EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 2, Article Number 5967 ISSN 1307-5543 – ejpam.com Published by New York Business Global The Double Sumudu-Sawi Transform Raed R. Abu Awwad1, Monther Al-Momani2, Baha’ Abughazaleh3,∗, Ali Jaradat4, Abdulkarim Farah3 1 Department of Mathematics, University of Petra, Amman, Jordan 2 Department of Basic Sciences, Al-Ahliyya Amman University, Amman, Jordan 3 Department of Mathematics, Isra University, Amman, Jordan 4 Department of Mathematics, Amman Arab University, Amman, Jordan Abstract. The paper explores integral transforms and their broader generalizations. The primary aim is to develop a new integral transform that combines the Hybrid Sumudu and Sawi transforms, investigating their properties, existence, and inversion theorem. We introduce recent findings on partial differential equations in higher dimensions and broaden the scope of the double convolution theorem to encompass two-dimensional scenarios. Utilizing these novel properties and theorems, we solve particular types of differential equations, showcasing their practical applications in physics and various scientific domains. 2020 Mathematics Subject Classifications: 44A05 Key Words and Phrases: Sumudu transform, Sawi transform, The Double Sumudu-Sawi trans- form. 1. Introduction Integral transforms are powerful mathematical tools that convert functions into dif- ferent domains, making them easier to analyze and manipulate. Once transformed, a function can be reverted to its original form using the inverse transform. These trans- formations play a crucial role in engineering, economics, physics, and chemistry, helping simplify complex real-world problems. As mathematical challenges grow, researchers continue to develop more general classes of differential equations and innovative analytical techniques. One of the most widely used integral transforms is the Laplace transform, introduced in 1780. In recent years, new transforms such as the Sumudu transform in [1] and Sawi transform in [2] have gained attention for their unique properties and diverse applications. ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i2.5967 Email addresses: rabuawwad@uop.edu.jo (R. Abu Awwad), montheralmomani72@gmail.com (M. Al-Momani), baha.abughazaleh@iu.edu.jo (B. Abughazaleh), a.jaradat@aau.edu.jo (A. Jaradat), karim.farah@iu.edu.jo (A. Farah) 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 (2) (2025), 5967 2 of 17 Beyond single-variable transforms, double transforms have been developed to handle multi-variable differential equations. Among these are the Double Laplace Transform in [3], Double Mellin-ARA Transform in [4], the Double Sumudu Transform [5], the Double Sumudu-Shehu Transform [6], Double Laplace-Sawi Transform [7], Double Sawi Transform in [8], and the Double ARA-Sawi Transform in [9]. These methods provide new ways to solve high-dimensional equations, making them essential tools in modern mathematical analysis. In this work, we introduce a novel Double Sumudu-Sawi Transform (DS-SWT) de- signed to extend the scope of differential equation analysis. We explore its fundamental properties, establishing the conditions for its existence and demonstrating its power in convolution theory and derivative operations. By applying this new transform, we un- cover innovative approaches to solving partial differential equations and integro-differential equations, paving the way for more efficient mathematical tools. 2. Sumudu and Sawi transforms This section offers an overview of the individual transforms, emphasizing the key prop- erties of the Sumudu and Sawi transforms, and highlighting their distinct features and applications. 2.1. Sumudu transform Definition 1. The Sumudu transform of a continuous function r(ξ) on (0,∞) is defined as follows R(δ) = S(r(ξ)) = 1 δ ∞∫ 0 e− ξ δ r(ξ)dξ, δ ∈ C. The fundamental properties of the Sumudu transform are presented as follows: Let R(δ) = S(r(ξ)), then for nonzero constants β and γ, we have S(βr1(ξ) + γr2(ξ)) = βS(r1(ξ)) + γS(r2(ξ)), (1) where r1(ξ) and r2(ξ) are continuous functions on (0,∞). S(ξβ) = Γ(β + 1)δβ, (2) S(eβξ) = 1 1− δβ , β ∈ R, (3) S(r′(ξ)) = R(δ) δ − r(0) δ , (4) R. Abu Awwad et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5967 3 of 17 S(r′′(ξ)) = R(δ) δ2 − r(0) δ2 − r′(0) δ . (5) 2.2. The Sawi transform Definition 2. The Sawi transform of a continuous function f(χ) on (0,∞) expressed as follows F (ϵ) = W (f(χ)) = 1 ϵ2 ∞∫ 0 e− χ ϵ f(χ)dχ. Let us examine the fundamental properties that characterize the Sawi transform. Suppose that F1(ϵ) = W (f1(χ)) and F2(ϵ) = W (f2(χ)), with u and v as nonzero real numbers, the following properties hold W (uf1(χ) + vf2(χ)) = uW (f1(χ)) + vW (f2(χ)), (6) W (χu) = Γ(u+ 1)ϵu−1, (7) W (evχ) = 1 ϵ (1− vϵ) , (8) W (f ′(χ)) = 1 ϵ F (ϵ)− 1 ϵ2 f(0), (9) W (f ′′(χ)) = 1 ϵ2 F (ϵ)− 1 ϵ3 f(0)− 1 ϵ2 f ′(0). (10) 3. The Double Sumudu-Sawi transform This section introduces the Double Sumudu-Sawi Transformation (DS-SWT). We be- gin by outlining its fundamental properties, including linearity and inversion. Next, we present a new result related to partial derivatives, as well as a novel outcome regarding the convolution theorem. Additionally, we explain how these results are applied to compute the DS-SWT of several basic functions. The definition of the DS-SWT is: H(δ, ϵ) = SξWχ(h(ξ, χ)) = 1 δϵ2 ∞∫ 0 ∞∫ 0 e− ξ δ −χ ϵ h(ξ, χ) dξdχ, (11) where h(ξ, χ) is a continuous function on (0,∞)× (0,∞). Clearly, SξWχ(h(ξ, χ)) is linear transformation. In fact, for nonzero constants u and v, we have SξWχ(uh1(ξ, χ)+vh2(ξ, χ)) R. Abu Awwad et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5967 4 of 17 = 1 δϵ2 ∞∫ 0 ∞∫ 0 e− ξ δ −χ ϵ (uh1(ξ, χ) + vh2(ξ, χ)) dξdχ = u δϵ2 ∞∫ 0 ∞∫ 0 e− ξ δ −χ ϵ h1(ξ, χ) dξdχ+ v δϵ2 ∞∫ 0 ∞∫ 0 e− ξ δ −χ ϵ h2(ξ, χ) dξdχ = uSξWχ(h1(ξ, χ)) + vSξWχ(h2(ξ, χ)). If h(ξ, χ) can be written as h(ξ, χ) = g(ξ)f(χ) for some continuous functions g and f , then SξWχ(h(ξ, χ)) = S(g(ξ))W (f(χ)). In fact SξWχ(h(ξ, χ)) = SξWχ(g(ξ)f(χ)) = 1 δϵ2 ∞∫ 0 ∞∫ 0 e− ξ δ −χ ϵ g(ξ)f(χ)dξdχ = 1 δ ∞∫ 0 e− ξ δ g(ξ)dξ  1 ϵ2 ∞∫ 0 e− χ ϵ f(χ)dχ  = S(g(ξ))W (f(χ)). 3.1. DS-SWT for some basic functions (i) SξWχ(1) = 1 δϵ2 ∞∫ 0 ∞∫ 0 e− ξ δ −χ ϵ dξdχ = 1 δ ∞∫ 0 e− ξ δ dξ  1 ϵ2 ∞∫ 0 e− χ ϵ dχ  = 1× 1 ϵ = 1 ϵ , Re( 1 δ ) > 0. (ii) SξWχ(ξ uχv) = 1 δϵ2 ∞∫ 0 ∞∫ 0 e− ξ δ −χ ϵ ξuχvdξdχ = 1 δ ∞∫ 0 ξue− ξ δ dξ  1 ϵ2 ∞∫ 0 χve− χ ϵ dχ  = Γ(v + 1)δu × Γ(v + 1)ϵv−1 = δuϵv−1Γ(u+ 1)Γ(v + 1), Re( 1 δ ) > 0 and Re(u) > −1. R. Abu Awwad et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5967 5 of 17 (iii) SξWχ(e uξ+vχ) = 1 δϵ2 ∞∫ 0 ∞∫ 0 e− ξ δ −χ ϵ euξ+vχdξdχ = 1 δ ∞∫ 0 euξ− ξ δ dξ  1 ϵ2 ∞∫ 0 evχ− χ ϵ dχ  = 1 1− δu × 1 ϵ (1− vϵ) = 1 ϵ (1− δu) (1− vϵ) , Re( 1 δ ) > Re(u). 3.2. Existence condition for DS-SWT Definition 3. A function h(ξ, χ) is said to be of exponential orders u and v on 0 ≤ ξ < ∞ and 0 ≤ χ < ∞. If there exist K,X, Y > 0 such that |h(ξ, χ)| ≤ Keuξ+vχ, for all ξ > X, χ > Y. Theorem 1. Let h(ξ, χ) be a continuous function on the region [0,∞) × [0,∞) of ex- ponential orders u and v. Then H(δ, ϵ) exists for δ, ϵ and γ whenever Re ( 1 δ ) > u and Re ( 1 ϵ ) > v. Proof. We have |H(δ, ϵ)| = ∣∣∣∣∣∣ 1 δϵ2 ∞∫ 0 ∞∫ 0 e− ξ δ −χ ϵ h(ξ, χ) dξdχ ∣∣∣∣∣∣ ≤ 1 δϵ2 ∞∫ 0 ∞∫ 0 e− ξ δ −χ ϵ |h(ξ, χ)| dξdχ ≤ K 1 δϵ2 ∞∫ 0 ∞∫ 0 e− ξ δ −χ ϵ euξ+vχdξdχ = K ∞∫ 0 e−(δ−u)ξdξ  1 ϵ2 ∞∫ 0 e−( 1 ϵ −v)χdχ  = K ϵ (δ − u) (1− vϵ) . where Re (δ) > u and Re ( 1 ϵ ) > v. 3.3. Derivatives properties Now, we present some basic properties of the DS-SWT Let H(δ, ϵ) = SξWχ(h(ξ, χ)) where h(ξ, χ) is a continuous function on (0,∞)× (0,∞). Then R. Abu Awwad et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5967 6 of 17 (i) SξWχ ( ∂h(ξ, χ) ∂ξ ) = 1 δ H(δ, ϵ)− 1 δ W (h(0, χ)), (12) (ii) SξWχ ( ∂2h(ξ, χ) ∂ξ2 ) = 1 δ2 H(δ, ϵ)− 1 δ2 W (h(0, χ))− 1 δ W (hξ(0, χ)), (iii) SξWχ ( ∂h(ξ, χ) ∂χ ) = 1 ϵ H(δ, ϵ)− 1 ϵ2 S(h(ξ, 0)), (13) (iv) SξWχ ( ∂2h(ξ, χ) ∂χ2 ) = 1 ϵ2 H(δ, ϵ)− 1 ϵ3 S(h(ξ, 0))− 1 ϵ2 S(hχ(ξ, 0)), (14) (v) SξWχ ( ∂2h(ξ, χ) ∂ξ∂χ ) = 1 δϵ H(δ, ϵ)− 1 δϵ2 S(h(ξ, 0))− 1 δϵ W (h(0, χ)) + 1 δϵ2 h(0, 0). (15) Proof. (1) SξWχ ( ∂h(ξ,χ) ∂ξ ) = 1 δϵ2 ∞∫ 0 ∞∫ 0 e− ξ δ −χ ϵ ∂h(ξ,χ) ∂ξ dξdχ = 1 δϵ2 ∞∫ 0 e− χ ϵ ∞∫ 0 e− ξ δ ∂h(ξ,χ) ∂ξ dξdχ. By integrating by parts, we get SξWχ ( ∂h(ξ,χ) ∂ξ ) = 1 δϵ2 ∞∫ 0 e− χ ϵ ( −h(0, χ) + 1 δ ∞∫ 0 e− ξ δ h(ξ, χ) dξ ) dχ = − 1 δϵ2 ∞∫ 0 e− χ ϵ h(0, χ)dχ+ 1 δ2ϵ2 ∞∫ 0 ∞∫ 0 e− ξ δ −χ ϵ h(ξ, χ) dξdχ = 1 δH(δ, ϵ)− 1 δW (h(0, χ)). The proof of Equations 2, 13, 14 and 15 can be obtained in the same manner. 3.4. Convolution Theorem of DS-SWT Let F (ξ, χ) represent the Heaviside unit step function, which is defined as follows: F (ξ − u, χ− v) = { 1, ξ > u and χ > v 0, otherwise Then we have the following lemma R. Abu Awwad et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5967 7 of 17 Lemma 1. Let h(ξ, χ) be a continuous function on (0,∞)×(0,∞) and F (ξ, χ) be the Heav- iside unit step function. Then SξWχ(h(ξ−u, χ−v)F (ξ−u, χ−v)) = e− u δ − v ϵ SξWχ(h(ξ, χ). Proof. We have SξWχ(h(ξ − u, χ− v)F (ξ − u, χ− v)) (16) = 1 δϵ2 ∞∫ 0 ∞∫ 0 e− ξ δ −χ ϵ h(ξ − u, χ− v)F (ξ − u, χ− v)dξdχ = 1 δϵ2 ∞∫ u ∞∫ v e− ξ δ −χ ϵ h(ξ − u, χ− v)dξdχ. Now, by making the substitution z = ξ − u and w = χ− v, equation 16 becomes: SξWχ(h(ξ − u, χ− v)F (ξ − u, χ− v)) = 1 δϵ2 ∞∫ 0 ∞∫ 0 e− (z+u) δ − (w+v) ϵ h(z, w)dzdw = e− u δ − v ϵ SξWχ(h(ξ, χ)). Definition 4. Let h(ξ, χ) and k(ξ, χ) be continuous functions. We define the convolution in the DS-SWT as (h ∗ ∗k)(ξ, χ) = ξ∫ 0 χ∫ 0 h(ξ − u, χ− v)k(u, v)dudv. In the following theorem, we compute DS-SWT of the convolution of two functions Theorem 2. Let H(δ, ϵ) = SξWχ(h(ξ, χ)) and K(δ, ϵ) = SξWχ(k(ξ, χ)). Then SξWχ((h ∗ ∗k)(ξ, χ)) = δϵ2H(δ, ϵ)K(δ, ϵ). Proof. SξWχ((h∗∗k)(ξ, χ)) = 1 δϵ2 ∞∫ 0 ∞∫ 0 e− ξ δ −χ ϵ (h ∗ ∗k)(ξ, χ)dξdχ = 1 δϵ2 ∞∫ 0 ∞∫ 0 e− ξ δ −χ ϵ  ξ∫ 0 χ∫ 0 h(ξ − u, χ− v)k(u, v)dudv  dξdχ. (17) R. Abu Awwad et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5967 8 of 17 Using the Heaviside unit step function, We can write equation 17 as SξWχ((h∗∗h)(ξ, χ)) = 1 δϵ2 ∞∫ 0 ∞∫ 0 e− ξ δ −χ ϵ ∞∫ 0 ∞∫ 0 h(ξ − u, χ− v)F (ξ − u, χ− v)k(u, v))dudv  dξdχ = ∞∫ 0 ∞∫ 0 k(u, v)  1 δϵ2 ∞∫ 0 ∞∫ 0 e− ξ δ −χ ϵ h(ξ − u, χ− v)F (ξ − u, χ− v)dξdχ  dudv. So by Lemma 1, We have SξWχ((h ∗ ∗k)(ξ, χ)) = H(δ, ϵ) ∞∫ 0 ∞∫ 0 k(u, v)e− u δ − v ϵ dudv = δϵ2H(δ, ϵ)K(δ, ϵ). In Table 1, we have the DAHT of some basic functions. Table 1: Table of DAHT h(ξ, χ) SξWχ(h(ξ, χ)) 1 1 ϵ , Re(1δ ) > 0 ξuχv δuϵv−1Γ(u+ 1)Γ(v + 1), Re(1δ ) > 0 and Re(u) > −1 euξ+vχ 1 ϵ(1−δu)(1−vϵ) , Re(1δ ) > Re(u). ei(uξ+vχ) −1 ϵ(δu+i)(i+vϵ) , Im(u) + Re(1δ ) > 0 sin (uξ + vχ) uδ+ϵv ϵ(1+u2δ2)(1+v2ϵ2) , |Im(u)| < Re(1δ ) cos (uξ + vχ) 1−δϵuv ϵ(1+u2δ2)(1+v2ϵ2) , |Im(u)| < Re(1δ ) sinh (uξ + vχ) uδ+ϵv ϵ(1−δ2u2)(1−v2ϵ2) , Re(1δ ) > Re(u) and Re(1δ + u) > 0 cosh (uξ + vχ) 1+δϵuv ϵ(1−δ2u2)(1−v2ϵ2) , Re(1δ ) > Re(u) and Re(1δ + u) > 0 g(ξ)f(χ) S(g(ξ))W (f(χ)) h(ξ − u, χ− v)H(ξ − u, χ− v) e− u δ − v ϵ SξWχ(h(ξ, χ) (h ∗ ∗k)(ξ, χ) δϵ2SξWχ(h(ξ, χ))SξWχ(k(ξ, χ)) J0 ( c √ ξχ ) 4 ϵ(4+c2δϵ) , Re ( 1 δ + c2ϵ 4 ) > 0 4. Applications In this section, we use the DS-SWT for solving PDEs and Integro PDEs R. Abu Awwad et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5967 9 of 17 4.1. DS-SWT for solving PDEs Consider the PDE of the form A1hξξ +A2hξχ +A3hχχ +A4hξ +A5hχ +A6h (ξ, χ) = k (ξ, χ) (18) With ICs h(ξ, 0) = g1 (ξ), hχ(ξ, 0) = g2 (ξ) and BCs h (0, χ) = f1 (χ), hξ (0, χ) = f2 (χ) and assuming h (0, 0) = Φ Given that h (ξ, χ) is the unknown function, k (ξ, χ) is the source term, andA1, A2, ..., A6 and Φ are constants, we aim to apply the DS-SWT to Equation 18. To achieve this, we first apply the single Sumudu transform to the ICs and the single Sawi transform to the BCs. S (g1 (ξ)) = G1(ξ), S (g2 (ξ)) = G2(ξ), W (f1 (χ)) = F1(χ) and W (f2 (χ)) = F2(χ) By applying the DS-SWT to Equation (18), we have A1SξWχ (hξξ) +A2SξWχ (hξχ) +A3SξWχ (hχχ) +A4SξWχ (hξ) (19) +A5SξWχ (hχ) +A6SξWχ (h (ξ, χ)) = SξWχ (k (ξ, χ)) By the properties of the derivatives in Equations (12)− (15), we get A1 ( 1 δ2 H(δ, ϵ)− 1 δ2 F1(χ)− 1 δ F2(χ) ) (20) +A2 ( 1 δϵ H(δ, ϵ)− 1 δϵ2 G1(ξ)− 1 δϵ F1(χ) + 1 δϵ2 Φ ) +A3 ( 1 ϵ2 H(δ, ϵ)− 1 ϵ3 G1(ξ)− 1 ϵ2 G2(ξ) ) +A4 ( 1 δ H(δ, ϵ)− 1 δ F1(χ) ) +A5 ( 1 ϵ H(δ, ϵ)H(δ, ϵ)− 1 ϵ2 G1(ξ) ) +A6H(δ, ϵ) = K(δ, ϵ) R. Abu Awwad et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5967 10 of 17 Simplify Equation 20 as follows H(δ, ϵ) = ( A1 1 δ2 +A2 1 δϵ +A4 1 δ ) F1 +A1 1 δF2 + ( A2 1 δϵ2 +A3 1 ϵ3 +A5 1 ϵ2 ) G1 +A3 1 ϵ2 G2 −A2 1 δϵ2 Φ+K A1 1 δ2 +A2 1 δϵ +A3 1 ϵ2 +A4 1 δ +A5 1 ϵ +A6 (21) Example 1. Consider the Klein-Gordon equation 2hξξ − hχχ − h(ξ, χ) = 5 sinh ξ cos 2χ, where ξ, χ ≥ 0, With ICs h(ξ, 0) = sinh ξ, hχ(ξ, 0) = 0, and BCs h (0, χ) = 0, hξ (0, χ) = cos 2χ. Solution 1. By applying the single Sumudu transform to the ICs and the single Sawi transform to the BCs, I get G1 = δ 1−δ2 , G2 = 0, F1 = 0, F2 = 1 ϵ(1+4ϵ2) , and K = SξWχ (5 sinh ξ cos 2χ) = 5δ ϵ(1−δ2)(1+4ϵ2) . Substitute in Equation (21) A1 = 2, A3 = −1, A6 = −1, A2 = A4 = A5 = 0 and the values of G1, G2, F1, F2 and K, we get H(δ, ϵ) = 2 δϵ(1+4ϵ2) − δ ϵ3(1−δ2) + 5δ ϵ(1−δ2)(1+4ϵ2) 2 δ2 − 1 ϵ2 − 1 (22) = 2ϵ2(1−δ2)−δ2(1+4ϵ2)+5δ2ϵ2 δϵ3(1−δ2)(1+4ϵ2) 2ϵ2−δ2−δ2ϵ2 δ2ϵ2 . (23) By simplify, we get H(δ, ϵ) = δ ϵ (1− δ2) (1 + 4ϵ2) . So, h(ξ, χ) = S−1 ξ W−1 χ ( δ ϵ (1− δ2) (1 + 4ϵ2) ) = sinh ξ cos 2χ. Its graph is R. Abu Awwad et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5967 11 of 17 Figure 1: The solution of Example 4.1 Example 2. Consider the telegraph equation hξξ − 2hχχ − hξ = 4h(ξ, χ), where ξ, χ ≥ 0, With ICs h(ξ, 0) = 0, hχ(ξ, 0) = e2ξ, and BCs h (0, χ) = sinχ, hξ (0, χ) = 2 sinχ. Solution 2. By applying the single Sumudu transform to the ICs and the single Sawi transform to the BCs, we get G1 = 0, G2 = 1 1−2δ , F1 = 1 1+ϵ2 , F2 = 1 1+4ϵ2 . Substitute in Equation (21) A1 = 1, A3 = −2, A4 = −1, A6 = −4, A2 = A5 = 0 and the values of G1, G2, F1 and F2, we get H(δ, ϵ) = ( 1 δ2 − 1 δ ) ( 1 1+ϵ2 ) + 2 δ(1+ϵ2) − 2 ϵ2(1−2δ) 1 δ2 − 2 ϵ2 − 1 δ − 4 (24) R. Abu Awwad et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5967 12 of 17 = ϵ2(1−δ)(1−2δ)+δϵ2(1−2δ)−2δ2(1+ϵ2) δ2ϵ2(1−2δ)(1+ϵ2) ϵ2−2δ2−δϵ2−4δ2ϵ2 δ2ϵ2 . By simplify, we get H(δ, ϵ) = 1 (1− 2δ) (1 + ϵ2) . So, h(ξ, χ) = S−1 ξ W−1 χ ( 1 (1− 2δ) (1 + ϵ2) ) = e2ξ sinχ. Its graph is Figure 2: The solution of Example 4.2 4.2. DS-SWT for solving Integro PDEs Example 3. Consider the equation of Volterra Integro PDE. hξ + hχ − cosχ+ ξ sinχ+ 2ξ2 sinχ = 4 ξ∫ 0 χ∫ 0 h(u, v))dudv, where ξ, χ ≥ 0, (25) R. Abu Awwad et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5967 13 of 17 With ICs h(ξ, 0) = ξ, h(0, χ) = 0. Solution 3. By applying the single Sumudu transform and the single Sawi transform to the ICs, we get G1 = δ, F1 = 0. By Definition 4 and Theorem 2, we have ξ∫ 0 χ∫ 0 h(u, v))dudv = (1 ∗ ∗h) (ξ, χ) . (26) Apply the DS-SWT to Equation 26, we get 1 δ H(δ, ϵ) + 1 ϵ H(δ, ϵ)− δ ϵ2 − 1 ϵ (1 + ϵ2) + δ (1 + ϵ2) + 4δ2 (1 + ϵ2) = 4δϵH(δ, ϵ). So, ϵ+ δ − 4δ2ϵ2 δϵ ×H(δ, ϵ) = δ ( 1 + ϵ2 ) + ϵ− δϵ2 − 4δ2ϵ2 ϵ2 (1 + ϵ2) . Thus, H(δ, ϵ) = δ ( ϵ+ δ − 4δ2ϵ2 ) ϵ (1 + ϵ2) (ϵ+ δ − 4δ2ϵ2) = δ ϵ (1 + ϵ2) . Therefore, h(ξ, χ) = S−1 ξ W−1 χ ( δ ϵ (1 + ϵ2) ) = ξ cosχ. Its graph is R. Abu Awwad et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5967 14 of 17 Figure 3: The solution of Example 4.3 Example 4. Consider the equation of Integro PDE. hξχ − 2hχ + 2eχ sinh ξ − cosh ξ − eχ + 1 = ξ∫ 0 χ∫ 0 h(u, v))dudv, where ξ, χ ≥ 0, (27) With ICs h(ξ, 0) = sinh ξ, h(0, χ) = 0 Solution 4. By applying the single Sumudu transform and the single Sawi transform to the ICs, we get G1 = δ 1−δ2 , F1 = 0 Apply the DS-SWT to Equation 27, we get 1 δ H(δ, ϵ)− 2 ϵ H(δ, ϵ) + 2δ ϵ2 (1− δ2) + 2δ ϵ (1− ϵ) (1− δ2) − 1 ϵ (1− δ2) − 1 ϵ (1− ϵ) + 1 ϵ = δϵH(δ, ϵ). R. Abu Awwad et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5967 15 of 17 So, ϵ− 2δ − δ2ϵ2 δϵ ×H(δ, ϵ) = −2δ (1− ϵ)− 2δϵ+ ϵ (1− ϵ) + ϵ ( 1− δ2 ) − ϵ (1− ϵ) ( 1− δ2 ) ϵ2 (1− ϵ) (1− δ2) Thus, H(δ, ϵ) = δ ( ϵ− 2δ − δ2ϵ2 ) ϵ (1− ϵ) (1− δ2) (ϵ− 2δ − δ2ϵ2) = δ ϵ (1− ϵ) (1− δ2) Therefore, h(ξ, χ) = S−1 ξ W−1 χ ( δ ϵ (1− ϵ) (1− δ2) ) = ξeχ Its graph is Figure 4: The solution of Example 4.4 R. Abu Awwad et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5967 16 of 17 5. Conclusion In this paper, we introduce the Double Sumudu-Sawi Transform (DS-SWT) and ex- plore its fundamental properties, shedding light on the key features that define this novel double transform. Several examples are presented to demonstrate the successful appli- cation of the DS-SWT in solving various partial and integral equations exactly. Our discussion is grounded in practical applications, where, where appropriate, we reference earlier numerical procedures that benefited from our previous research, while emphasizing the key advantages of the DS-SWT in solving complex problems. We believe that the future of the DS-SWT holds significant promise in the realm of conformable partial differ- ential equations and integro-PDEs, particularly those with varying coefficients. Additional results related to conformable PDEs and Integro PDEs are available in references [10, 11]. Author contribution statement All authors listed have significantly contributed to the development and the writing of this article. 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] G. K. Watugala. Sumudu transform: a new integral transform to solve differential equations and control engineering problems. International Journal of Mathematical Education in Science and Technology, 24(1):35–43, 1993. [2] M. M. A. Mahgoub and M. Mohand. The new integral transform Sawi transform. Advances in Theoretical and Applied Mathematics, 14(1):81–87, 2019. [3] A. Aghili and B. Parsa Moghaddam. Certain theorems on two dimensional Laplace transform and non-homogeneous parabolic partial differential equations. Surveys in Mathematics and its Applications, 6:165–174, 2011. [4] B. Abughazaleh, M. A. Amleh, A. Al-Natoor, and R. Saadeh. Double Mellin-ARA transform. 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