EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 2, Article Number 5898 ISSN 1307-5543 – ejpam.com Published by New York Business Global Solving Partial Differential Equations Via The Double Sumudu-Shehu 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 introduces a new double hybrid transform yielding single integral trans- forms and their generalizations. The main purpose of this study is to propose the most common form for generalized transformations in terms of Hybrid Sumudu and Shehu transforms. In this paper, we introduce a just invented transform and research its basic characteristics such as exis- tence, inversion, along with related theorems. The study also introduces novel results with respect to partials and generalizes the double convolution theorem. Furthermore, it uses the developed properties and theorems to solve specific kinds of differential equations that have very important applications in physics and science. The purpose of this research is to show the applicability and efficiency of a novel transform in solving differential equations with multiple variable to solve. 2020 Mathematics Subject Classifications: 44A05 Key Words and Phrases: Sumudu transform, Shehu transform, The Double Sumudu-Shehu transform. 1. Introduction Integral transforms are a class of mathematical operators that map functions from one space to another through the process of integration. These transforms simplify the manipulation of certain properties of the original functions by moving them to a new functional space. After transformation the function can be changed back to its original space using inverse of integral transformation. They are an integral part of physics, chemistry, engineering and economy since they help in modeling real world phenomena. Thus, mathematicians keep coming up with new techniques to solve a more and more wider group of differential equations. ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i2.5898 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), 5898 2 of 18 Integral transformations are known for their effectiveness and simplicity, particularly when applied to differential equations with initial or boundary conditions. They simplify the process by converting differential equations, reducing the complexity from derivative operations to algebraic ones. By carefully selecting the appropriate integral transforma- tion, it becomes easier to manage not only the derivatives in intricate differential equations but also the boundary conditions, leading to a form of the equation that is more straight- forward to solve. One of the most well-known transforms is the Laplace transform, which was introduced in 1780. It is used in various fields such as science and engineering. In 1993, the Sumudu transform was defined by [1]. More recently, the Shehu transform was introduced by [2] in 2019, which represents a generalization of the Laplace and Sumudu transforms. For additional details on the Shehu transform. Additionally, Double transforms have been defined for solving differential equations involving more than one variable. Examples of Double transforms include the Double Laplace transform [3], the Double Sumudu transform [4], Double Mellin-ARA Transform [5], for more details about integral transform see [6], [7], [8], [9], [10] and [11]. In this research, we define the Double Sumudu-Shehu transform(DSHT). We explore its proper- ties, including the conditions for its existence, linearity. The study employs this hybrid transform across various fundamental functions, revealing its potential in the realms of convolution theory and derivative operations. We also apply the Double Sumudu-Shehu transform to solve partial differential equations. 2. Sumudu and Shehu transforms This section provides a brief overview and fundamental properties of the single trans- forms: Sumudu, and Shehu transforms. 2.1. Sumudu transform Definition 1. For a continuous function r(τ) defined on (0,∞), the Sumudu is defined as follows: R(κ) = S(r(τ)) = 1 κ ∞∫ 0 e− τ κ r(τ)dτ, κ ∈ C. Here, we present some fundamental properties of the Sumudu transform. Let R(κ) = S(r(τ)), then for nonzero constants β and γ, we have S(βr1(τ) + γr2(τ)) = βS(r1(τ)) + γS(r2(τ)), (1) where r1(τ) and r2(τ) are continuous functions on (0,∞). M. Al-Momani et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5898 3 of 18 S(τβ) = Γ(β + 1)κβ (2) S(eβτ ) = 1 1− κβ , β ∈ R (3) S(r′(τ)) = R(κ) κ − r(0) κ (4) S(r′′(τ)) = R(κ) κ2 − r(0) κ2 − r′(0) κ . (5) 2.2. The Shehu transform Definition 2. For a continuous function t(υ) defined on (0,∞), the Shehu is defined as follows: T (λ, µ) = H(t(υ)) = ∞∫ 0 e −λυ µ t(υ)dυ. We now outline the fundamental properties of the Shehu transform. Suppose that T1(λ, µ) = H(t1(υ)) and T2(λ, µ) = H(t2(υ)), and β and γ are nonzero real numbers, then the following properties hold: H(βt1(υ) + γt2(υ)) = βH(t1(υ)) + γH(t2(υ)) (6) H(υβ) = Γ(β + 1) (µ λ )β+1 (7) H(eγυ) = µ λ− γµ (8) H(t′(υ)) = λ µ T (λ, µ)− t(0) (9) H(t′′(υ)) = λ2 µ2 T (λ, µ)− λ µ t(0)− t′(0). (10) 3. The Double Sumudu-Shehu transform This section introduces DSHT, a novel mathematical tool combining the Sumudu and Shehu transforms. It outlines its core properties linearity, invertibility, and behavior with partial derivatives and establishes a dedicated convolution theorem. Practical examples demonstrate its application to fundamental functions. M. Al-Momani et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5898 4 of 18 The DSHT transform is defined as follows: Q(κ, λ, µ) = SτHυ(q(τ, υ)) = 1 κ ∞∫ 0 ∞∫ 0 e − τ κ −λυ µ q(τ, υ) dτdυ, (11) where q(τ, υ) is a continuous function on (0,∞)× (0,∞). If q(τ, υ) can be written as q(τ, υ) = w(τ)z(υ) for some continuous functions w and z, then SτHυ(q(τ, υ)) = S(w(τ))H(z(υ)). In fact SτHυ(q(τ, υ)) = SτHυ(w(τ)z(υ)) = 1 κ ∞∫ 0 ∞∫ 0 e − τ κ −λυ µ w(τ)z(υ)dτdυ = 1 κ ∞∫ 0 e− τ κw(τ)dτ ∞∫ 0 e −λυ µ z(υ)dυ  = S(w(τ))H(z(υ)). 3.1. The DSHT for some basic functions (i) SτHυ(1) = 1 κ ∞∫ 0 ∞∫ 0 e − τ κ −λυ µ dτdυ = 1 κ ∞∫ 0 e− τ κ ∞∫ 0 e −λυ µ dυ  = 1× µ λ = µ λ , Re(κ) > 0. (ii) SτHυ(τ βυγ) = 1 κ ∞∫ 0 ∞∫ 0 e − τ κ −λυ µ τβυγdτdυ = 1 κ ∞∫ 0 τβe− τ κdτ  ∞∫ 0 υγe −λυ µ dυ  = Γ(β + 1)κβ × Γ(γ + 1) (µ λ )γ+1 = κβµγ+1 λγ+1 Γ(β + 1)Γ(γ + 1), Re(κ) > 0 and Re(β) > −1. M. Al-Momani et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5898 5 of 18 (iii) SτHυ(e βτ+γυ) = 1 κ ∞∫ 0 ∞∫ 0 e − τ κ −λυ µ eβτ+γυdτdυ = 1 κ ∞∫ 0 eβτ− τ κdτ ∞∫ 0 e γυ−λυ µ dυ  = 1 1− κβ × µ λ− γµ = µ (1− κβ) (λ− γµ) , Re( 1 κ ) > Re(β). 3.2. Existence condition for the DSHT Definition 3. A function q(τ, υ) is said to be of exponential orders β and γ on 0 ≤ τ < ∞ and 0 ≤ υ < ∞. If there exist B,X, Y > 0 such that |q(τ, υ)| ≤ Beβτ+γυ, for all τ > X, υ > Y. Theorem 1. Let q(τ, υ) be a continuous function on the region [0,∞) × [0,∞) of expo- nential orders β and γ. Then Q(κ, λ, µ) exists for κ, λ and µ whenever Re( 1κ) > β and Re ( λ µ ) > γ. Proof. |Q(κ, λ, µ)| = ∣∣∣∣∣∣1κ ∞∫ 0 ∞∫ 0 e − τ κ −λυ µ q(τ, υ) dτdυ ∣∣∣∣∣∣ ≤ 1 κ ∞∫ 0 ∞∫ 0 e − τ κ −λυ µ |q(τ, υ)| dτdυ ≤ B κ ∞∫ 0 ∞∫ 0 e − τ κ −λυ µ eβτ+γυdτdυ = B κ ∞∫ 0 e−( 1 κ −β)τdτ ∞∫ 0 e −(λ µ −γ)υ dυ = B κ( 1κ − β)(λµ − γ) = Bµ (1− κβ)(λ− γ µ) where Re( 1κ) > β and Re ( λ µ ) > γ. 3.3. Linearity The transform SτHυ(q(τ, υ)) exhibits linearity. For any nonzero constants β and γ, this property is expressed as: SτHυ(βq1(τ, υ)+γq2(τ, υ)) M. Al-Momani et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5898 6 of 18 = 1 κ ∞∫ 0 ∞∫ 0 e − τ κ −λυ µ (βq1(τ, υ) + γq2(τ, υ)) dτdυ, = β × 1 κ ∞∫ 0 ∞∫ 0 e − τ κ −λυ µ q1(τ, υ) dτdυ + γ × 1 κ ∞∫ 0 ∞∫ 0 e − τ κ −λυ µ q2(τ, υ) dτdυ = βSτHυ(q1(τ, υ)) + γSτHυ(q2(τ, υ)). 4. Properties of the DSHT In this section, we explore the fundamental properties of the DSHT 4.1. Derivatives properties Let Q(κ, λ, µ) = SτHυ(q(τ, υ)). Then (i) SτHυ ( ∂q(τ, υ) ∂τ ) = Q(κ, λ, µ) κ − H(q(0, υ)) κ (12) (ii) SτHυ ( ∂2q(τ, υ) ∂τ2 ) = Q(κ, λ, µ) κ2 − H(q(0, υ)) κ2 − H(qτ (0, υ)) κ (13) (iii) SτHυ ( ∂q(τ, υ) ∂υ ) = λ µ Q(κ, λ, µ)− S(q(τ, 0)) (14) (iv) SτHυ ( ∂2q(τ, υ) ∂υ2 ) = λ2 µ2 Q(κ, λ, µ)− λ µ S(q(τ, 0))− S(qυ(τ, 0)) (15) (v) SτHυ ( ∂2q(τ, υ) ∂τ∂υ ) = λ κµ Q(κ, λ, µ)− 1 κ S(q(τ, 0))− λ κµ H(q(0, υ)) + 1 κ q(0, 0) (16) Proof. (1) SτHυ ( ∂q(τ,υ) ∂τ ) = 1 κ ∞∫ 0 ∞∫ 0 e − τ κ −λυ µ ∂q(τ,υ) ∂τ dτdυ = 1 κ ∞∫ 0 e −λυ µ ∞∫ 0 e− τ κ ∂q(τ,υ) ∂τ dτdυ. By integrating by parts, we get SτHυ ( ∂q(τ,υ) ∂τ ) = 1 κ ∞∫ 0 e −λυ µ ( −q(0, υ) + 1 κ ∞∫ 0 e− τ κ q(τ, υ) dτ ) dυ M. Al-Momani et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5898 7 of 18 = − 1 κ ∞∫ 0 e −λυ µ q(0, υ)dυ + 1 κ × 1 κ ∞∫ 0 ∞∫ 0 e − τ κ −λυ µ q(τ, υ) dτdυ = Q(κ,λ,µ) κ − H(q(0,υ)) κ . (2) SτHυ ( ∂2q(τ,υ) ∂τ2 ) = 1 κ ∞∫ 0 ∞∫ 0 e − τ κ −λυ µ ∂2q(τ,υ) ∂τ2 dτdυ = 1 κ ∞∫ 0 e −λυ µ ∞∫ 0 e− τ κ ∂2q(τ,υ) ∂τ2 dτdυ. By integrating by parts, we get SτHυ ( ∂2q(τ,υ) ∂τ2 ) = 1 κ ∞∫ 0 e −λυ µ ( −qτ (0, υ)− 1 κq(0, υ) + 1 κ2 ∞∫ 0 e− τ κ q(τ, υ)dτ ) dυ = − 1 κ ∞∫ 0 e −λυ µ qτ (0, υ)dυ − 1 κ2 ∞∫ 0 e −λυ µ q(0, υ)dυ + 1 κ2 × 1 κ ∞∫ 0 ∞∫ 0 e − τ κ −λυ µ q(τ, υ)dτdυ = Q(κ,λ,µ) κ2 − H(q(0,υ)) κ2 − H(qτ (0,υ)) κ . (3) SτHυ ( ∂q(τ,υ) ∂υ ) = 1 κ ∞∫ 0 ∞∫ 0 e − τ κ −λυ µ ∂q(τ,υ) ∂υ dτdυ = 1 κ ∞∫ 0 e− τ κ ∞∫ 0 e −λυ µ ∂q(τ,υ) ∂υ dυdτ. By integrating by parts, we get SτHυ ( ∂q(τ,υ) ∂υ ) = 1 κ ∞∫ 0 e− τ κ ( −q(τ, 0) + λ µ ∞∫ 0 e −λυ µ q(τ, υ)dυ ) dτ = − 1 κ ∞∫ 0 e− τ κ q(τ, 0)dτ + λ µ × 1 κ ∞∫ 0 ∞∫ 0 e − τ κ −λυ µ q(τ, υ) dυdτ = λ µQ(κ, λ, µ)− S(q(τ, 0)). (4) SτHυ ( ∂2q(τ,υ) ∂υ2 ) = 1 κ ∞∫ 0 ∞∫ 0 e − τ κ −λυ µ ∂2q(τ,υ) ∂υ2 dτdυ = 1 κ ∞∫ 0 e− τ κ ∞∫ 0 e −λυ µ ∂2q(τ,υ) ∂υ2 dυdτ. By integrating by parts, we get SτHυ ( ∂2q(τ,υ) ∂υ2 ) = 1 κ ∞∫ 0 e− τ κ ( −qυ(τ, 0)− λ µq(τ, 0) + λ2 µ2 ∞∫ 0 e −λυ µ q(τ, υ)dυ ) dτ = − 1 κ ∞∫ 0 e− τ κ qυ(τ, 0)dτ − λ µ × 1 κ ∞∫ 0 e− τ κ q(τ, 0)dτ + λ2 µ2 × 1 κ ∞∫ 0 ∞∫ 0 e − τ κ −λυ µ q(τ, υ)dυdτ So, SτHυ ( ∂2q(τ,υ) ∂υ2 ) = λ2 µ2Q(κ, λ, µ)− λ µS(q(τ, 0))− S(qυ(τ, 0)). (5) SτHυ ( ∂2q(τ,υ) ∂τ∂υ ) = 1 κ ∞∫ 0 ∞∫ 0 e − τ κ −λυ µ ∂2q(τ,υ) ∂τ∂υ dτdυ = 1 κ ∞∫ 0 e −λυ µ ∞∫ 0 e− τ κ ∂2q(τ,υ) ∂τ∂υ dτdυ By integrating by parts, we get M. Al-Momani et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5898 8 of 18 SτHυ ( ∂2q(τ,υ) ∂τ∂υ ) = 1 κ ∞∫ 0 e −λυ µ ( −qυ(0, υ) + 1 κ ∞∫ 0 e− τ κ qυ(τ, υ) dτ ) dυ = − ∞ 1 κ ∫ 0 e −λυ µ qυ(0, υ)dυ + 1 κ × 1 κ ∞∫ 0 ∞∫ 0 e − τ κ −λυ µ qυ(τ, υ)dτdυ = − 1 κH(qυ(0, υ)) + 1 κSτHυ (qυ(τ, υ)) Using Equations 9 and 14, we get SτHυ ( ∂2q(τ,υ) ∂τ∂υ ) = λ κµQ(κ, λ, µ)− 1 κS(q(τ, 0))− λ κµH(q(0, υ)) + 1 κq(0, 0). 4.2. Convolution Theorem of the DSHT The Heaviside unit step function M(τ, υ) is defined as M(τ − β, υ − γ) = { 1, τ > β and υ > γ 0, otherwise Then we have the following lemma Lemma 1. SτHυ(q(τ − β, υ − γ)M(τ − β, υ − γ)) = e −β κ −λγ µ SτHυ(q(τ, υ) Proof. We have SτHυ(q(τ − β, υ − γ)M(τ − β, υ − γ)) = 1 κ ∞∫ 0 ∞∫ 0 e − τ κ −λυ µ q(τ − β, υ − γ)M(τ − β, υ − γ)dτdυ = 1 κ ∞∫ β ∞∫ γ e − τ κ −λυ µ q(τ − β, υ − γ)dτdυ. (17) Now, by making the substitution s = τ − β and r = υ − γ, equation (17) becomes: SτHυ(q(τ − β, υ − γ)M(τ − β, υ − γ)) = 1 κ ∞∫ 0 ∞∫ 0 e − (s+β) κ −λ(r+γ) µ q(s, r)dsdr = e −β κ −λγ µ SτHυ(q(τ, υ)). M. Al-Momani et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5898 9 of 18 Definition 4. Let q(τ, υ) and p(τ, υ) be continuous functions. We define the convolution in the DSHT as (q ∗ ∗p)(τ, υ) = τ∫ 0 υ∫ 0 q(τ − β, υ − γ)p(β, γ))dβdγ. The following theorem provides the computation of the DSHT for the convolution of two functions Theorem 2. Let Q(κ, λ, µ) = SτHυ(q(τ, υ)) and P (κ, λ, µ) = SτHυ(p(τ, υ)). Then SτHυ((q ∗ ∗p)(τ, υ)) = κQ(κ, λ, µ)P (κ, λ, µ). Proof. SτHυ((q∗∗p)(τ, υ)) = 1 κ ∞∫ 0 ∞∫ 0 e − τ κ −λυ µ (q ∗ ∗p)(τ, υ)dτdυ = 1 κ ∞∫ 0 ∞∫ 0 e − τ κ −λυ µ  τ∫ 0 υ∫ 0 q(τ − β, υ − γ)p(β, γ))dβdγ  dτdυ. (18) By incorporating the Heaviside unit step function, equation (18) can be rewritten as: SτHυ((q∗∗p)(τ, υ)) = 1 κ ∞∫ 0 ∞∫ 0 e − τ κ −λυ µ ∞∫ 0 ∞∫ 0 q(τ − β, υ − γ)M(τ − β, υ − γ)p(β, γ))dβdγ  dτdυ = ∞∫ 0 ∞∫ 0 p(β, γ) 1 κ ∞∫ 0 ∞∫ 0 e − τ κ −λυ µ q(τ − β, υ − γ)M(τ − β, υ − γ)dτdυ  dβdγ So by Lemma 1, we have SτHυ((q ∗ ∗p)(τ, υ)) = Q(κ, λ, µ) ∞∫ 0 ∞∫ 0 p(β, γ)e −β κ −λγ µ dβdγ = κQ(κ, λ, µ)P (κ, λ, µ). In Table 1, we have the DSHT of some basic functions M. Al-Momani et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5898 10 of 18 Table 1: Table of the DSHT q(τ, υ) SτHυ(q(τ, υ)) w(τ)z(υ) S(w(τ))H(z(υ)) 1 µ λ , Re(κ) > 0 τβυγ κβµγ+1 λγ+1 Γ(β + 1)Γ(γ + 1), Re(κ) > 0 and Re(β) > −1 eβτ+γυ µ (1−κβ)(λ−γµ) , Re( 1κ) > Re(β) ei(βτ+γυ) iµ (i+κβ)(λ−iγµ) , Im(β) + Re( 1κ) > 0 sin (βτ + γυ) µ(κλβ+µγ) (1+κ2β2)(λ2+γ2µ2) , |Im(β)| < Re( 1κ) cos (βτ + γυ) µ(λ−κµβγ) (1+κ2β2)(λ2+γ2µ2) , |Im(β)| < Re( 1κ) sinh (βτ + γυ) µ(κλβ+µγ) (κ2β2−1)(λ2−γ2µ2) , Re( 1κ) > Re(β) and Re( 1κ + β) > 0 cos (βτ + γυ) µ(λ+κµβγ) (κ2β2−1)(λ2−γ2µ2) , Re( 1κ) > Re(β) and Re( 1κ + β) > 0 J0 (c √ τυ) 4µ 4λ+c2κµ , Re ( 1 κ + c2µ 4λ ) > 0 q(τ − β, υ − γ)M(τ − β, υ − γ) e −β κ −λγ µ SτHυ(q(τ, υ)) (q ∗ ∗p)(τ, υ) κSτHυ(q(τ, υ))SτHυ(p(τ, υ)) 5. Applications In this section, we use the DHST for solving PDEs Consider the PDE of the form B1qττ +B2qτυ +B3qυυ +B4qτ +B5qυ +B6q (τ, υ) = k (τ, υ) (19) With ICs q(τ, 0) = r1 (τ), qυ(τ, 0) = r2 (τ) and BCs q (0, υ) = t1 (υ), qτ (0, υ) = t2 (υ) and assuming q (0, 0) = Ψ Given that q (τ, υ) is the unknown function, k (τ, υ) is the source term, and B1, B2, ..., B6 and Ψ are constants, we aim to apply the DHST to Equation (19). To do this, we begin by applying the single Sumudu transform to the ICs and the single Shehu transform to the BCs M. Al-Momani et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5898 11 of 18 S (r1 (τ)) = R1(τ), S (r2 (τ)) = R2(τ), H (t1 (υ)) = T1(υ) and H (t2 (υ)) = T2(υ) By applying the DHST to Equation (19), we have B1SτHυ (qττ ) +B2SτHυ (qτυ) +B3SτHυ (qυυ) +B4SτHυ (qτ ) +B5SτHυ (qυ) +B6SτHυ (q (τ, υ)) = SτHυ (k (τ, υ)) (20) By the properties of the derivatives in Equations (12)− (16), we get B1 ( 1 κ2 Q(κ, λ, µ)− 1 κ2 T1(υ)− 1 κ T2(υ) ) +B2 ( λ κµ Q(κ, λ, µ)− 1 κ R1(τ)− λ µ T1(υ) + Ψ ) +B3 ( λ2 µ2 Q(κ, λ, µ)− λ µ R1(τ)−R2(τ) ) +B4 ( 1 κ Q(κ, λ, µ)− 1 κ T1(υ) ) +B5 ( λ µ Q(κ, λ, µ)Q(κ, λ, µ)−R1(τ) ) +B6Q(κ, λ, µ) = K(κ, λ, µ) (21) Simplify Equation 21 as following Q(κ, λ, µ) =( B1 1 κ2 +B2 λ µ +B4 1 κ ) T1 +B1 1 κT2 + ( B2 1 κ +B3 λ µ +B5 ) R1 +B3R2 −B2Ψ+K B1 1 κ2 +B2 λ κµ +B3 λ2 µ2 +B4 1 κ +B5 λ µ +B6 (22) Example 1. Consider the heat equation qττ = 2qυ − 3q(τ, υ) + 3, where τ, υ ≥ 0 With IC q(τ, 0) = 1− 2 sin τ and BCs q (0, υ) = 1, qτ (0, υ) = −2eυ Solution 1. By applying the single Sumudu transform to the IC and the single Shehu transform to the BCs, we get R1 = 1− 2κ 1+κ2 , T1 = µ λ , T2 = −2µ λ−µ and K(κ, λ, µ) = SτHυ (3) = 3µ λ M. Al-Momani et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5898 12 of 18 Substitute in Equation (22) B1 = 1, B5 = −2, B6 = 3, B2 = B3 = B4 = 0 and the values of R1, T1, T2 and K, we get Q(κ, λ, µ) = µ κ2λ − 2µ κ(λ−µ) − 2 + 4κ 1+κ2 + 3µ λ 1 κ2 − 2λ µ + 3 = 3κ2µ−2κ2λ+µ κ2λ − 2(3κ2µ−2κ2λ+µ) κ(1+κ2)(λ−µ) 3κ2µ−2κ2λ+µ κ2µ = µ λ − 2κµ (1 + κ2) (λ− µ) So, q(τ, υ) = S−1 τ H−1 υ ( µ λ − 2κµ (1 + κ2) (λ− µ) ) = 1− 2eυ sin τ The graph of the exact solution is Figure 1: The solution q(τ, υ) of Example 1 M. Al-Momani et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5898 13 of 18 Example 2. Consider the Advection-Diffusion equation qυ = qττ − 2qτ , where τ, υ ≥ 0 With IC q(τ, 0) = e2τ − τ and BCs q (0, υ) = 2υ + 1, qτ (0, υ) = 1 Solution 2. By applying the single Sumudu transform to the IC and the single Shehu transform to the BCs, we get R1 = 1 1−2κ − κ, T1 = 2µ2 λ2 + µ λ , T2 = µ λ Substitute in Equation (22) B1 = 1, B4 = −2, B5 = −1, B2 = B3 = B6 = 0 and the values of R1, T1 and T2, we get Q(κ, λ, µ) = −4κµ2+λµ−2κλµ+2µ2 κ2λ2 + µ κλ − 1 1−2κ + κ 1 κ2 − 2 κ − λ µ By simplify, Q(κ, λ, µ) = 2µ2 λ2 + µ (1− 2κ)λ − κµ λ q(τ, υ) = S−1 τ H−1 υ ( 2µ2 λ2 + µ (1− 2κ)λ − κµ λ ) = υ + e2τ − τ The graph of the exact solution is M. Al-Momani et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5898 14 of 18 Figure 2: The solution q(τ, υ) of Example 2 Example 3. Consider the Klein-Gordon equation qττ − qυυ − 4q(τ, υ) = −2 sinh τ cos υ With ICs q(τ, 0) = sinh τ , qυ(τ, 0) = 0 and BCs q (0, υ) = 0, qτ (0, υ) = cos υ Solution 3. By applying the single Sumudu transform to the ICs and the single Shehu transform to the BCs, we get R1 = κ 1−κ2 , R2 = 0, T1 = 0, T2 = λµ λ2+µ2 and K(κ, λ, µ) = SτHυ (−2 sinh τ cos υ) = −2κλµ (1−κ2)(λ2+µ2) Substitute in Equation (22) B1 = 1, B3 = −1, B6 = −4, B2 = B4 = B5 = 0 and the values of R1, R2, T1, T2 and K, we get Q(κ, λ, µ) = λµ κ(λ2+µ2) − κλ (1−κ2)µ − 2κλµ (1−κ2)(λ2+µ2) 1 κ2 − λ2 µ2 − 4 M. Al-Momani et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5898 15 of 18 = λ(µ2−κ2λ2−4κ2µ2) κµ(1−κ2)(λ2+µ2) µ2−κ2λ2−4κ2µ2 κ2µ2 = κλµ (1− κ2) (λ2 + µ2) So, q(τ, υ) = S−1 τ H−1 υ ( κλµ (1− κ2) (λ2 + µ2) ) = sinh τ cos υ The graph of the exact solution is Figure 3: The solution q(τ, υ) of Example 3 Example 4. Consider the telegraph equation qττ = qυυ − 2qυ − q(τ, υ), where τ, υ ≥ 0 With ICs q(τ, 0) = sin τ, qυ(τ, 0) = 2 sin τ and BCs q (0, υ) = 0, qτ (0, υ) = e2υ M. Al-Momani et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5898 16 of 18 Solution 4. By applying the single Sumudu transform to the ICs and the single Shehu transform to the BCs, we get R1 = κ 1+κ2 , R2 = 2κ 1+κ2 , T1 = 0, T2 = µ λ−2µ Substitute in Equation (22) B1 = 1, B3 = −1, B5 = 2, B6 = 1, B2 = B4 = 0 and the values of R1, R2, T1 and T2, we get Q(κ, λ, µ) = µ κ(λ−2µ) − κλ (1+κ2)µ + 2κ 1+κ2 − 2κ 1+κ2 1 κ2 − λ2 µ2 + 2λ µ + 1 = κ2µ2−κ2λ2+2κ2λµ+µ2 κ(1+κ2)(λ−2µ)µ κ2µ2−κ2λ2+2κ2λµ+µ2 κ2µ2 = κµ (1 + κ2) (λ− 2µ) So, q(τ, υ) = S−1 τ H−1 υ ( κµ (1 + κ2) (λ− 2µ) ) = sin τe2υ The graph of the exact solution is Figure 4: The solution q(τ, υ) of Example 4 M. Al-Momani et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5898 17 of 18 6. Conclusion In this research, we introduce a novel approach termed the DSHT (Double Sumudu- Shehu Transform), offering a fresh perspective in the field of mathematical analysis. We explore the fundamental properties of this innovative double transform and demonstrate its application in solving partial differential equations and integral equations. 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