have L(up1(η) + vp2(η)) = uL(p1(η)) + vL(p2(η)), (1) where p1(η) and p2(η) are continuous functions on (0,∞). L(ηu) = Γ(u+ 1) δu+1 , (2) L(euη) = 1 δ − u , u ∈ R, (3) M. Al-Momani, A. Jaradat , B. Abughazaleh / Eur. J. Pure Appl. Math, 18 (1) (2025), 5619 3 of 19 L(p′(η)) = δP (δ)− p(0), (4) L(p′′(η)) = δ2P (δ)− δp(0)− p′(0). (5) 2.2. The Sawi transform Definition 2. The Sawi transform of a continuous function q(θ) on (0,∞) expressed as follows Q(ϵ) = W (q(θ)) = 1 ϵ2 ∞∫ 0 e− θ ϵ q(θ)dθ. Let us now explore the core properties that define the Sawi transform. Suppose that Q1(ϵ) = W (q1(θ)) and Q2(ϵ) = W (q2(θ)),with u and v as nonzero real numbers, the following properties hold W (uq1(θ) + vq2(θ)) = uW (q1(θ)) + vW (q2(θ)), (6) W (θu) = Γ(u+ 1)ϵu−1, (7) W (evθ) = 1 ϵ (1− vϵ) , (8) W (q′(θ)) = 1 ϵ Q(ϵ)− 1 ϵ2 q(0), (9) W (q′′(θ)) = 1 ϵ2 Q(ϵ)− 1 ϵ3 q(0)− 1 ϵ2 q′(0). (10) 3. Double Laplace-Sawi transform This section announces the Double Laplace-Sawi Transformation (DLSWT). We start by stating the basic properties of the DLSWT, such as linearity and inversion. Then we state a new result regarding the partial derivatives and another new result regarding the convolution theorem. We also state how we use these results to compute the DLSWT of some basic functions. The definition of the DLSWT is: G(δ, ϵ) = LηWθ(g(η, θ)) = 1 ϵ2 ∞∫ 0 ∞∫ 0 e−δη− θ ϵ g(η, θ) dηdθ, (11) where g(η, θ) is a continuous function on (0,∞)× (0,∞). M. Al-Momani, A. Jaradat , B. Abughazaleh / Eur. J. Pure Appl. Math, 18 (1) (2025), 5619 4 of 19 Clearly, LηWθ(g(η, θ)) is linear transformation. In fact, for nonzero constants u and v, we have LηWθ(ug1(η, θ)+vg2(η, θ)) = 1 ϵ2 ∞∫ 0 ∞∫ 0 e−δη− θ ϵ (ug1(η, θ) + vg2(η, θ)) dηdθ = u 1 ϵ2 ∞∫ 0 ∞∫ 0 e−δη− θ ϵ g1(η, θ) dηdθ + v 1 ϵ2 ∞∫ 0 ∞∫ 0 e−δη− θ ϵ g2(η, θ) dηdθ = uLηWθ(g1(η, θ)) + vLηWθ(g2(η, θ)). If g(η, θ) can be written as g(η, θ) = p(η)q(θ) for some continuous functions p and q, then LηWθ(g(η, θ)) = L(p(η))W (q(θ)). In fact LηWθ(g(η, θ)) = LηWθ(p(η)q(θ)) = 1 ϵ2 ∞∫ 0 ∞∫ 0 e−δη− θ ϵ p(η)q(θ)dηdθ = ∞∫ 0 e−δηp(η)dη  1 ϵ2 ∞∫ 0 e− θ ϵ q(θ)dθ  = L(p(η))W (q(θ)). 3.1. Double Laplace-Sawi transform for some basic functions (i) LηWθ(1) = 1 ϵ2 ∞∫ 0 ∞∫ 0 e−δη− θ ϵ dηdθ = ∞∫ 0 e−δηdη  1 ϵ2 ∞∫ 0 e− θ ϵ dθ  = 1 δ × 1 ϵ = 1 δϵ , Re(δ) > 0. (ii) LηWθ(η uθv) = 1 ϵ2 ∞∫ 0 ∞∫ 0 e−δη− θ ϵ ηuθvdηdθ = ∞∫ 0 ηue−δηdη  1 ϵ2 ∞∫ 0 θve− θ ϵ dθ  M. Al-Momani, A. Jaradat , B. Abughazaleh / Eur. J. Pure Appl. Math, 18 (1) (2025), 5619 5 of 19 = Γ(u+ 1) δu+1 × Γ(v + 1)ϵv−1 = ϵv−1 δu+1 Γ(u+ 1)Γ(v + 1), Re(δ) > 0 and Re(u) > −1. (iii) LηWθ(e uη+vθ) = 1 ϵ2 ∞∫ 0 ∞∫ 0 e−δη− θ ϵ euη+vθdηdθ = ∞∫ 0 euη−δηdη  1 ϵ2 ∞∫ 0 evθ− θ ϵ dθ  = 1 δ − u × 1 ϵ (1− vϵ) = 1 ϵ (δ − u) (1− vϵ) , Re(δ) > Re(u). 3.2. Existence condition for Double Laplace-Sawi transform Definition 3. A function g(η, θ) is said to be of exponential orders u and v on 0 ≤ η < ∞ and 0 ≤ θ < ∞. If there exist K,X, Y > 0 such that |g(η, θ)| ≤ Keuη+vθ, for all η > X, θ > Y. Theorem 1. Let g(η, θ) be a continuous function on the region [0,∞) × [0,∞) of ex- ponential orders u and v. Then G(δ, ϵ) exists for δ, ϵ and γ whenever Re (δ) > u and Re ( 1 ϵ ) > v. Proof. We have |G(δ, ϵ)| = ∣∣∣∣∣∣ 1ϵ2 ∞∫ 0 ∞∫ 0 e−δη− θ ϵ g(η, θ) dηdθ ∣∣∣∣∣∣ ≤ 1 ϵ2 ∞∫ 0 ∞∫ 0 e−δη− θ ϵ |g(η, θ)| dηdθ ≤ K 1 ϵ2 ∞∫ 0 ∞∫ 0 e−δη− θ ϵ euη+vθdηdθ = K ∞∫ 0 ∞∫ 0 ( e−(δ−u)η )( 1 ϵ2 e−( 1 ϵ −v)θ ) dηdθ = K ∞∫ 0 e−(δ−u)ηdη  1 ϵ2 ∞∫ 0 e−( 1 ϵ −v)θdθ  = K ϵ (δ − u) (1− vϵ) , where Re (δ) > u and Re ( 1 ϵ ) > v. M. Al-Momani, A. Jaradat , B. Abughazaleh / Eur. J. Pure Appl. Math, 18 (1) (2025), 5619 6 of 19 3.3. Derivatives properties Now, we present some basic properties of the DLSWT Let G(δ, ϵ) = LηWθ(g(η, θ)) where g(η, θ) is a continuous function on (0,∞)× (0,∞). Then (i) LηWθ ( ∂g(η, θ) ∂η ) = δG(δ, ϵ)−W (g(0, θ)), (12) (ii) LηWθ ( ∂2g(η, θ) ∂η2 ) = δ2G(δ, ϵ)− δW (g(0, θ))−W (gη(0, θ)), (iii) LηWθ ( ∂g(η, θ) ∂θ ) = 1 ϵ G(δ, ϵ)− 1 ϵ2 L(g(η, 0)), (13) (iv) LηWθ ( ∂2g(η, θ) ∂θ2 ) = 1 ϵ2 G(δ, ϵ)− 1 ϵ3 L(g(η, 0))− 1 ϵ2 L(gθ(η, 0)), (14) (v) LηWθ ( ∂2g(η, θ) ∂η∂θ ) = δ ϵ G(δ, ϵ)− δ ϵ2 L(g(η, 0))− 1 ϵ W (g(0, θ)) + 1 ϵ2 g(0, 0). (15) Proof. (1) LηWθ ( ∂g(η,θ) ∂η ) = 1 ϵ2 ∞∫ 0 ∞∫ 0 e−δη− θ ϵ ∂g(η,θ) ∂η dηdθ = 1 ϵ2 ∞∫ 0 e− θ ϵ ∞∫ 0 e−δη ∂g(η,θ) ∂η dηdθ. By integrating by parts, we get LηWθ ( ∂g(η,θ) ∂η ) = 1 ϵ2 ∞∫ 0 e− θ ϵ ( −g(0, θ) + δ ∞∫ 0 e−δηg(η, θ) dη ) dθ = − 1 ϵ2 ∞∫ 0 e− θ ϵ g(0, θ)dθ + δ ϵ2 ∞∫ 0 ∞∫ 0 e−δη− θ ϵ g(η, θ) dηdθ = δG(δ, ϵ)−W (g(0, θ)). (2) LηWθ ( ∂2g(η,θ) ∂η2 ) = 1 ϵ2 ∞∫ 0 ∞∫ 0 e−δη− θ ϵ ∂2g(η,θ) ∂η2 dηdθ = 1 ϵ2 ∞∫ 0 e− θ ϵ ∞∫ 0 e−δη ∂2g(η,θ) ∂η2 dηdθ. By integrating by parts, we get M. Al-Momani, A. Jaradat , B. Abughazaleh / Eur. J. Pure Appl. Math, 18 (1) (2025), 5619 7 of 19 LηWθ ( ∂2g(η,θ) ∂η2 ) = 1 ϵ2 ∞∫ 0 e− θ ϵ ( −gη(0, θ)− δg(0, θ) + δ2 ∞∫ 0 e−δηg(η, θ)dη ) dθ = − 1 ϵ2 ∞∫ 0 e− θ ϵ gη(0, θ)dθ − δ ϵ2 ∞∫ 0 e− θ ϵ g(0, θ)dθ + δ2 ϵ2 ∞∫ 0 ∞∫ 0 e−δη− θ ϵ g(η, θ)dηdθ = δ2G(δ, ϵ)− δW (g(0, θ))−W (gη(0, θ)). (3) LηWθ ( ∂g(η,θ) ∂θ ) = 1 ϵ2 ∞∫ 0 ∞∫ 0 e−δη− θ ϵ ∂g(η,θ) ∂θ dηdθ = 1 ϵ2 ∞∫ 0 e−δη ∞∫ 0 e− θ ϵ ∂g(η,θ) ∂θ dθdη. By integrating by parts, we get LηWθ ( ∂g(η,θ) ∂θ ) = 1 ϵ2 ∞∫ 0 e−δη ( −g(η, 0) + 1 ϵ ∞∫ 0 e− θ ϵ g(η, θ)dθ ) dη = − 1 ϵ2 ∞∫ 0 e−δηg(η, 0)dη + 1 ϵ3 ∞∫ 0 ∞∫ 0 e−δη− θ ϵ g(η, θ) dθdη = 1 ϵG(δ, ϵ)− 1 ϵ2 L(g(η, 0)). (4) LηWθ ( ∂2g(η,θ) ∂θ2 ) = 1 ϵ2 ∞∫ 0 ∞∫ 0 e−δη− θ ϵ ∂2g(η,θ) ∂θ2 dηdθ = 1 ϵ2 ∞∫ 0 e−δη ∞∫ 0 e− θ ϵ ∂2g(η,θ) ∂θ2 dθdη. By integrating by parts, we get LηWθ ( ∂2g(η,θ) ∂θ2 ) = 1 ϵ2 ∞∫ 0 e−δη ( −gθ(η, 0)− 1 ϵ g(η, 0) + 1 ϵ2 ∞∫ 0 e− θ ϵ g(η, θ)dθ ) dη = − 1 ϵ2 ∞∫ 0 e−δηgθ(η, 0)dη − 1 ϵ3 ∞∫ 0 e−δηg(η, 0)dη + 1 ϵ4 ∞∫ 0 ∞∫ 0 e−δη− θ ϵ g(η, θ)dθdη LηWθ ( ∂2g(η,θ) ∂θ2 ) = 1 ϵ2 G(δ, ϵ)− 1 ϵ3 L(g(η, 0))− 1 ϵ2 L(gθ(η, 0)). (5) LηWθ ( ∂2g(η,θ) ∂η∂θ ) = 1 ϵ2 ∞∫ 0 ∞∫ 0 e−δη− θ ϵ ∂2g(η,θ) ∂η∂θ dηdθ = 1 ϵ2 ∞∫ 0 e− θ ϵ ∞∫ 0 e−δη ∂2g(η,θ) ∂η∂θ dηdθ By integrating by parts, we get LηWθ ( ∂2g(η,θ) ∂η∂θ ) = 1 ϵ2 ∞∫ 0 e− θ ϵ ( −gθ(0, θ) + δ ∞∫ 0 e−δηgθ(η, θ) dη ) dθ = − 1 ϵ2 ∞∫ 0 e− θ ϵ gθ(0, θ)dθ + δ ϵ2 ∞∫ 0 ∞∫ 0 e−δη− θ ϵ gθ(η, θ)dηdθ = −W (gθ(0, θ)) + δLηWθ (gθ(η, θ)) Using Equations 9 and 13 we get = δ ϵG(δ, ϵ)− δ ϵ2 L(g(η, 0))− 1 ϵW (g(0, θ)) + 1 ϵ2 g(0, 0). M. Al-Momani, A. Jaradat , B. Abughazaleh / Eur. J. Pure Appl. Math, 18 (1) (2025), 5619 8 of 19 3.4. Convolution Theorem of Double Laplace-Sawi transform Let H(η, θ) represent the Heaviside unit step function, which is defined as follows: H(η − u, θ − v) = { 1, η > u and θ > v 0, otherwise Then we have the following lemma Lemma 1. Let g(η, θ) be a continuous function on (0,∞)×(0,∞) and H(η, θ) be the Heav- iside unit step function. Then LηWθ(g(η−u, θ−v)H(η−u, θ−v)) = e−δu− v ϵLηWθ(g(η, θ). Proof. We have LηWθ(g(η − u, θ − v)H(η − u, θ − v)) (16) = 1 ϵ2 ∞∫ 0 ∞∫ 0 e−δη− θ ϵ g(η − u, θ − v)H(η − u, θ − v)dηdθ = 1 ϵ2 ∞∫ u ∞∫ v e−δη− θ ϵ g(η − u, θ − v)dηdθ. Now, by making the substitution z = η − u and w = θ − v, equation 16 becomes: LηWθ(g(η − u, θ − v)H(η − u, θ − v)) = 1 ϵ2 ∞∫ 0 ∞∫ 0 e−δ(z+u)− (w+v) ϵ g(z, w)dzdw = e−δu− v ϵLηWθ(g(η, θ)). Definition 4. Let g(η, θ) and k(η, θ) be continuous functions. We define the convolution in the DLSWT as (g ∗ ∗k)(η, θ) = η∫ 0 θ∫ 0 g(η − u, θ − v)k(u, v)dudv. In the following theorem, we compute DLSWT of the convolution of two functions Theorem 2. Let G(δ, ϵ) = LηWθ(g(η, θ)) and K(δ, ϵ) = LηWθ(k(η, θ)). Then LηWθ((g ∗ ∗k)(η, θ)) = ϵ2G(δ, ϵ)K(δ, ϵ). M. Al-Momani, A. Jaradat , B. Abughazaleh / Eur. J. Pure Appl. Math, 18 (1) (2025), 5619 9 of 19 Proof. LηWθ((g∗∗k)(η, θ)) = 1 ϵ2 ∞∫ 0 ∞∫ 0 e−δη− θ ϵ (g ∗ ∗k)(η, θ)dηdθ = 1 ϵ2 ∞∫ 0 ∞∫ 0 e−δη− θ ϵ  η∫ 0 θ∫ 0 g(η − u, θ − v)k(u, v)dudv  dηdθ. (17) Using the Heaviside unit step function, We can write equation 17 as LηWθ((g∗∗g)(η, θ)) = 1 ϵ2 ∞∫ 0 ∞∫ 0 e−δη− θ ϵ ∞∫ 0 ∞∫ 0 g(η − u, θ − v)H(η − u, θ − v)k(u, v))dudv  dηdθ = ∞∫ 0 ∞∫ 0 k(u, v)  1 ϵ2 ∞∫ 0 ∞∫ 0 e−δη− θ ϵ g(η − u, θ − v)H(η − u, θ − v)dηdθ  dudv. So by Lemma 1, We have LηWθ((g ∗ ∗k)(η, θ)) = G(δ, ϵ) ∞∫ 0 ∞∫ 0 k(u, v)e−δu− v ϵ dudv = ϵ2G(δ, ϵ)K(δ, ϵ). In Table 1, we have the DAHT of some basic functions. Table 1: Table of DAHT g(η, θ) LηWθ(g(η, θ)) 1 1 δϵ , Re(δ) > 0 ηuθv ϵv−1 δu+1Γ(u+ 1)Γ(v + 1), Re(δ) > 0 and Re(u) > −1 euη+vθ 1 ϵ(δ−u)(1−vϵ) , Re(δ) > Re(u) ei(uη+vθ) i ϵ(δ−iu)(i+vϵ) , Im(u) + Re(δ) > 0 sin (uη + vθ) u+δϵv ϵ(δ2+u2)(1+v2ϵ2) , |Im(u)| < Re(δ) cos (uη + vθ) δ−ϵuv ϵ(δ2+u2)(1+v2ϵ2) , |Im(u)| < Re(δ) sinh (uη + vθ) u+δϵv ϵ(δ2−u2)(1−v2ϵ2) , Re(δ) > Re(u) and Re(δ) + Re(u) > 0 cosh (uη + vθ) δ+ϵuv ϵ(δ2−u2)(1−v2ϵ2) , Re(δ) > Re(u) and Re(δ) + Re(u) > 0 p(η)q(θ) L(p(η))W (q(θ)) g(η − u, θ − v)H(η − u, θ − v) e−δu− v ϵLηWθ(g(η, θ) (g ∗ ∗k)(η, θ) ϵ2LηWθ(g(η, θ))LηWθ(k(η, θ)) J0 ( c √ ηθ ) 4 ϵ(4δ+c2ϵ) , Re ( δ + c2ϵ 4 ) > 0 M. Al-Momani, A. Jaradat , B. Abughazaleh / Eur. J. Pure Appl. Math, 18 (1) (2025), 5619 10 of 19 4. Applications In this section, we use the DLSWT for solving PDEs and Integro PDEs 4.1. Double Laplace-Sawi transform for solving partial differential equa- tions Consider the PDE of the form A1gηη +A2gηθ +A3gθθ +A4gη +A5gθ +A6g (η, θ) = k (η, θ) , (18) With ICs g(η, 0) = p1 (η), gθ(η, 0) = p2 (η) , and BCs g (0, θ) = q1 (θ), gη (0, θ) = q2 (θ) , and assuming g (0, 0) = Φ. Given that g (η, θ) is the unknown function, k (η, θ) is the source term, andA1, A2, ..., A6 and Φ are constants, we aim to apply the DLSWT to Equation 18. To achieve this, we first apply the single Laplace transform to the ICs and the single Sawi transform to the BCs. L (p1 (η)) = P1(η), L (p2 (η)) = P2(η), W (q1 (θ)) = Q1(θ) and W (q2 (θ)) = Q2(θ). By applying the DLSWT to Equation (18), we have A1LηWθ (gηη) +A2LηWθ (gηθ) +A3LηWθ (gθθ) +A4LηWθ (gη) (19) +A5LηWθ (gθ) +A6LηWθ (g (η, θ)) = LηWθ (k (η, θ)) . By the properties of the derivatives in Equations (12)− (15), we get A1 ( δ2G(δ, ϵ)− δ2Q1(θ)− δQ2(θ) ) (20) +A2 ( δϵ γ G(δ, ϵ)− δP1(η)− δϵ γ Q1(θ) + δΦ ) M. Al-Momani, A. Jaradat , B. Abughazaleh / Eur. J. Pure Appl. Math, 18 (1) (2025), 5619 11 of 19 +A3 ( 1 ϵ G(δ, ϵ)− 1 ϵ P1(η)− P2(η) ) +A4 (δG(δ, ϵ)− δQ1(θ)) +A5 ( 1 ϵ G(δ, ϵ)G(δ, ϵ)− P1(η) ) +A6G(δ, ϵ) = K(δ, ϵ). Simplify Equation 20 as follows G(δ, ϵ) = ( A1δ 2 +A2 δϵ γ +A4δ ) Q1 +A1δQ2 + ( A2δ +A3 1 ϵ +A5 ) P1 +A3P2 −A2δΦ+K A1δ2 +A2 δϵ γ +A3 1 ϵ +A4δ +A5 1 ϵ +A6 . (21) Example 1. Consider the wave equation gηη − gθθ = 0, where η, θ ≥ 0, With ICs g(η, 0) = 5η, gθ(η, 0) = cos η, and BCs g (0, θ) = sin θ, gη (0, θ) = 5. Solution 1. By applying the single Laplace transform to the ICs and the single Sawi transform to the BCs, I get P1 = 5 δ2 , P2 = δ 1+δ2 , Q1 = 1 1+ϵ2 , Q2 = 5 ϵ Substitute in Equation (21) A1 = 1, A3 = −1, A2 = A4 = A5 = A6 = 0 and the values of P1, P2, Q1 and Q2, we get G(δ, ϵ) = δ 1+ϵ2 + 5 ϵ − 5 δ2ϵ3 − δ ϵ2(1+δ2) δ2 − 1 ϵ2 (22) = 5(δ2ϵ2−1) δ2ϵ + δ(δ2ϵ2−1) (1+δ2)(1+ϵ2) δ2ϵ2 − 1 = 5 δ2ϵ + δ (1 + δ2) (1 + ϵ2) . So, g(η, θ) = L−1 η W−1 θ ( 5 δ2ϵ + δ (1 + δ2) (1 + ϵ2) ) = 5η + cos η sin θ. Its graph is M. Al-Momani, A. Jaradat , B. Abughazaleh / Eur. J. Pure Appl. Math, 18 (1) (2025), 5619 12 of 19 Figure 1: The solution of Example 1 Example 2. Consider the Advection-Diffusion equation gδ + 2gϵϵ = 2gϵ, where η, θ ≥ 0, With IC g(η, 0) = 2δ − 1, gϵ (δ, 0) = 0, and BCs g (0, θ) = ϵ− eϵ. Solution 2. By applying the single Laplace transform to the ICs and the single Sawi transform to the BCs, we get P1 = 1 δ−1 , P2 = 0, Q1 = 1 ϵ(1+2ϵ) Substitute in Equation (21) A3 = 2, A4 = 1, A5 = −2, A1 = A2 = A6 = 0 and the values of P1, P2, Q1 and Q2, we get G(δ, ϵ) = 1− 1 ϵ(1−ϵ) + ( 2 ϵ3 − 2 ϵ2 ) × ( 2 δ2 − 1 δ ) 2 ϵ2 + δ − 2 ϵ . By simplifying, we get G(δ, ϵ) = 2 δ2ϵ − 1 δϵ (1− ϵ) + 1 δ . M. Al-Momani, A. Jaradat , B. Abughazaleh / Eur. J. Pure Appl. Math, 18 (1) (2025), 5619 13 of 19 So, g(η, θ) = L−1 η W−1 θ ( 2 δ2ϵ − 1 δϵ (1− ϵ) + 1 δ ) = 2η − eθ + θ. Its graph is Figure 2: The solution of Example 2 Example 3. Consider the telegraph equation 2gηη + gθθ − gη = 5g(η, θ), where η, θ ≥ 0, With ICs g(η, 0) = eη, gθ(η, 0) = −2eη, and BCs g (0, θ) = e−2θ, gη (0, θ) = e−2θ. Solution 3. By applying the single Laplace transform to the ICs and the single Sawi transform to the BCs, we get P1 = 1 δ−1 , P2 = −2 δ−1 , Q1 = 1 ϵ(1+2ϵ) , Q2 = 1 ϵ(1+2ϵ) . Substitute in Equation (21) A1 = 2, A3 = 1, A4 = −1, A6 = −5, A2 = A5 = 0 and the M. Al-Momani, A. Jaradat , B. Abughazaleh / Eur. J. Pure Appl. Math, 18 (1) (2025), 5619 14 of 19 values of P1, P2, Q1 and Q2, we get G(δ, ϵ) = 2δ−1 ϵ(1+2ϵ) + 2 ϵ(1+2ϵ) + 1 ϵ3(δ−1) − 2 ϵ2(δ−1) 2δ2 − 1 ϵ2 − δ − 5 (23) = ϵ2(δ−1)(2δ+1)+(1+2ϵ)−2ϵ(1+2ϵ) ϵ3(δ−1)(1+2ϵ) 2δ2ϵ2−δϵ2−5ϵ2+1 ϵ2 . By simplify, G(δ, ϵ) = 1 ϵ (δ − 1) (1 + 2ϵ) . So, g(η, θ) = L−1 η W−1 θ ( 1 ϵ (δ − 1) (1 + 2ϵ) ) = eη−2θ. Its graph is Figure 3: The solution of Example 3 M. Al-Momani, A. Jaradat , B. Abughazaleh / Eur. J. Pure Appl. Math, 18 (1) (2025), 5619 15 of 19 4.2. Double Laplace-Sawi transform for solving Integro partial differential equations Example 4. Consider the equation of Volterra Integro PDE. gη + gθ − e2η − eθ − 2e2η+θ + 1 = 2 η∫ 0 θ∫ 0 g(u, v))dudv, where η, θ ≥ 0, (24) With ICs g(η, 0) = e2η, g (0, θ) = eθ. Solution 4. By applying the single Laplace transform and the single Sawi transform to the ICs, we get P1 = 1 δ−2 , Q1 = 1 ϵ(1−ϵ) . By Definition 4 and Theorem 2, we have η∫ 0 θ∫ 0 g(u, v))dudv = (1 ∗ ∗g) (η, θ) . (25) Apply the DLSWT to Equation 25, we get δG(δ, ϵ)− 1 ϵ (1− ϵ) + 1 ϵ G(δ, ϵ)− 1 ϵ2 (δ − 2) − 1 ϵ (δ − 2) − 1 δϵ (1− ϵ) − 2 ϵ (δ − 2) (1− ϵ) + 1 δϵ = 2ϵ δ G(δ, ϵ). So, δ2ϵ+ δ − 2ϵ2 δϵ ×G(δ, ϵ) = δϵ (δ − 2) + δ (1− ϵ) + δϵ (1− ϵ) + ϵ (δ − 2) + 2δϵ− ϵ (δ − 2) (1− ϵ) δϵ2 (δ − 2) (1− ϵ) . Thus, G(δ, ϵ) = δ2ϵ+ δ − 2ϵ2 ϵ (δ − 2) (1− ϵ) (δ2ϵ+ δ − 2ϵ2) = 1 ϵ (δ − 2) (1− ϵ) . Therefore, g(η, θ) = L−1 η W−1 θ ( 1 ϵ (δ − 2) (1− ϵ) ) = e2η+θ. M. Al-Momani, A. Jaradat , B. Abughazaleh / Eur. J. Pure Appl. Math, 18 (1) (2025), 5619 16 of 19 Its graph is Figure 4: The solution of Example 4 Example 5. Consider the equation of Integro PDE. gηθ + gη − 2eθ + η2eθ − η2 = 2 η∫ 0 θ∫ 0 g(u, v))dudv, where η, θ ≥ 0, (26) With ICs g(η, 0) = η, g (0, θ) = 0. Solution 5. By applying the single Laplace transform and the single Sawi transform to the ICs, we get P1 = 1 δ2 , Q1 = 0. Apply the DLSWT to Equation 26, we get δ ϵ G(δ, ϵ)− 1 δϵ2 + δG(δ, ϵ)− 2 δϵ (1− ϵ) M. Al-Momani, A. Jaradat , B. Abughazaleh / Eur. J. Pure Appl. Math, 18 (1) (2025), 5619 17 of 19 + 2 δ3ϵ (1− ϵ) − 2 δ3ϵ − 1 ϵ (δ − 2) = 2ϵ δ G(δ, ϵ). So, δ2ϵ+ δ2 − 2ϵ2 δϵ ×G(δ, ϵ) = δ2 (1− ϵ) + 2δ2ϵ− 2ϵ+ 2ϵ (1− ϵ) δ3ϵ (1− ϵ) . Thus, G(δ, ϵ) = δ2ϵ+ δ2 − 2ϵ2 δ2ϵ (1− ϵ) (δ2ϵ+ δ2 − 2ϵ2) = 1 δ2ϵ (1− ϵ) . Therefore, g(η, θ) = L−1 η W−1 θ ( 1 δ2ϵ (1− ϵ) ) = ηeθ. Its graph is M. Al-Momani, A. Jaradat , B. Abughazaleh / Eur. J. Pure Appl. Math, 18 (1) (2025), 5619 18 of 19 Figure 5: The solution of Example 5 5. Conclusion In this paper, we introduce the Double Laplace-Sawi Transform (DLSWT) and we have delved deeply into the foundational properties of the proposed hybrid double trans- form, rigorously characterizing the necessary conditions for its existence. Through this exploration, we have demonstrated the transformative power of these properties in the realms of convolution theory and derivative operations. By establishing the theoretical framework and validating its applicability. Our discussion is realistic in that where appro- priate we specify earlier numerical procedures that benefitted from our previous research while highlighting the key advantages of the DLSWT in problem solving. We believe that the future of the DLSWT is profound in the area of fractional and conformable PDEs and Integro PDEs with coefficients that vary. More related results on fractional and con- formable PDEs and Integro PDEs can be found in [11–14]. Author contribution statement The authors listed have significantly contributed to the development and the writing of this article. M. Al-Momani, A. Jaradat , B. Abughazaleh / Eur. J. Pure Appl. Math, 18 (1) (2025), 5619 19 of 19 Data availability statement No data was used for the research described in the article. Conflict of interest The authors declare that they have no conflict of interest. References [1] M Mahgoub and MMohand. The new integral transform “Sawi Transform”. Advances in Theoretical and Applied Mathematics, 14(1): 81-87, 2019. [2] M Higazy and S Aggarwal. Sawi transformation for system of ordinary differential equations with application. Ain Shams Engineering Journal, 12: 3173-3182, 2021. [3] GK Watugala. Sumudu transform: a new integral transform to solve differential equa- tions and control engineering problems. International Journal of Mathematical Edu- cation in Science and Technology, 24(1): 35-43, 1993. [4] R Saadeh A Qazza and A Burqan. A new integral transform: ARA transform and its properties and applications. Symmetry, 12: 925, 2020. [5] K Aboodh I Abdullahi and R Nuruddeen. On the Aboodh transform connections with some famous integral transforms. Int. J.Eng. Inform. Syst., 1: 143-151, 2017. [6] A Aghili and B Parsa Moghaddam. Certain theorems on two dimensional Laplace transform and non-homogeneous parabolic partial differential equations. Surv. Math. Its Appl., 6: 165-174, 2011. [7] AK Sedeeg ZI Mahamoud and R Saadeh. Using Double Integral Transform (Laplace- ARA Transform) in Solving Partial Differential Equations. Symmetry, 14(11): 2418, 2022. [8] M Hunaiber and A Al-Aati. On Double Laplace-Shehu Transform and its Properties with Applications. Turkish Journal of Mathematics and Computer Science, 15(2): 218-226, 2023. [9] S Khan A Ullah M De la Sen and S Ahmad. Double Sawi Transform: Theory and Applications to Boundary Values Problems. Symmetry, 15(4): 921, 2023. [10] B Abughazaleh MA Amleh A Al-Natoor and R Saadeh. Double Mellin-ARA Trans- form. Springer Proceedings in Mathematics and Statistics, 466: 383-394, 2024. [11] HK Jassim and H Kadhim. Fractional Sumudu decomposition method for solving PDEs of fractional order. J. Appl. Comput. Mech., 7, 302-311, 2021. [12] H Jafari HK Jassim and C & Ãœnlãœ. Laplace decomposition method for solving the two-dimensional diffusion problem in fractal heat transfer. FRACTALS (fractals), 32(4): 1-6, 2024. [13] MA Amleh B Abughazaleh and A Al-Natoor. Conformable fractional Lomax proba- bility distribution. J. Math. Comput. Sci., 12, Article-ID 130, 2022. [14] A Qazza A Burqan R Saadeh and R Khalil. Applications on double ARA–Sumudu transform in solving fractional partial differential equations. Symmetry, 14(9): 1817, 2022.