EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 4, Article Number 6890 ISSN 1307-5543 – ejpam.com Published by New York Business Global The Double Sawi-Shehu Transform Monther Al-Momani1,∗, Baha’ Abughazaleh2, Abdulkarim Farah2 1 Department of Basic Sciences, Al-Ahliyya Amman University, Amman, Jordan 2 Department of Mathematics, Isra University, Amman, Jordan Abstract. This research combines the Sawi and Shehu transforms into a unified framework called the Double Sawi-Shehu Transform. We study its core features such as when it exists and how to recover the original function. We also develop improved methods for solving partial differential equations in multiple dimensions and extend the double convolution theorem to two-dimensional problems. Practical examples show how this approach simplifies complex calculations in physics and other sciences proving its effectiveness. 2020 Mathematics Subject Classifications: 44A05 Key Words and Phrases: Sawi transform, Shehu transform, double integral transform, double Sawi-Shehu transform 1. Introduction Integral transforms simplify mathematical problems by converting functions into easier forms. Engineers and physicists widely use these tools to study complex phenomena leading researchers to create new transforms like the Sawi [1] and Shehu [2] transforms in recent years. For equations with multiple variables specialized double transforms are needed. Exam- ples include the Double Laplace Transform [3] Double Shehu Transform [? ] and others, see [4–10]. In this work, we introduce a new transform that merges the strengths of the Sawi and Shehu transforms. This combination solves a wider range of partial and integral differential equations. Its simplicity makes it particularly useful in physics saving time and effort compared to traditional methods. 2. Sawi and Shehu transforms This section gives a brief description and some basic properties of the single transforms: Sawi, and Shehu transforms. ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.6890 Email addresses: montheralmomani72@gmail.com (M. Al-Momani), 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 (4) (2025), 6890 2 of 15 2.1. Sawi transform Definition 1. The Sawi transform of a continuous function b(ε) on [0,∞) is defined as follows B(σ) = W (b(ε)) = 1 σ2 ∞∫ 0 e−σεb(ε)dε, σ > 0. Some basic properties of the Sawi transform are now given. Let B(σ) = W (b(ε)), then for nonzero constants β and γ, we have W (βb1(ε) + γb2(ε)) = βW (b1(ε)) + γW (b2(ε)), (1) where b1(ε) and b2(ε) are continuous functions on [0,∞). W (εβ) = Γ(β + 1)σβ−1, (2) W (eβε) = 1 σ (1− σβ) , β ∈ R, (3) W (b′(ε)) = B(σ) σ − b(0) σ2 , (4) W (b′′(ε)) = B(σ) σ2 − b(0) σ3 − b′(0) σ2 . (5) 2.2. The Shehu transform Definition 2. The Shehu transform of a continuous function p(ζ) on [0,∞) is defined as follows P (ϕ, δ) = H(p(ζ)) = ∞∫ 0 e− ϕζ δ p(ζ)dζ. We now outline the fundamental properties of the Shehu transform. Suppose that P1(ϕ, δ) = H(p1(ζ)) and P2(ϕ, δ) = H(p2(ζ)), and β and γ are nonzero real numbers, then the following properties hold: H(βp1(ζ) + γp2(ζ)) = βH(p1(ζ)) + γH(p2(ζ)), (6) H(ζβ) = Γ(β + 1) ( δ ϕ )β+1 , (7) H(eγζ) = δ ϕ− γδ , (8) M. Al-Momani et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6890 3 of 15 H(p′(ζ)) = ϕ δ P (ϕ, δ)− p(0), (9) H(p′′(ζ)) = ϕ2 δ2 P (ϕ, δ)− ϕ δ p(0)− p′(0). (10) 3. The Double Sawi-Shehu transform This section introduces the Double Sawi-Shehu Transformation (DSW-SHT), which combines the Sawi and Shehu transforms. The fundamental properties of this new dou- ble transform, including linearity and inversion, are presented. Additionally, new results related to partial derivatives and the convolution theorem are established. These results are implemented to compute the DSW-SHT for some basic functions. We define the DSW-SHT transform as follows: U(σ, ϕ, δ) = WεHζ(u(ε, ζ)) = 1 σ2 ∞∫ 0 ∞∫ 0 e− ε σ −ϕζ δ u(ε, ζ) dεdζ, (11) where u(ε, ζ) is a continuous function on [0,∞)×[0,∞). If u(ε, ζ) can be written as u(ε, ζ) = w(ε)x(ζ) for some continuous functions w and σ, then WεHζ(u(ε, ζ)) = W (w(ε))H(x(ζ)). In fact WεHζ(u(ε, ζ)) = WεHζ(w(ε)x(ζ)) = 1 σ2 ∞∫ 0 ∞∫ 0 e− ε σ −ϕζ δ w(ε)x(ζ)dεdζ =  1 σ2 ∞∫ 0 e− ε σw(ε)dε ∞∫ 0 e− ϕζ δ x(ζ)dζ  = W (w(ε))H(x(ζ)). 3.1. The Double Sawi-Shehu Transform for some basic functions (i) WεHζ(1) = 1 σ2 ∞∫ 0 ∞∫ 0 e− ε σ −ϕζ δ dεdζ =  1 σ2 ∞∫ 0 e− ε σ ∞∫ 0 e− ϕζ δ dζ  = 1 σ × δ ϕ = δ σϕ , Re(σ) > 0. M. Al-Momani et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6890 4 of 15 (ii) WεHζ(ε βζγ) = 1 σ2 ∞∫ 0 ∞∫ 0 e− ε σ −ϕζ δ εβζγdεdζ =  1 σ2 ∞∫ 0 εβe− ε σ dε  ∞∫ 0 ζγe− ϕζ δ dζ  = Γ(β + 1)σβ−1 × Γ(γ + 1) ( δ ϕ )γ+1 = σβ−1δγ+1 ϕγ+1 Γ(β + 1)Γ(γ + 1), Re(σ) > 0 and Re(β) > −1. (iii) WεHζ(e βε+γζ) = 1 σ2 ∞∫ 0 ∞∫ 0 e− ε σ −ϕζ δ eβε+γζdεdζ =  1 σ2 ∞∫ 0 eβε− ε σ dε ∞∫ 0 eγζ− ϕζ δ dζ  = 1 σ (1− σβ) × δ ϕ− γδ = δ σ (1− σβ) (ϕ− γδ) , Re( 1 σ ) > Re(β). 3.2. Existence condition for Double Sawi-Shehu Transform Definition 3. A function u(ε, ζ) is said to be of exponential orders β and γ on 0 ≤ ε < ∞ and 0 ≤ ζ < ∞. If there exist B,X, Y > 0 such that |u(ε, ζ)| ≤ Beβε+γζ , for all ε > X, ζ > Y. Theorem 1. Let u(ε, ζ) be a continuous function on the region [0,∞)×[0,∞) of expo- nential orders β and γ. Then U(σ, ϕ, δ) exists for σ, ϕ and δ whenever Re( 1σ ) > β and Re ( ϕ δ ) > γ. Proof. |U(σ, ϕ, δ)| = ∣∣∣∣∣∣ 1σ2 ∞∫ 0 ∞∫ 0 e− ε σ −ϕζ δ u(ε, ζ) dεdζ ∣∣∣∣∣∣ ≤ 1 σ2 ∞∫ 0 ∞∫ 0 e− ε σ −ϕζ δ |u(ε, ζ)| dεdζ ≤ B σ2 ∞∫ 0 ∞∫ 0 e− ε σ −ϕζ δ eβε+γζdεdζ = B σ2 ∞∫ 0 e−( 1 σ −β)εdε ∞∫ 0 e−(ϕ δ −γ)ζdζ M. Al-Momani et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6890 5 of 15 = B σ(1− σβ)(ϕδ − γ) = Bδ σ(1− σβ)(ϕ− γ δ ) where Re( 1σ ) > β and Re ( ϕ δ ) > γ. 3.3. Linearity The transform WεHζ(u(ε, ζ)) is linear transformation. In fact, for nonzero constants β and γ, we have WεHζ(βu1(ε, ζ)+γu2(ε, ζ)) = 1 σ2 ∞∫ 0 ∞∫ 0 e− ε σ −ϕζ δ (βu1(ε, ζ) + γu2(ε, ζ)) dεdζ, = β × 1 σ2 ∞∫ 0 ∞∫ 0 e− ε σ −ϕζ δ u1(ε, ζ) dεdζ + γ × 1 σ2 ∞∫ 0 ∞∫ 0 e− ε σ −ϕζ δ u2(ε, ζ) dεdζ = βWεHζ(u1(ε, ζ)) + γWεHζ(u2(ε, ζ)). 4. Properties of the Double Sawi-Shehu Transform Now, we present some basic properties of the DSW-SHT 4.1. Derivatives properties Let U(σ, ϕ, δ) = WεHζ(u(ε, ζ)). Then (i) WεHζ ( ∂u(ε, ζ) ∂ε ) = U(σ, ϕ, δ) σ − H(u(0, ζ)) σ2 , (12) (ii) WεHζ ( ∂2u(ε, ζ) ∂ε2 ) = U(σ, ϕ, δ) σ2 − H(u(0, ζ)) σ3 − H(uε(0, ζ)) σ2 , (13) (iii) WεHζ ( ∂u(ε, ζ) ∂ζ ) = ϕ δ U(σ, ϕ, δ)−W (u(ε, 0)), (14) (iv) WεHζ ( ∂2u(ε, ζ) ∂ζ2 ) = ϕ2 δ2 U(σ, ϕ, δ)− ϕ δ W (u(ε, 0))−W (uζ(ε, 0)), (15) M. Al-Momani et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6890 6 of 15 (v) WεHζ ( ∂2u(ε, ζ) ∂ε∂ζ ) = ϕ σδ U(σ, ϕ, δ)− 1 σ W (u(ε, 0))− ϕ σ2δ H(u(0, ζ))+ 1 σ2 u(0, 0). (16) Proof. (1) WεHζ ( ∂u(ε,ζ) ∂ε ) = 1 σ2 ∞∫ 0 ∞∫ 0 e− ε σ −ϕζ δ ∂u(ε,ζ) ∂ε dεdζ = 1 σ2 ∞∫ 0 e− ϕζ δ ∞∫ 0 e− ε σ ∂u(ε,ζ) ∂ε dεdζ. By integrating by parts, we get WεHζ ( ∂u(ε,ζ) ∂ε ) = 1 σ2 ∞∫ 0 e− ϕζ δ ( −u(0, ζ) + 1 σ ∞∫ 0 e− ε σ u(ε, ζ) dε ) dζ = − 1 σ2 ∞∫ 0 e− ϕζ δ u(0, ζ)dζ + 1 σ2 × 1 σ ∞∫ 0 ∞∫ 0 e− ε σ −ϕζ δ u(ε, ζ) dεdζ = U(σ,ϕ,δ) σ − H(u(0,ζ)) σ2 . (2) WεHζ ( ∂2u(ε,ζ) ∂ε2 ) = 1 σ2 ∞∫ 0 ∞∫ 0 e− ε σ −ϕζ δ ∂2u(ε,ζ) ∂ε2 dεdζ = 1 σ2 ∞∫ 0 e− ϕζ δ ∞∫ 0 e− ε σ ∂2u(ε,ζ) ∂ε2 dεdζ. By integrating by parts, we get WεHζ ( ∂2u(ε,ζ) ∂ε2 ) = 1 σ2 ∞∫ 0 e− ϕζ δ ( −uε(0, ζ)− 1 σu(0, ζ) + 1 σ2 ∞∫ 0 e− ε σ u(ε, ζ)dε ) dζ = − 1 σ2 ∞∫ 0 e− ϕζ δ uε(0, ζ)dζ − 1 σ3 ∞∫ 0 e− ϕζ δ u(0, ζ)dζ + 1 σ2 × 1 σ2 ∞∫ 0 ∞∫ 0 e− ε σ −ϕζ δ u(ε, ζ)dεdζ = U(σ,ϕ,δ) σ2 − H(u(0,ζ)) σ3 − H(uε(0,ζ)) σ2 . (3) WεHζ ( ∂u(ε,ζ) ∂ζ ) = 1 σ2 ∞∫ 0 ∞∫ 0 e− ε σ −ϕζ δ ∂u(ε,ζ) ∂ζ dεdζ = 1 σ2 ∞∫ 0 e− ε σ ∞∫ 0 e− ϕζ δ ∂u(ε,ζ) ∂ζ dζdε. By integrating by parts, we get WεHζ ( ∂u(ε,ζ) ∂ζ ) = 1 σ2 ∞∫ 0 e− ε σ ( −u(ε, 0) + ϕ δ ∞∫ 0 e− ϕζ δ u(ε, ζ)dζ ) dε = − 1 σ2 ∞∫ 0 e− ε σ u(ε, 0)dε+ ϕ δ × 1 σ2 ∞∫ 0 ∞∫ 0 e− ε σ −ϕζ δ u(ε, ζ) dζdε = ϕ δU(σ, ϕ, δ)−W (u(ε, 0)). (4) WεHζ ( ∂2u(ε,ζ) ∂ζ2 ) = 1 σ2 ∞∫ 0 ∞∫ 0 e− ε σ −ϕζ δ ∂2u(ε,ζ) ∂ζ2 dεdζ = 1 σ2 ∞∫ 0 e− ε σ ∞∫ 0 e− ϕζ δ ∂2u(ε,ζ) ∂ζ2 dζdε. By integrating by parts, we get M. Al-Momani et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6890 7 of 15 WεHζ ( ∂2u(ε,ζ) ∂ζ2 ) = 1 σ2 ∞∫ 0 e− ε σ ( −uζ(ε, 0)− ϕ δ u(ε, 0) + ϕ2 δ2 ∞∫ 0 e− ϕζ δ u(ε, ζ)dζ ) dε = − 1 σ2 ∞∫ 0 e− ε σ uζ(ε, 0)dε− ϕ δ × 1 σ2 ∞∫ 0 e− ε σ u(ε, 0)dε+ ϕ2 δ2 × 1 σ2 ∞∫ 0 ∞∫ 0 e− ε σ −ϕζ δ u(ε, ζ)dζdε So, WεHζ ( ∂2u(ε,ζ) ∂ζ2 ) = ϕ2 δ2 U(σ, ϕ, δ)− ϕ δW (u(ε, 0))−W (uζ(ε, 0)). (5) WεHζ ( ∂2u(ε,ζ) ∂ε∂ζ ) = 1 σ2 ∞∫ 0 ∞∫ 0 e− ε σ −ϕζ δ ∂2u(ε,ζ) ∂ε∂ζ dεdζ = 1 σ2 ∞∫ 0 e− ϕζ δ ∞∫ 0 e− ε σ ∂2u(ε,ζ) ∂ε∂ζ dεdζ By integrating by parts, we get WεHζ ( ∂2u(ε,ζ) ∂ε∂ζ ) = 1 σ2 ∞∫ 0 e− ϕζ δ ( −uζ(0, ζ) + 1 σ ∞∫ 0 e− ε σ uζ(ε, ζ) dε ) dζ = − 1 σ2 ∞∫ 0 e− ϕζ δ uζ(0, ζ)dζ + 1 σ2 × 1 σ ∞∫ 0 ∞∫ 0 e− ε σ −ϕζ δ uζ(ε, ζ)dεdζ = − 1 σ2H(uζ(0, ζ)) + 1 σWεHζ (uζ(ε, ζ)) Using Equations 9 and 14, we get WεHζ ( ∂2u(ε,ζ) ∂ε∂ζ ) = ϕ σδU(σ, ϕ, δ)− 1 σW (u(ε, 0))− ϕ σ2δ H(u(0, ζ)) + 1 σ2u(0, 0). 4.2. Convolution Theorem of DSW-SHT The Heaviside unit step function M(ε, ζ) defined as M(ε− β, ζ − γ) = { 1, ε > β and ζ > γ 0, otherwise Then we have the following lemma Lemma 1. WεHζ(u(ε− β, ζ − γ)M(ε− β, ζ − γ)) = e− β σ −ϕγ δ WεHζ(u(ε, ζ) Proof. We have WεHζ(u(ε− β, ζ − γ)M(ε− β, ζ − γ)) = 1 σ2 ∞∫ 0 ∞∫ 0 e− ε σ −ϕζ δ u(ε− β, ζ − γ)M(ε− β, ζ − γ)dεdζ = 1 σ2 ∞∫ β ∞∫ γ e− ε σ −ϕζ δ u(ε− β, ζ − γ)dεdζ. (17) M. Al-Momani et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6890 8 of 15 Now, by making the substitution s = ε− β and r = ζ − γ, equation (17) becomes: WεHζ(u(ε− β, ζ − γ)M(ε− β, ζ − γ)) = 1 σ2 ∞∫ 0 ∞∫ 0 e− (s+β) σ −ϕ(r+γ) δ u(s, r)dsdr = e− β σ −ϕγ δ WεHζ(u(ε, ζ)). Definition 4. Let u(ε, ζ) and p(ε, ζ) be continuous functions. We define the convolution in the DSW-SHT as (u ∗ ∗p)(ε, ζ) = ε∫ 0 ζ∫ 0 u(ε− β, ζ − γ)p(β, γ))dβdγ. In the following theorem, we compute DSW-SHT of the convolution of two functions Theorem 2. Let U(σ, ϕ, δ) = WεHζ(u(ε, ζ)) and P (σ, ϕ, δ) = WεHζ(p(ε, ζ)). Then WεHζ((u ∗ ∗p)(ε, ζ)) = σ2U(σ, ϕ, δ)P (σ, ϕ, δ). Proof. WεHζ((u∗∗p)(ε, ζ)) = 1 σ2 ∞∫ 0 ∞∫ 0 e− ε σ −ϕζ δ (u ∗ ∗p)(ε, ζ)dεdζ = 1 σ2 ∞∫ 0 ∞∫ 0 e− ε σ −ϕζ δ  ε∫ 0 ζ∫ 0 u(ε− β, ζ − γ)p(β, γ))dβdγ  dεdζ. (18) Using the Heaviside unit step function, we can write equation (18) as WεHζ((u∗∗p)(ε, ζ)) = 1 σ2 ∞∫ 0 ∞∫ 0 e− ε σ −ϕζ δ ∞∫ 0 ∞∫ 0 u(ε− β, ζ − γ)M(ε− β, ζ − γ)p(β, γ))dβdγ  dεdζ = ∞∫ 0 ∞∫ 0 p(β, γ)  1 σ2 ∞∫ 0 ∞∫ 0 e− ε σ −ϕζ δ u(ε− β, ζ − γ)M(ε− β, ζ − γ)dεdζ  dβdγ So by Lemma 1, we have WεHζ((u ∗ ∗p)(ε, ζ)) = U(σ, ϕ, δ) ∞∫ 0 ∞∫ 0 p(β, γ)e− β σ −ϕγ δ dβdγ M. Al-Momani et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6890 9 of 15 = σ2U(σ, ϕ, δ)P (σ, ϕ, δ). In Table 1, we have the DSW-SHT of some basic functions Table 1: Table of the Double Sawi-Shehu Transform u(ε, ζ) WεHζ(u(ε, ζ)) w(ε)x(ζ) W (w(ε))H(x(ζ)) 1 δ σϕ , Re(σ) > 0 εβζγ σβ−1δγ+1 ϕγ+1 Γ(β + 1)Γ(γ + 1), Re(σ) > 0 and Re(β) > −1 eβε+γζ δ σ(1−σβ)(ϕ−γδ) , Re( 1σ ) > Re(β) ei(βε+γζ) δ σ(1−iσβ)(ϕ−iγδ) , Im(β) + Re( 1σ ) > 0 sin (βε+ γζ) δ(σϕβ+δγ) σ(1+σ2β2)(ϕ2+γ2δ2) , |Im(β)| < Re( 1σ ) cos (βε+ γζ) δ(ϕ−σδβγ) σ(1+σ2β2)(ϕ2+γ2δ2) , |Im(β)| < Re( 1σ ) sinh (βε+ γζ) δ(σϕβ+δγ) σ(σ2β2−1)(ϕ2−γ2δ2) , Re( 1σ ) > Re(β) and Re( 1σ + β) > 0 cosh (βε+ γζ) δ(ϕ+σδβγ) σ(σ2β2−1)(ϕ2−γ2δ2) , Re( 1σ ) > Re(β) and Re( 1σ + β) > 0 J0 ( c √ εζ ) 4δ σ(4ϕ+c2σδ) , Re ( 1 σ + c2δ 4ϕ ) > 0 u(ε− β, ζ − γ)M(ε− β, ζ − γ) e− β σ −ϕγ δ WεHζ(u(ε, ζ)) (u ∗ ∗p)(ε, ζ) σ2WεHζ(u(ε, ζ))WεHζ(p(ε, ζ)) 5. Applications In this section, we use the DSW-SHT for solving PDEs and Integro PDEs Example 1. Consider the Advection-Diffusion equation uζ + uεε = uε + 2, where ε, ζ ≥ 0, (19) With initial conditions(ICs) u(ε, 0) = −eε, and boundary conditions(BCs) u (0, ζ) = 2ζ − 1, uε (0, ζ) = −1. Solution 1. By applying the single Sawi transform to the ICs and the single Shehu trans- form to the BCs, we get W (u(ε, 0)) = −1 σ(1−σ) , H (u (0, ζ)) = 2δ2 ϕ2 − δ ϕ , H (uε (0, ζ)) = − δ ϕ . Apply the DSW-SHT to Equation 19, we get ϕ δ U(σ, ϕ, δ)−W (u(ε, 0)) + U(σ, ϕ, δ) σ2 M. Al-Momani et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6890 10 of 15 −H(u(0, ζ)) σ3 − H(uε(0, ζ)) σ2 = U(σ, ϕ, δ) σ − H(u(0, ζ)) σ2 + 2δ σϕ . So, σ2ϕ+ δ − σδ σ2δ × U(σ, ϕ, δ) = −1 σ (1− σ) + 1 σ3 × ( 2δ2 ϕ2 − δ ϕ ) − δ2 ϕ − 1 σ2 × ( 2δ2 ϕ2 − δ ϕ ) + 2δ σϕ . By simplifying, we get, U(σ, ϕ, δ) = 2δ2 σϕ2 − δ σ (1− σ)ϕ . Therefore, u(ε, ζ) = W−1 ε H−1 ζ ( 2δ2 σϕ2 − δ σ (1− σ)ϕ ) = 2ζ − eε. Its graph is Figure 1: The solution of Example 1 M. Al-Momani et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6890 11 of 15 Example 2. Consider the telegraph equation uεε − uε + uζζ = u(ε, ζ), where ε, ζ ≥ 0, (20) With ICs u(ε, 0) = 0, uζ(ε, 0) = e2ε, and BCs u (0, ζ) = sin ζ, uε (0, ζ) = 2 sin ζ. Solution 2. By applying the single Sawi transform and the single Shehu transform to the ICs, we get W (u(ε, 0)) = 0, W (uζ(ε, 0)) = 1 σ(1−2σ) , H (u (0, ζ)) = δ2 ϕ2+δ2 , H (uε (0, ζ)) = 2δ2 ϕ2+δ2 . Apply the DSW-SHT to Equation 20, we get U(σ, ϕ, δ) σ2 − H(u(0, ζ)) σ3 − H(uε(0, ζ)) σ2 − U(σ, ϕ, δ) σ + H(u(0, ζ)) σ2 + ϕ2 δ2 U(σ, ϕ, δ)− ϕ δ W (u(ε, 0)) −W (uζ(ε, 0)) = U. So, δ2 − σδ2 + σ2ϕ2 − σ2δ2 σ2δ2 × U(σ, ϕ, δ) = δ2 σ3 (ϕ2 + δ2) + δ2 σ2 (ϕ2 + δ2) + 1 σ (1− 2σ) . By simplifying, we get, U(σ, ϕ, δ) = δ2 σ (1− 2σ) (ϕ2 + δ2) . Therefore, u(ε, ζ) = W−1 ε H−1 ζ ( δ2 σ (1− 2σ) (ϕ2 + δ2) ) = e2ε sin ζ. Its graph is M. Al-Momani et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6890 12 of 15 Figure 2: The solution of Example 2 Example 3. Consider the equation of Volterra Integro PDE. uε + uζ − cosh ε cos ζ + eε sin ζ − sin ζ = ε∫ 0 ζ∫ 0 u(γ, δ))dγdδ, where ε, ζ ≥ 0, (21) With ICs u(ε, 0) = sinh ε, u(0, ζ) = 0. Solution 3. By applying the single Sawi transform and the single Shehu transform to the ICs, we get W (u(ε, 0)) = 1 1−σ2 , H (u (0, ζ)) = 0. By Definition 4 and Theorem 2, we have ε∫ 0 ζ∫ 0 u(γ, δ))dγdδ = (1 ∗ ∗u) (ε, ζ) . (22) Apply the DSW-SHT to Equation 21, we get U(σ, ϕ, δ) σ − H(u(0, ζ)) σ2 + ϕ δ U(σ, ϕ, δ) M. Al-Momani et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6890 13 of 15 −W (u(ε, 0))− δϕ ϕ2 + δ2 × 1 σ (1− σ2) + δ2 ϕ2 + δ2 × 1 σ (1− σ) − 1 σ × δ2 ϕ2 + δ2 = σ2 × δ σϕ × U(σ, ϕ, δ). So, δϕ+ σϕ2 − σ2δ2 σδ × U(σ, ϕ, δ) = 1 1− σ2 + δϕ σ (1− σ2) (ϕ2 + δ2) − δ2 σ (1− σ) (ϕ2 + δ2) + δ2 σ (ϕ2 + δ2) . By simplifying, we get, U(σ, ϕ, δ) = δϕ (1− σ2) (ϕ2 + δ2) . Therefore, u(ε, ζ) = W−1 ε H−1 ζ ( δϕ (1− σ2) (ϕ2 + δ2) ) = sinh ε cos ζ. Its graph is Figure 3: The solution of Example 3 M. Al-Momani et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6890 14 of 15 6. Conclusion This paper introduced DSW-SHT and explored its main properties while establishing the conditions needed for its existence. The results showed that this hybrid double trans- form can serve as a powerful tool in convolution theory and in dealing with derivative operations. The theoretical framework that has been developed confirms the robustness of the transform and demonstrates that it can be applied to a wide range of mathematical problems. The study also connected the proposed transform with earlier numerical procedures and results from related works to underline its practical relevance. Through these connections it became clear that the DSW-SHT does not only extend the family of integral transforms but also provides more flexibility in analyzing equations that are otherwise difficult to handle. The advantages of the DSW-SHT appear in its ability to simplify calculations, unify different approaches, and improve the analysis of complex models. Looking forward, the DSW-SHT can be a basis for further research directions. It shows strong potential in the study of fractional and conformable partial differential equations and in integro-partial differential equations that involve variable coefficients. These areas remain rich with open problems where new approaches are still needed. We believe that the extension of the DSW-SHT to conformable PDEs and its applications in other branches of applied mathematics will lead to deeper insights and more effective methods for solving challenging equations. Future studies may also focus on numerical implementations and computational aspects of the transform to test its efficiency in real applications. Another promising direction is to investigate how the DSW-SHT interacts with other transforms and whether hybrid structures can be created to address specialized problems. Such efforts will strengthen the role of the DSW-SHT in both theoretical and applied mathematics and confirm its place as a valuable tool for ongoing and future research. Further developments and applications in this field, including extensions to conformable PDEs, are available in [11–13]. References [1] M. Mahgoub and M. Mohand. The sawi transform: A new integral transform. Ad- vances in Theoretical and Applied Mathematics, 14(1):81–87, 2019. [2] S. Maitam and W. Zhao. The shehu transform: A generalization of sumudu and laplace transform for solving differential equations. International Journal of Analysis and Applications, 17(2):167–190, 2019. [3] A. Aghili and B. Parsa Moghaddam. Certain theorems on two-dimensional laplace transform and non-homogeneous parabolic partial differential equations. 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Symmetry, 15(4):921, 2023. [9] B. Abughazaleh, M. A. Amleh, A. Al-Natoor, and R. Saadeh. Double mellin-ara transform. In Springer Proceedings in Mathematics and Statistics, volume 466, pages 383–394, 2024. [10] R. Abu Awwad, M. Al-Momani, B. Abughazaleh, A. Jaradat, and A. Farah. The double sumudu-sawi transform. European Journal of Pure and Applied Mathematics, 18(2):5967, 2025. [11] R. Abu Awwad, M. Al-Momani, B. Abughazaleh, A. Jaradat, and A. Farah. The conformable double laplace-sawi transform. European Journal of Pure and Applied Mathematics, 18(2):6034, 2025. [12] M. Al-Momani, A. Jaradat, B. Abughazaleh, and A. Farah. Solving partial differential equations via the conformable double ara-sawi transform. European Journal of Pure and Applied Mathematics, 18(2):6099, 2025. [13] M. Al-Momani and B. Abughazaleh. The conformable double sumudu-shehu trans- form and its properties with applications. European Journal of Pure and Applied Mathematics, 18(4):6384, 2025.