EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 17, No. 1, 2024, 385-409 ISSN 1307-5543 – ejpam.com Published by New York Business Global The Mohanad Transforms and Their Applications for Solving Systems of Differential Equations Rania Saadeh1, Al-anoud Alshawabkeh1,Raed Khalil2, Mohamed A. Abdoon3,4, Nidal E. Taha5, Dalal Khalid Almutairi6,∗ 1 Department of Mathematics, Faculty of Science, Jordan , Zarqa University , Zarqa 13110, Jordan 2 Faculty of Computer Information, Salt 19117, Al-Balqa’ Applied University, Salt 19117, Jordan 3 Department of Basic Sciences, Common First Year Deanship, King Saud University, Riyadh 12373, Saudi Arabia 4 Department of Mathematics, Faculty of Science, Bakht Al-Ruda University, Duwaym 28812, Sudan 5 Department of Mathematics, Faculty of Science Arts (Dhariah), Qassim University, Qassim, KSA 6 Department of Mathematics, College of Education (Majmaah), Majmaah University, P.O.Box 66, Al-5 Majmaah, 11952, Saudi Arabia Abstract. In recent years, Mohanad transform, a mathematical approach, has drawn a lot of interest from researchers. It is useful for solving many engineering and scientific problems, such as those involving electric circuits, population growth, vibrational beams, and heat conduction. The Mohanad transform is defined and introduced in this study, along with its fundamental qualities, including linearity and convolution. It is also discussed in connection with other integral transforms and how it is used in derivatives. Additionally, we use the Mohanad transform to solve a few systems of ordinary differential equations (ODEs) and review its properties in this paper. Determining the concentration of a chemical reactant (material) in a series is a physical chemistry problem that we use in the application part. We achieve this by developing a model based on ordinary differential equations (ODEs) and then solving them using the Mohanad transform. This research proves that, with little computational effort, we can get the exact solutions of ordinary differential equations (ODEs) via the Mohanad transform. We used graphs and tables to show our answer. 2020 Mathematics Subject Classifications: 97M40 Key Words and Phrases: Mohanad transform, Laplace transform, Convolution, System of differential equations ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v17i1.5005 Email addresses: rsaadeh@zu.edu. (Saadeh),20219299@zu.edu.jo ( Alshawabkeh), mabdoon.c@ksu.edu.sa ( A. Abdoon) n.taha@qu.edu.sa (N Taha), dk.almutairi@mu.edu.sa (D. K. Almutairi) https://www.ejpam.com 385 © 2024 EJPAM All rights reserved. D. K. Almutairi et al. / Eur. J. Pure Appl. Math, 17 (1) (2024), 385-409 386 1. Introduction In actuality, mathematics is a universal language and a vital resource for comprehend- ing the world we live in [57]. Particularly, differential equations are essential to many areas of mathematics and its applications [35]. They give us the ability to explain the connections between variables and offer mathematical models that help us comprehend scientific, engineering, natural, and economic events [19]. In physics, differential equations are frequently used to describe the motion of objects and the changes in dynamic systems [16]. They have the ability to foresee and analyze the behavior of a wide range of physical systems, including celestial planets and subatomic particles. In engineering, differential equations are utilized to solve problems related to dynamics, structural analysis, and de- sign. They provide the mathematical framework required to erect frameworks, optimize processes, and ensure system stability [25, 44]. Furthermore, industrial design, control systems, and electrical engineering all heavily rely on differential equations [46, 59]. They are necessary for industrial processes, control sys- tems, and electrical circuit analysis and design. Engineers can maximize the efficiency and stability of these systems by creating differential equations that characterize their behav- ior. Differential equations are fundamental to many branches of science and technology [30, 46, 54]. It is used for expressing relationships, creating models, and resolving issues in physics, engineering, economics, the natural sciences and many others fields, they offer a potent mathematical tool for handling these issues and its explanation [26]. By converting functions from one domain to another, integral transformations are strong mathematical tools that help us in solving problems more easily and creatively in variety of domains [20]. For example, Fourier transform [21]. is a transformation that breaks down a function into its frequency components, allowing for analysis in the frequency domain. It was solved by many researchers to get the solutions of many differential equations [17, 55] The Laplace transform [18] is another significant integral transformation that transforms a function of time into a function of complex frequency, facilitating the solution of differ- ential equations and system analysis. It has a great application in the fields of science and engineering [33, 56]. The Sumudu transform [37, 58] offered an alternative to the conventional Fourier trans- form and attracted attention for its exceptional capacity to accurately capture transient signals and non-stationary phenomena [14]. Numerous domains, such as pattern recog- nition, biomedical signal processing, image de-noising, and others wares the motivation for providing many integral transforms later, for instances: ARA integral transform [49], Aboodh transform [1], and Formable transform [52]. Moreover, double integral transforms have been effective in solving differential equations such as: double Laplace transform [22], double Laplace ARA [42, 53], double formable [51] double ARA [45] and others [7, 17, 48]. They are essential in many scientific and technological fields, improving our comprehen- sion and facilitating effective problem-solving [10, 23, 38–40]. Developed by Mohand [37], M.T. is a revolutionary mathematical technique that has at- tracted a lot of attention recently. This transform provides a strong tool for signal process- ing and analysis. Applications including data encryption, audio compression, and image D. K. Almutairi et al. / Eur. J. Pure Appl. Math, 17 (1) (2024), 385-409 387 processing benefit greatly from the transform. Signals can be transformed mathemati- cally from the time domain to the frequency domain and back again using the Mohandas approach. By analyzing signals and comprehending their many properties, including the frequencies contained in the signal and the energy contained in each frequency, this trans- formation is used [2]. The Laplace transform is the foundation of the M.T. approach, however it is enhanced and modified in some ways [3]. When handling non-infinite signals or signals with restricted frequency, M.T. performs better [5, 6, 8]. A wide range of prob- lems involving differential equations, partial differential equations, integral equations, and population development and decay are among the many uses of M.T. [24, 43]. It is also used to solve linear Volterra integral equations [4, 50] of the second sort. The novelty in this research lies in the utilization of the Mohanad transform, a relatively recent mathematical approach, as a powerful tool for addressing a broad spectrum of sci- entific and engineering problems. The distinctive aspects of this study can be highlighted as follows: The study contributes to the evolving interest in the Mohanad transform, showcasing its versatility across diverse domains such as electric circuits, population growth, vibrational beams, and heat conduction. By exploring its application in varied fields, the research expands the understanding of where and how this transform can be effectively employed. Comprehensive Analysis: This work not only introduces the Mohanad transform but also delves into its foundational properties—highlighting linearity, convolution, and its con- nections with other integral transforms. The comprehensive analysis provides a robust understanding of the transform’s capabilities and its relation to established mathematical tools. Solver for ODEs: The emphasis on solving systems of ordinary differential equations (ODEs) using the Mohanad transform is a key contribution. Demonstrating its efficacy in solving these equations with minimal computational burden underscores its potential as an efficient solver for complex mathematical problems arising in various scientific and engineering disciplines. Cross-disciplinary Application: The application of the Mohanad transform to determine chemical reactant concentrations in a series reaction bridges the gap between mathemat- ics and physical chemistry. This cross-disciplinary application showcases the transform’s ability to address real-world chemical problems, demonstrating its practical utility beyond theoretical constructs. Efficiency and Precision: The research underscores the transformative power of the Mo- hanad transform by showcasing its ability to yield exact solutions to ODEs without re- quiring extensive computational efforts. The use of graphs and tables to present these solutions further emphasizes the precision and ease of interpretation of the results ob- tained. In essence, the novelty of this research lies in the comprehensive exploration and practical application of the Mohanad transform across various scientific and engineering realms, showcasing its efficiency, precision, and applicability in solving real-world problems. Additionally; we go over how to solve systems of ODEs with the suggested transform by a simple algorithm, we discuss a physical chemistry problem model and solve it using D. K. Almutairi et al. / Eur. J. Pure Appl. Math, 17 (1) (2024), 385-409 388 M.T. the obtained results are used to examine the concentration of chemical reactants in a series of chemical reactions of reactants in a series of reactions. To obtain the numeri- cal findings and create the figures, and create the figures, we utilize the Python software package. The novelty of this work is obvious in the new algorithm presented to solve systems of ODEs[14, 28, 47], M.T. is presented for the first time to solve these equations. The proposed method shows its simplicity and applicability to solve problems in physical chemistry[9, 15]. The utilization of the Mohanad transform (M.T) in solving a chemical application, specif- ically in determining chemical reactant concentrations within a series reaction, can be attributed to several reasons: Complexity of Chemical Reactions: Chemical reactions, especially in series, can involve intricate kinetics and complex equations describing the change in concentrations over time. Oftentimes, these reactions lead to systems of ordinary differential equations (ODEs) that are challenging to solve directly. The M.T, known for its efficacy in solving ODEs, provides an alternative and efficient method to address these complexities. Mathematical Modeling of Chemical Systems: When trying to understand chemical reac- tions, it’s common to model them using differential equations. These equations describe how the concentrations of reactants change over time. By utilizing the M.T, researchers can translate these models into solvable equations, enabling a deeper understanding of the reaction kinetics and concentrations involved. Accuracy and Efficiency: The M.T offers an advantage in providing exact solutions to ODEs without necessitating extensive computational efforts. Its application streamlines the process of solving these equations, making it an appealing choice when dealing with chemical systems, where precise solutions are crucial. Visualization and Interpretation: Presenting the solutions in tables and graphs, as men- tioned in the study, is beneficial for visualizing and interpreting the concentration changes of reactants over time. This graphical representation aids in comprehending the behavior of the chemical system and allows for clearer communication of results. Broader Applicability: The versatility of the M.T extends beyond specific scientific fields. Its application in solving ODEs allows for a cross-disciplinary approach, enabling re- searchers from various domains, such as mathematics, physics, engineering, and chemistry, to collaborate and leverage this mathematical tool for problem-solving. The choice of utilizing the M.T to solve a chemical application involving the determination of chemical reactant concentrations in a series reaction showcases its ability to simplify complex systems, offer accurate solutions, and facilitate a deeper understanding of chem- ical kinetics through mathematical modeling and analysis. The structure of this article is as follows: The structure of this article is as follows: the primary definitions and characteristics of M.T. are presented in Section 2. In Section 3, the applications of M.T. to solve ODEs. Section 4 presents a chemical application of system of ODEs that is solved by the proposed method, finally, we get the conclusion in Section 5. D. K. Almutairi et al. / Eur. J. Pure Appl. Math, 17 (1) (2024), 385-409 389 2. Basics about M.T. In this section, we present the definition of M.T. [1, 49], and some basic properties and relations to other integral transforms. For more details about M.T.. Definition 1. The Mohanad transform (M.T.) is defined by the integral equation: M{φ(ξ)} = ζ2 ∫ ∞ 0 e−ζξφ(ξ)dξ = Φ(ζ). (1) Definition 2. The inverse M.T. is given by: φ(ξ) = M−1{Φ(ζ)} = 1 2πi ∫ c+i∞ c−i∞ 1 ζ2 eζξΦ(ζ)dζ, c ∈ R, (2) where M−1 is the inverse operator of M.T.. 2.1. Properties of M.T. Some properties of M.T. are presented in this section. (i) The linearity property of M.T. If M {φ1(ξ)} = Φ1(ζ) and M {φ2(ξ)} = Φ2(ζ) then M {qφ1(ξ) + pφ2(ξ)} = qΦ1(ζ) + pΦ2(ζ), where q and p are constants. Proof. By the definition 1 of M.T., we obtain M {qφ1(ξ) + pφ2(ξ)} = ζ2 ∫ ∞ 0 e−ζξ [qφ2(ξ) + pφ2(ξ)] dξ (3) = ζ2 ∫ ∞ 0 e−ζξqφ1(ξ)dξ + ζ2 ∫ ∞ 0 e−ζξpφ2(ξ)dξ = qζ2 ∫ ∞ 0 e−ζξφ1(ξ)dξ + pζ2 ∫ ∞ 0 e−ζξφ2(ξ)dξ = qΦ1(ζ) + pΦ2(ζ). (4) Moreover, the inverse M.T. is linear; If M−1 {Φ1(ζ)} = φ1(ξ) and M−1 {Φ2(ζ)} = φ2(ξ). (5) Then, M−1 {qΦ1(ζ) + pΦ2(ζ)} = qM−1 {Φ1(ζ)}+ pM−1 {Φ1(ζ)} = qφ1(ξ) + pφ2(ξ) (6) D. K. Almutairi et al. / Eur. J. Pure Appl. Math, 17 (1) (2024), 385-409 390 (ii) Change of scale property If M.T. of a function φ(ξ) is Φ(ζ), then M.T. of the function φ(aξ) is given by aΦ ( ζ a ) , where a is non zero constant. Proof. By the definition 1 of M.T., we get M{φ(aξ)} = ζ2 ∫ ∞ 0 e−ζξφ(aξ)dξ. (7) Letting aξ = u, then adξ = du, then we substitute in equation (7). M{φ(aξ)} = ζ2 ∫ ∞ 0 e−( ζ a)uφ(u) du a = a [ ζ2 a2 ∫ ∞ 0 e−( ζ a)uφ(u)du ] = aΦ ( ζ a ) . (8) (iii) Shifting property of M.T. If M.T. of a function φ(ξ) is Φ(ζ), then M.T. of the function ekξφ(ξ) is given by ζ2 (ζ − k) Φ(ζ − k), where k ∈ R. Proof. By the definition 1 of M.T., we get M { ekξφ(ξ) } = ζ2 ∫ ∞ 0 e−ζξekξφ(ξ)dξ = ζ2 ∫ ∞ 0 e−(ζ−k)ξφ(ξ)dξ = (ζ −K)2 (ζ −K)2 ζ2 ∫ ∞ 0 e−(ζ−k)ξφ(ξ)dξ = ζ2 (ζ −K) Φ(ζ − k). (9) Now, we introduce the M.T. of some basic functions in Table 1. 2.2. Relations to other integral transforms In this section, we discuss the duality between M.T. and some other famous transforms. • Laplace transform If L{φ(ξ)} = ∫∞ 0 e−ζξφ(ξ)dξ is the Laplace transform of φ(ξ), then M{φ(ξ)} = ζ2L{φ(ξ)}. Proof. M{φ(ξ)} = ζ2 [∫ ∞ 0 e−ζξφ(ξ)dξ ] = ζ2L{φ(ξ)}. (10) D. K. Almutairi et al. / Eur. J. Pure Appl. Math, 17 (1) (2024), 385-409 391 Table 1: M.T. of some elementary functions. Functions φ(ξ) M{φ(ξ)} = Φ(ζ) 1 ζ ξ 1 ξ2 2! ζ ξα, α > 0 Γ(α+1) ζα−1 eαξ ζ2 ζ−α sin(αξ) αζ2 ζ2+α2 cos(αξ) ζ3 ζ2+α2 sinh(αξ) αζ2 ζ2−α2 cosh(αξ) ζ3 ζ2−α2 • Sumudu transform If S{φ(ξ)} = 1 ζ ∫∞ 0 e − ξ ζφ(ξ)dξ is the Sumudu transform of φ(ξ), then M{φ(ξ)} = 1 ζ S{φ(ξ)}. Proof. M{φ(ξ)} = ζ2 ∫ ∞ 0 e−ζξφ(ξ)dξ = Φ(ζ). Moreover, Φ ( 1 ζ ) = 1 ζ2 ∫ ∞ 0 e − 1 ζ ξ φ(ξ)dξ = 1 ζ [ 1 ζ ∫ ∞ 0 e −ξ ( 1 ζ ) φ(ξ)dξ ] = 1 ζ S{φ(ξ)}. (11) • Aboodh transform If A{φ(ξ)} = 1 ζ ∫∞ 0 e−ζξφ(ξ)dξ is the Aboodh transform of φ(ξ), then M{φ(ξ)} = ζ3A{φ(ξ)}. Proof. M{φ(ξ)} = ζ2 ∫ ∞ 0 e−ζξφ(ξ)dξ = Φ(ζ). Moreover, M{φ(ξ)} = ζ3 [ 1 ζ ∫ ∞ 0 e−ξζφ(ξ)dξ ] = ζ3A{φ(ξ)} (12) • Formable transform D. K. Almutairi et al. / Eur. J. Pure Appl. Math, 17 (1) (2024), 385-409 392 If B(ζ, u) = ζ ∫∞ 0 e−ζξφ(uξ)dξ is the Formable transform of φ(uξ), then M{φ(ξ)} = ζB(ζ, 1). Proof. M{φ(ξ)} = ζ2 ∫ ∞ 0 e−ζξφ(ξ)dξ = ζ [ ζ ∫ ∞ 0 e−ζξφ(ξ)dξ ] = ζB(ζ, 1) (13) • ARA transform If G(n, ζ) = ζ ∫∞ 0 ξn−1e−ζξφ(ξ)dξ is the ARA transform of φ(ξ), then M{φ(ξ)} = ζG { ξn−1φ(ξ) } Proof. M{φ(ξ)} = ζ2 ∫ ∞ 0 e−ζtφ(ξ)dξ = ζ [ ζ ∫ ∞ 0 e−ζξξn−1φ(ξ)dξ ] = ζG { ξn−1φ(ξ) } (14) 2.3. M.T. for derivatives If M{φ(ξ)} = Φ(ζ), then (i) M { φ′(ξ) } = ζΦ(ζ)− ζ2φ(0). (15) (ii) M { φ′′(ξ) } = ζ2Φ(ζ)− ζ3φ(0)− ζ2φ′(0). (16) (iii) M { φ(n)(ξ) } = ζnΦ(ζ)− n−1∑ k=0 ζn−k+1φ(k)(0). (17) Proof. To proof (i), by definition 1 of the M.T., we have M { φ′(ξ) } = ζ2 ∫ ∞ 0 e−ζξφ′(ξ)dξ. (18) Using integration by parts, we have u = e−ζξ and dv = φ′(ξ), du = −ζe−ζξ, v = φ(ξ). Then the equation (18) becomes, M { φ′(ξ) } = ζΦ(ζ)− ζ2φ(0). (19) To proof (ii), we use the definition 1 of M.T., M {φ′′(ξ)} = M { (φ′(ξ))′ } . D. K. Almutairi et al. / Eur. J. Pure Appl. Math, 17 (1) (2024), 385-409 393 Using part (i). M { φ′′(ξ) } = ζM { φ′(ξ) } − ζ2φ′(0) = ζ [ ζΦ(ζ)− ζ2φ(0) ] − ζ2φ′(0) = ζ2Φ(ζ)− ζ3φ(0)− ζ2φ′(0). (20) Proof (iii). By induction, for n = 1 its true from part (i). Now, assume that the equation (17) is true for n = k, then M { φ(k)(ξ) } = ζkΦ(ζ)− ζk+1φ(0)− ζkφ′(0)− · · · − ζ2φ(k−1)(0). (21) Now, we prove it for n = k + 1, thus M { φ(k+1)(ξ) } = M {( φ(k)(ξ) )′ } . Using part (i), we get M { φ(k+1)(ξ) } = ζk [ ζΦ(ζ)− ζ2φ(0) ] − ζk+1φ′(0)− ζkφ′′(0)− · · · − ζ2φ(k)(0) = ζnΦ(ζ)− n−1∑ k=0 ζn−k+1φ(k)(0). (22) 3. M.T. for solving system of ODEs In this part, we solve systems of ordinary differential equations by applying M.T.. Consider the system of first order of ODEs: dφ1 dξ = h11φ1(ξ) + h12φ2(ξ) + h13φ3(ξ) + · · ·+ h1nφn(ξ) + ρ1(ξ), dφ2 dξ = h21φ1(ξ) + h22φ2(ξ) + h23φ3(ξ) + · · ·+ h2nφn(ξ) + ρ2(ξ), · · dφn dξ = hn1φ1(ξ) + hn2φ2(ξ) + hn3φ3(ξ) + · · ·+ hnnφn(ξ) + ρn(ξ).  (23) with the initial conditions: φ1(0) = b1, φ2(0) = b2, . . . , φn(0) = bn, (24) where h11, h12, h13, . . . , hnn are constants, and φ1(ξ), φ2(ξ), . . . , φn(ξ) are unknown continuous functions, and ρ1(ξ), ρ2(ξ), . . . , ρn(ξ) are known continuous functions. The matrix representation of system (23) with (24) is D. K. Almutairi et al. / Eur. J. Pure Appl. Math, 17 (1) (2024), 385-409 394 dΦ dξ = Hφ(ξ) + ρ(ξ), with φ(0) = B, (25) where dφ dξ =  dφ1 dξ dφ2 dξ ... dφn dξ  , H =  h11 h12 · · · h1n h21 h22 · · · h2n ... ... . . . ... hn1 hn2 · · · hnn  , φ(ξ) =  φ1(ξ) φ2(ξ) ... φn(ξ)  , ρ(ξ) =  ρ1(ξ) ρ2(ξ) ... ρn(ξ)  , φ(0) =  φ1(0) φ2(0) ... φn(0)  and B =  b1 b2 ... bn  . By applying M.T. to system (23), we have M {φ′ 1(ξ)} = h11M {φ1(ξ)}+ h12M {φ2(ξ)}+ · · ·+ h1nM {φn(ξ)}+M {ρ1(ξ)} , M {φ′ 2(ξ)} = h21M {φ1(ξ)}+ h22M {φ2(ξ)}+ · · ·+ h2nM {φn(ξ)}+M {ρ2(ξ)} , · · M {φ′ n(ξ)} = hn1M {φ1(ξ)}+ hn2M {φ2(ξ)}+ · · ·+ hnnM {φn(ξ)}+M {ρn(ξ)} .  (26) Then, we get ζM {φ1(ξ)} −ζ2φ1(0) = h11M {φ1(ξ)}+ h12M {φ2(ξ)}+ · · ·+ h1nM {φn(ξ)}+M {ρ1(ξ)} , ζM {φ2(ξ)} −ζ2φ2(0) = h21M {φ1(ξ)}+ h22M {φ2(ξ)}+ · · ·+ h2nM {φn(ξ)}+M {ρ2(ξ)} , · · ζM {φn(ξ)} −ζ2φn(0) = hn1M {φ1(ξ)}+ hn2M {φ2(ξ)}+ · · ·+ hnnM {φn(ξ)}+M {ρn(ξ)} .  (27) Then, using the initial conditions (24), the system (27) becomes (ζ − h11)Mρ {φ1(ξ)} − h12M {φ2(ξ)} − · · · − h1nM {φn(ξ)} = M {ρ1(ξ)}+ b1ζ 2, −h21M {φ1(ξ)}+ (ζ − h22)M {φ2(ξ)} − · · · − h1nM {φn(ξ)} = M {ρ2(ξ)}+ b2ζ 2 · · −hn1M {φn(ξ)} − hn2M {φ2(ξ)} − · · ·+ (ζ − hnn)M {φn(ξ)} = M {ρn(ξ)}+ bnζ 2  (28) Then, the solution of system (26) can be obtained using Cramer’s rule as D. K. Almutairi et al. / Eur. J. Pure Appl. Math, 17 (1) (2024), 385-409 395 M {φ1(ξ)} = ∣∣∣∣∣∣∣∣∣∣∣∣ M {ρ1(ξ)}+ b1ζ 2 −h12 · · · −h1n M {ρ2(ξ)}+ b2ζ 2 (ζ − h22) · · · −h2n ... ... . . . ... M{ρ(ξ)}+ bnζ 2 −hnn · · · (ζ − hnn) ∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣ (ζ − h11) −h12 · · · −h1n −h21 (ζ − h22) · · · −h2n ... ... . . . ... −hn1 −hn2 · · · (ζ − hnn) ∣∣∣∣∣∣∣∣∣∣∣∣ M {φ2(ξ)} = ∣∣∣∣∣∣∣∣∣∣∣∣ (ζ − h11) M {ρ1(ξ)}+ b1ζ 2 · · · −h1n −h21 M {ρ2(ξ)}+ b2ζ 2 · · · −h2n ... ... . . . ... −hn1 M {ρn(ξ)}+ bnζ 2 · · · (ζ − hnn) ∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣ (ζ − h11) −h12 · · · −h1n −h21 (ζ − h22) · · · −h2n ... ... . . . ... −hn1 −hn2 · · · (ζ − hnn) ∣∣∣∣∣∣∣∣∣∣∣∣ ... M {φn(ξ)} = ∣∣∣∣∣∣∣∣∣∣∣∣ (ζ − h11) −h1n · · · M {ρ1(ξ)}+ b1ζ 2 −h21 (ζ − h22) · · · M {ρ2(ξ)}+ b2ζ 2 ... ... . . . ... −hn1 −hn2 · · · M {ρn(ξ)}+ bnζ 2 ∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣ (ζ − h11) −h12 · · · −h1n −h21 (ζ − h22) · · · −h2n ... ... . . . ... −hn1 −hn2 · · · (ζ − hnn) . ∣∣∣∣∣∣∣∣∣∣∣∣ Now, applying the inverse M.T. of M {φ1(ξ)} , M {φ2(ξ)} , . . . , M {φn(ξ)}, then we get the values of φ1(ξ), φ2(ξ), . . . , φn(ξ). Now, we introduce some examples of systems ODEs and solve them by M.T.. Example 1. Consider the following system of ODEs dφ1 dξ = φ3(ξ), dφ2 dξ = −φ3(ξ), dφ3 dξ = −φ1(ξ)− φ2(ξ),  (29) D. K. Almutairi et al. / Eur. J. Pure Appl. Math, 17 (1) (2024), 385-409 396 with the initial conditions: φ1(0) = 0, φ2(0) = 1 and φ3(0) = 0. (30) The matrix form of the system (29) with the initial conditions (30) is given by: dφ dξ = Hφ(ξ) + ρ(ξ), with φ(0) = B, (31) where: dφ dξ =  dφ1 dξ dφ2 dξ dφ3 dξ  , H =  0 0 1 0 0 −1 −1 −1 0  , φ(ξ) =  φ1(ξ) φ2(ξ) φ3(ξ)  , ρ(ξ) =  0 0 0  , φ(0) =  φ1(0) φ2(0) φ3(0)  and B =  0 1 0  . By applying M.T. to the system (29), we get M {φ′ 1(ξ)} −M {φ3(ξ)} = 0, M {φ′ 2(ξ)}+M {φ3(ξ)} = 0, M {φ1(ξ)}+M {φ2(ξ)}+M {φ3 ′(ξ)} = 0.  (32) Operating M.T. on (32) and using the initial condition (30) ζM {φ1(ξ)} − ζ2φ1(0)−M {φ3(ξ)} = 0, ζM {φ2(ξ)} − ζ2φ2(0) +M {φ3(ξ)} = 0, M {φ1(ξ)}+M {φ2(ξ)}+ ζM {φ3(ξ)} − ζ2φ3(0) = 0.  (33) Simplifying the system (33), we obtain ζM {φ1(ξ)} −M {φ3(ξ)} = 0, ζM {φ2(ξ)}+M {φ3(ξ)} = ζ2, M {φ1(ξ)}+M {φ2(ξ)}+ ζM {φ3(ξ)} = 0.  (34) Using Cramer’s rule to solve M {φ1(ξ)} ,M {φ2(ξ)} and M {φ3(ξ)} on the system (34) M {φ1(ξ)} = ∣∣∣∣∣∣∣∣ 0 0 −1 ζ2 ζ 1 0 1 ζ ∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣ ζ 0 −1 0 ζ 1 1 1 ζ ∣∣∣∣∣∣∣∣ = −1 ζ , (35) D. K. Almutairi et al. / Eur. J. Pure Appl. Math, 17 (1) (2024), 385-409 397 M {φ2(ξ)} = ∣∣∣∣∣∣∣∣ ζ 0 −1 0 ζ2 1 1 0 ζ ∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣ ζ 0 −1 0 ζ 1 1 1 ζ ∣∣∣∣∣∣∣∣ = ζ + ( 1 ζ ) , (36) M {φ3(ξ)} = ∣∣∣∣∣∣∣∣ ζ 0 −0 0 ζ ζ2 1 1 0 ∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣ ζ −1 0 ζ 1 1 1 ζ ∣∣∣∣∣∣∣∣ = −1. (37) By applying the inverse M.T. on the equations (35), (36) and (37), then we have φ1(ξ) = M−1 { −1 ζ } = −ξ2 2 , (38) φ2(ξ) = M−1 { ζ + 1 ζ } = 1 + ξ2 2 , (39) φ3(ξ) = M−1{−1} = −ξ. (40) Equations (38), (39) and (40) give the solution of the system (29) with the initial condition (30). 4. Chemical -Physical Application This section of the study includes a chemical-physical application to estimate the concentrations c1, c2 and c3 of three reactants X,Y and Z of a first-order chemical reaction in batches defined that is solved and discussed by the M.T.. reactant X −→ reactant Y −→ reactant Z. dC1 dt = −λ1C1, dC2 dt = λ1C1 − λ2C2, dC3 dt = λ2C2.  (41) with C1(0) = α, C2(0) = 0 and C3(0) = 0. (42) D. K. Almutairi et al. / Eur. J. Pure Appl. Math, 17 (1) (2024), 385-409 398 C1 = C1(t) = Concentration of achemical reactant X at time t, C2 = C2(t) = Concentration of achemical reactant Y at time t, C3 = C3(t) = Concentration of achemical reactant Z at time t, where λ1, λ2 = rate constant > 0. C1(0) = α = The initial concentration of achemical reactant X, C2(0) = 0 = The initial concentration of achemical reactant Y, C3(0) = 0 = The initial concentration of achemical reactant Z.  The matrix form of the system (41) with the initial conditions (42): dC dt = HC(t) + ρ(t), withC(0) = B, (43) where: dC dt =  dC1 dt dC2 dt dC3 dt  , H =  −λ1 0 0 λ1 −λ2 0 0 λ2 0  , C(t) =  C1(t) C2(t) C3(t)  , ρ(t) =  0 0 0  , C(0) =  C1(0) C2(0) C3(0)  , and B =  α 0 0  . By applying M.T. on the system (41), we get: M {C ′ 1(t)}+ λ1M {C1(t)} = 0, M {C ′ 2(t)} − λ1M {C1(t)}+ λ2M {C2(t)} = 0, M {C ′ 3(t)} − λ2M {C2(t)} = 0.  (44) Operating M.T. to (44) and using the initial conditions (42) ζM {C1(t)} − ζ2C1(0) + λ1M {C1(t)} = 0, ζM {C2(t)} − ζ2C2(0) + λ2M {C2(t)} − λ1M {C1(t)} = 0, ζM {C3(t)} − ζ2C3(0)− λ2M {C2(t)} = 0.  (45) Simplifying the system (45), we obtain (ζ + λ1)M {C1(t)} = ζ2α, (ζ + λ2)M {C2(t)} − λ1M {C1(t)} = 0, ζM {C3(t)} − λ2M {C2(t)} = 0.  (46) Using Cramer’s rule to solve M {C1(t)} ,M {C2(t)} and M {C3(t)} on the system (46) D. K. Almutairi et al. / Eur. J. Pure Appl. Math, 17 (1) (2024), 385-409 399 M {C1(t)} = ∣∣∣∣∣∣∣∣ ζ2α 0 0 0 ζ + λ2 0 0 −λ2 ζ ∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣ ζ + λ1 0 0 −λ1 ζ + λ2 0 0 −λ2 ζ ∣∣∣∣∣∣∣∣ = ζ3α(ζ+λ2) ζ(ζ+λ2)(ζ+λ1) = α ( ζ2 (ζ+λ1) ) , (47) M {C2(t)} = ∣∣∣∣∣∣∣∣ ζ + λ1 ζ2α 0 −λ1 0 0 0 0 ζ ∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣ ζ + λ1 0 0 −λ1 ζ + λ2 0 0 −λ2 ζ ∣∣∣∣∣∣∣∣ = ζ3αλ1 ζ(ζ+λ2)(ζ+λ1) = αλ1 ( ζ2 (ζ+λ2)(ζ+λ1) ) , (48) M {C3(t)} = ∣∣∣∣∣∣∣∣ ζ + λ1 0 ζ2α −λ1 ζ + λ2 0 0 −λ2 0 ∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣ ζ + λ1 0 0 −λ1 ζ + λ2 0 0 −λ2 ζ ∣∣∣∣∣∣∣∣ = αλ1λ2 ( ζ (ζ+λ2)(ζ+λ1) ) . (49) By applying the inverse M.T. on the equations (47), (48) and (49) then, we have C1(t) = M−1 { α ( ζ2 (ζ + λ1) )} = αM−1 { ζ2 (ζ + λ1) } = αe−λ1t. (50) C2(t) = M−1 { α ( ζ2 (ζ + λ2) (ζ + λ1) )} = αλ1M −1 { ζ2 (ζ + λ2) (ζ + λ1) } = ( αλ1 λ2 − λ1 )( e−λ1t − e−λ2t ) . (51) C3(t) = M−1 { αλ1λ2 ( ζ (ζ + λ2) (ζ + λ1) )} = α ( M−1{ζ} − λ2 λ2 − λ1 M−1 { λ2 1 ζ + λ1 } + λ1 λ2 − λ1 M−1 { λ2 2 ζ + λ2 }) = α ( 1− ( λ2 λ2 − λ1 ) e−λ1t + ( λ1 λ2 − λ1 ) e−λ2t ) .(52) The values of concentration C1, C2 and C3 conforming to diverse values of time t and for varied combinations of values of α, λ1 and λ2 are specified and displayed in Table 2, and Figure 1 presents graphical representations. Table 2 shows that with the increasing time t from 0 to 7 seconds, the concentration C1(t) of a chemical substance X declines for any combinations of values of α and λ1 namely D. K. Almutairi et al. / Eur. J. Pure Appl. Math, 17 (1) (2024), 385-409 400  α = 1 ( kg/m3 ) , λ1 = 1 ( sec−1 ) , α = 1 ( kg/m3 ) , λ1 = 1.2 ( sec−1 ) , α = 1 ( kg/m3 ) , λ1 = 1.3 ( sec−1 ) ,  where concentration C1(t) of a chemical substance X at time t ( kg/m3 ) . Table 2 further shows that as the value of rate constant λ1 intensifies from 1 to 1.2sec−1 and 1 to 1.3sec−1, the value of the concentration C1(t) of a chemical substance X depletes for time values t stretching from 1 to 5 seconds. What’s more, Table 2 determines that for high values of time, the concentration C1(t) of a chemical substance X transforms into 0 ( kg/m3 ) . The results exhibited in Table 2 are corroborated by the diagram of Figure 1. Table 2. Concentration C1(t) of a chemical substance X at time t of different combi- nations of values α and λ1. t(sec) C1(t) ( kg/m3 ) α = 1 ( kg/m3 ) , λ1 = 1 ( sec−1 ) α = 1 ( kg/m3 ) , λ1 = 1.2 ( sec−1 ) α = 1 ( kg/m3 ) , λ1 = 1.3 ( sec−1 ) 0 1.00 1.00 1.00 1.0 0.37 0.30 0.27 2.0 0.14 0.09 0.07 3.0 0.05 0.03 0.02 4.0 0.02 0.01 0.01 5.0 0.01 0.00 0.00 6.0 0.00 0.00 0.00 7.0 0.00 0.00 0.00 D. K. Almutairi et al. / Eur. J. Pure Appl. Math, 17 (1) (2024), 385-409 401 Figure 1: Concentration C1(t) of a chemical substance X at time t of different combinations of values α and λ1. Table 4 illustrates that the concentration C2(t) of a chemical substance Y decreases as time t increases from 0 to 6 seconds for every combination of values of α, λ1 and λ2. α = 1 ( kg/m3 ) , λ1 = 1.0 ( sec−1 ) , λ2 = 0.5 ( sec−1 ) , α = 1 ( kg/m3 ) , λ1 = 1.0 ( sec−1 ) , λ2 = 1.1 ( sec−1 ) , α = 1 ( kg/m3 ) , λ1 = 1.0 ( sec−1 ) , λ2 = 1.4 ( sec−1 ) , α = 1 ( kg/m3 ) , λ1 = 1.2 ( sec−1 ) , λ2 = 0.5 ( sec−1 ) , α = 1 ( kg/m3 ) , λ1 = 1.2 ( sec−1 ) , λ2 = 1.1 ( sec−1 ) , α = 1 ( kg/m3 ) , λ1 = 1.2 ( sec−1 ) , λ2 = 1.4 ( sec−1 ) , α = 1 ( kg/m3 ) , λ1 = 1.3 ( sec−1 ) , λ2 = 0.5 ( sec−1 ) , α = 1 ( kg/m3 ) , λ1 = 1.3 ( sec−1 ) , λ2 = 1.1 ( sec−1 ) , α = 1 ( kg/m3 ) , λ1 = 1.3 ( sec−1 ) , λ2 = 1.4 ( sec−1 ) .  From the following table, it is evident that when the value of rate constant λ1 raises from 1 to 1.3sec−1, the value of the concentration C2(t) of a chemical substance Y raises initially, and diminishes subsequently as time t increases from 0 to 7 seconds. In addition, this table manifests that since the value of rate constant λ2 goes from 0.5 to 1.4sec−1, the value of a chemical substance’s concentration Y drops for all time t values. Moreover, Table 3. displays that for high values of time t, the concentration C2(t) of a chemical D. K. Almutairi et al. / Eur. J. Pure Appl. Math, 17 (1) (2024), 385-409 402 substance Y becomes 0 kg/m3. The diagram of Figure 2 depicts the similar observations as the Table 3. Table 3 . Concentration C2(t) of a chemical substance Y at time t of different combi- nations of values α, λ1 and λ2. t(sec) C2(t) ( kg/m3 ) α = 1 ( kg/m3 ) , λ1 = 1 ( sec−1 ) α = 1 ( kg/m3 ) , λ1 = 1.2 ( sec−1 ) α = 1 ( kg/m3 ) , λ1 = 1.3 ( sec−1 ) λ2 = 0.5( sec−1 ) λ2 = 1.1( sec−1 ) λ2 = 1.4( sec−1 ) λ2 = 0.5( sec−1 ) λ2 = 1.1( sec−1 ) λ2 = 1.4( sec−1 ) λ2 = 0.5( sec−1 ) λ2 = 1.1( sec−1 ) 0 0.0 0.0 0.00 0.00 0.00 0.00 0.00 0.00 1.0 0.48 0.35 0.29 0.52 0.38 0.33 0.54 0.39 2.0 0.47 0.25 0.17 0.48 0.24 0.18 0.48 0.24 3.0 0.35 0.13 0.08 0.34 0.11 0.07 0.33 0.11 4.0 0.23 0.06 0.03 0.22 0.05 0.03 0.21 0.04 5.0 0.15 0.03 0.01 0.14 0.02 0.01 0.13 0.02 6.0 0.09 0.01 0.00 0.08 0.01 0.00 0.08 0.01 7.0 0.06 0.00 0.00 0.05 0.00 0.00 0.05 0.00 Figure 2: Concentration C2(t) of a chemical substance Y at time t of different combinations of values α, λ1 and λ2. Table 4 reveals how the concentration C3(t) of a chemical substance Z elevates as time elevates from 0 to 6 seconds for all combinations of values of α, λ1 and λ2, namly D. K. Almutairi et al. / Eur. J. Pure Appl. Math, 17 (1) (2024), 385-409 403  α = 1 ( kg/m3 ) , λ1 = 1.0 ( sec−1 ) , λ2 = 0.5 ( sec−1 ) , α = 1 ( kg/m3 ) , λ1 = 1.0 ( sec−1 ) , λ2 = 1.1 ( sec−1 ) , α = 1 ( kg/m3 ) , λ1 = 1.0 ( sec−1 ) , λ2 = 1.4 ( sec−1 ) , α = 1 ( kg/m3 ) , λ1 = 1.2 ( sec−1 ) , λ2 = 0.5 ( sec−1 ) , α = 1 ( kg/m3 ) , λ1 = 1.2 ( sec−1 ) , λ2 = 1.1 ( sec−1 ) , α = 1 ( kg/m3 ) , λ1 = 1.2 ( sec−1 ) , λ2 = 1.4 ( sec−1 ) , α = 1 ( kg/m3 ) , λ1 = 1.3 ( sec−1 ) , λ2 = 0.5 ( sec−1 ) , α = 1 ( kg/m3 ) , λ1 = 1.3 ( sec−1 ) , λ2 = 1.1 ( sec−1 ) , α = 1 ( kg/m3 ) , λ1 = 1.3 ( sec−1 ) , λ2 = 1.4 ( sec−1 ) .  Moreover, this table portrays that as the value of rate constant λ1 advances from 1 to 1.3sec−1, the value of the concentration C3(t) of a chemical substance Z increases for all time t values. In addition, this table unveils that while the value of rate constant λ2 expands from 0.5 to 1.4sec−1, the value of the concentration C3(t) of a chemical substance Z expands for all values of time t. In addition, Table 4 exhibits that for high time t values, the concentration C3(t) of a chemical substance Z develops into 1 kg/m3. The graph sketch in Figure 3 renders the equivalent findings as in Table 4. Table 4. Concentration C3(t) of a chemical substance Z at time t of different combi- nations of values α, λ1 and λ2. t(sec) C3(t) ( kg/m3 ) α = 1 ( kg/m3 ) , λ1 = 1 ( sec−1 ) α = 1 ( kg/m3 ) , λ1 = 1.2 ( sec−1 ) α = 1 ( kg/m3 ) , λ1 = 1.3 ( sec−1 ) λ2 = 0.5( sec−1 ) λ2 = 1.1( sec−1 ) λ2 = 1.4( sec−1 ) λ2 = 0.5( sec−1 ) λ2 = 1.1( sec−1 ) λ2 = 1.4 ( sec −1 ) λ2 = 0.5( sec−1 ) λ2 = 1.1( sec−1 ) 0 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 1.0 0.15 0.28 0.33 0.18 0.32 0.37 0.18 0.34 2.0 0.40 0.62 0.68 0.43 0.67 0.73 0.45 0.69 3.0 0.60 0.82 0.86 0.64 0.86 0.90 0.65 0.87 4.0 0.75 0.92 0.95 0.77 0.94 0.96 0.78 0.95 5.0 0.84 0.97 0.98 0.86 0.98 0.99 0.87 0.98 6.0 0.90 0.99 0.99 0.92 0.99 1.00 0.92 0.99 7.0 0.94 0.99 1.00 0.95 1.00 1.00 0.95 1.00 D. K. Almutairi et al. / Eur. J. Pure Appl. Math, 17 (1) (2024), 385-409 404 Figure 3: Concentration C3(t) of a chemical substance Z at time t of different combinations of values α, λ1 and λ2. 5. Conclusion In this comprehensive research, the Mohanad transform (M.T.) emerged as a pivotal mathematical tool, its versatility and efficiency evident in solving a wide array of scientific and engineering problems. The study showcased the foundational properties of M.T. and its relation to integral transforms, affirming its robustness in tackling complex differential equations. By presenting concrete examples of solving systems of ordinary differential equations (ODEs) and applying M.T. to determine chemical reactant concentrations in a series reaction, this work not only validated its efficacy but also demonstrated its practical utility in physical chemistry. The transformative aspect of M.T. lies in its ability to yield exact solutions with minimal computational load, underscoring its precision and efficiency in problem-solving. The novelty of this research extends beyond the mere introduction of M.T., shedding light on its application across diverse fields such as electric circuits, population growth, vibrational beams, and heat conduction. By delving into its interdis- ciplinary utility, this study broadens the understanding of where and how M.T. can be effectively employed. Moreover, the emphasis on its cross-disciplinary application between mathematics and physical chemistry exemplifies its real-world relevance, surpassing theo- retical constructs to address practical scientific challenges[11–13, 27, 29, 31, 32, 34, 36, 41]. The incorporation of graphs and tables further enhances the precision and interpretabil- REFERENCES 405 ity of results, highlighting M.T.’s practical significance in both academic and industrial settings. 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