PARADIGMA BARU PENDIDIKAN MATEMATIKA DAN APLIKASI ONLINE INTERNET PEMBELAJARAN CONTACT: Muhammad Amin Sadiq Murad muhammad.murad@uod. Department of Mathematics, College of Sciences, University of Duhok, Duhok, Iraq The article can be accessed here. https://doi.org/10.15642/mantik.2022.8.2.124-138 A New analytical Modeling for Fractional Telegraph Equation Arising in Electromagnetic Muhammad Amin Sadiq Murad1*, Mudhafar Hamed Hamadamen2 1*Department of Mathematics, College of Sciences, University of Duhok, Iraq 2Department of Mathematics, College of Education, Salahaddin University, Iraq Article history: Received Oct 26, 2022 Revised Dec 23, 2022 Accepted Dec 31, 2022 Kata Kunci: Persamaan telegraf pecahan, metode iterasi variasi, transformasi integral Elzaki, polinomial He, metode perturbasi homotopi. Abstrak. Pada artikel ini, metode iterasi variasi He (VIM) dan transformasi integral Elzaki diusulkan untuk menyelesaikan persamaan telegraf fraksional linier dan nonlinier yang muncul dalam elektromagnetik. Caputo sense digunakan untuk mendeskripsikan fractional derivatives. Salah satu keuntungan dari teknik ini adalah tidak perlu menghitung pengali Lagrange dengan menghitung integrasi dalam relasi perulangan atau dengan mengambil teorema konvolusi. Selanjutnya, untuk mengurangi istilah komputasi nonlinier, polinomial Adomian diidentifikasi dengan homotopy perturbation method (HPM). Metode yang diusulkan diterapkan pada beberapa contoh persamaan telegraf fraksional linier dan nonlinier. Solusi yang diperoleh dengan teknik komputasi baru menunjukkan bahwa metode ini efisien dan memfasilitasi proses penyelesaian time fractional differential equations. Keywords: Fractional telegraph equations, variation iteration method, Elzaki integral transform, He’s polynomial, homotopy perturbation method. Abstract. In this article, He’s variation iteration method (VIM) and Elzaki integral transform are proposed to analyze the time-fractional telegraph equations arising in electromagnetics. The Caputo sense is used to describe fractional derivatives. One of the advantages of this technique is that there is neither need to compute the Lagrange multiplier by calculating the integration in recurrence relations or via taking the convolution theorem. Further, to decrease nonlinear computational terms, the Adomian polynomial is identified with the homotopy perturbation method (HPM). The proposed method is applied to some examples of linear and nonlinear fractional telegraph equations. The solutions obtained by the new computational technique indicate that this method is efficient and facilitates the process of solving time fractional differential equations. How to cite: M. A. S. Murad and M. H. Hamadamen, “A New analytical Modeling for Fractional Telegraph Equation Arising in Electromagnetic”, J. Mat. Mantik, vol. 8, no. 2, pp. 124-138, December 2022. Jurnal Matematika MANTIK Vol. 8, No. 2, December 2022, pp. 124-138 ISSN: 2527-3159 (print) 2527-3167 (online) http://u.lipi.go.id/1458103791 http://u.lipi.go.id/1457054096 Muhammad Amin Sadiq Murad, Mudhafar Hamed Mamadamen A New Analytical Modeling for Fractional Telegraph Equation Arising in Electromagnetic 125 1. Introduction Differential equations of fractional orders can be used to simulate many scientific disciplines, which improves our understanding of how to characterize natural occurrences in a variety of scientific disciplines like engineering, electronics, biology, business, computer science, and physics. Indeed, in the improvement of fractional calculus, many scientists such as Bernoulli, Liouville, Euler, L’Hopital, and Wallis greatly contributed to this area of research. The numerical solutions are used to investigate the solutions of differential equations of fractional and integer orders, because the exact solutions of differential equations are quite difficult to be found. Telegraph equations have applied to many problems in different fields of science which developed by Heaviside in 1880. The difference and time are described on electric transmissions with current and voltage by telegraph equation, also the proposed equation is applied for investigating the wave propagation in the cable transmission and electric signals, and it is also applied in the field of telephone lines, wireless signals, and radio frequency [1]. Telegraph equations of fractional orders have been solved, using various numerical and analytical methods, the Adomian technique [2], homotopy perturbation technique[3], Laplace decomposition combined with HPM [4], modified Adomian decomposition method (MADM) [5], and reduced differential transform technique [6]. The VIM employed to study the solution of the proposed model and obtained the same result as obtained by (ADM) with fewer computations [7], and the hyperbolic telegraph equation is analyzed by Chebyshev tau technique [8]. The researcher Inokuti was the first who study the VIM [9][10], while the Lagrange multiplier was difficult to be identified. Then, variation iteration method developed by Chinese mathematician He [11], and was applied by many researchers, see [12][13][14][15]. The homotopy perturbation method (HPM) is another crucial method which is employed to solve PDEs [16][17][18]. The solution of Voltera-Fredhom is studied by HPM[19], also the hyperbolic PDEs and many other PDEs were solved by HPM, see [20][21][22]. In the last decade, different methods have been developed to analyze the solution of PDEs of fractional orders[23][24]. Recently, Elzaki homotopy transformation perturbation method is employed to solve a class of models such, see [25][26][27]. The Elzaki transform was proposed by the Jordanian mathematician Tarig Elzaki [28], and this transform has been applied on many models to acquire their solution, see [29][30][31][32][33][34][35]. In this paper, the Elzaki transform with a new method of VIM combined with the homotopy perturbation method is utilized to study the solution of time fractional telegraph equation. The object of the present work is to extend the implementations of EVIM and show the accuracy of the suggested technique. Therefore, the fractional telegraph equation is considered. Nanoelectromechanical systems are playing an enormous rule in the area of sensing and actuating. However, nonlinearity effects negatively on the Nanoelectromechanical systems devise. The nonlinear vibration systems have complex behaviors that are characterized by noise, instability in response, and bifurcation phenomena. Therefore, controlling the nonlinear vibrations of Nanoelectromechanical systems is essential to obtain stable vibrations. 𝜕𝛽𝑟 𝜕𝑥𝛽 + 𝐺 𝜕𝛼𝑤 𝜕𝑡 𝛼 + 𝐻𝑤 = 0 (1) 𝜕𝛽𝑤 𝜕𝑥𝛽 + 𝐿 𝜕𝛼𝑟 𝜕𝑡 𝛼 + 𝑅𝑟 = 0 (2) Differentiate the equation (1) with respect to 𝑡 and (2) with respect to 𝑥, then solving the system, the following equation is obtained Jurnal Matematika MANTIK Vol. 8, No. 2, December 2022, pp. 124-138 126 𝜕2𝛽𝑤 𝜕𝑥2𝛽 + 𝑅 [−𝐺 𝜕𝛼𝑤 𝜕𝑡 𝛼 − 𝐻𝑤] + 𝐿 [−𝐺 𝜕2𝛼𝑤 𝜕𝑡 2𝛼 − 𝐻 𝜕𝛼𝑤 𝜕𝑡 𝛼 ] = 0 (3) Assume that = 𝑅 𝐿 , 𝜖 = 𝐻 𝐺 , 𝛿 2 = 1 𝐿𝐺 , substituting these values in the equation (3), we obtain equation (4) is telegraph equation which arises in electromagnetic waves. 𝜕2𝛼𝑤 𝜕𝑡 2𝛼 + ( + 𝜖) 𝜕𝛼𝑤 𝜕𝑡 𝛼 + 𝜖𝑤 = 𝛿 2 𝜕2𝛽𝑤 𝜕𝑥2𝛽 (4) 2. Preliminaries Here, several basic definitions and characteristics of fractional calculus and the proposed transform are given. Definition2.1.[36] A function g(y), y > 0 is considered to be a real valued function and belong to the space ∁σ, σ ∈ R . assume that the real number d > σ, such that g(y) = ydg1(y) where g1(y) ∈ ∁(0, ∞), and it is said to be in the space ∁σ n if gn ∈ 𝑅σ, n ∈ N. Definition2.2.[37] The function f(u) is called Riemann-Liouvill fractional integral of order 𝛼 > 0 if it defines as: 𝐽𝛼 𝑓(𝑢) = 1 𝛤(𝛼) ∫ (𝑢 − 𝑡)𝛼−1 𝑢 0 𝑓(𝑡)𝑑𝑡 , 𝑡 > 0. In particular 𝐽0𝑓(𝑢) = 𝑓(𝑢). For θ ≥ 0 and ϑ ≥ −1, we have the following properties: 1. 𝐽𝛼 𝐽𝜃 𝑓(𝑢) = 𝐽𝛼+𝜃 𝑓(𝑢), 2. 𝐽𝛼 𝐽𝜃 𝑓(𝑢) = 𝐽𝜃 𝐽𝛼 𝑓(𝑢), 3. 𝐽𝛼 𝑥𝜗 = 𝛤(𝜗+1) 𝛤(𝛼+𝜗+1) 𝑥𝛼+𝜗. Definition2.3.[37] Assume that function f ∈ C−1 n , n ∈ N. The function f is called Caputo fractional derivative and defined by 𝐷𝛼 𝑓(𝑢) = 1 𝛤(𝑛 − 𝛼) ∫ (𝑢 − 𝑡)𝑛−𝛼−1 𝑢 0 𝑓 𝑛(𝑡)𝑑𝑡, 𝑛 − 1 < 𝛼 ≤ 𝑛. Definition2.4.[28] The function f(u) is called Elzaki-transform if defined as follows: 𝐸[𝑓(𝑢)] = 𝑇(𝑣) = 𝑣 ∫ 𝑓(𝑢)𝑒 −𝑢 𝑣 𝑑𝑢 ∞ 0 𝑢 > 0. Assume that f is piecewise continuous, then Elzaki transform of the Caputo derivative 𝐸 [ 𝜕𝑛𝑓(𝑥,𝑡) 𝜕𝑢𝑛 ] = 𝑇(𝑥,𝑣) 𝑣𝑛 − ∑ 𝑣 2−𝑛+𝑖𝑛−1𝑖=0 𝜕𝑖𝑓(𝑥,0) 𝜕𝑢𝑖 . (5) The Caputo fractional derivative of Laplace transform is defined as follows 𝐿(𝐷𝑥 𝛼 𝑔(𝑥, 𝑢)) = 𝑠𝛼 𝐺(𝑠) − ∑ 𝑠𝛼−1−𝑖 𝑔(𝑖)(𝑥, 0) 𝑛 − 1 < 𝛼 ≤ 𝑛𝑛−1𝑖=0 . (6) Where 𝐺(𝑠) represents the Laplace transform of 𝑔(𝑥). Theorem 2.1 Let 𝐵 = {𝑓(𝑥, 𝑢)| 𝑡ℎ𝑒𝑟𝑒 𝑒𝑥𝑖𝑠𝑡 𝑀, 𝑚1, 𝑚2 > 0 𝑠. 𝑡|𝑓(𝑥, 𝑢)| < 𝑀𝑒 |𝑢| 𝑚𝑗 , 𝑢 ∈ (−1)𝑗 × [0, ∞)} and let 𝑓(𝑥, 𝑢) ∈ 𝐵. The Elzak transform 𝑇(𝑣) of 𝑓(𝑢)is 𝑇(𝑣) = 𝑣𝐺 ( 1 𝑣 ), where 𝐺(𝑠) is the Laplace transform of 𝑔(𝑥). Muhammad Amin Sadiq Murad, Mudhafar Hamed Mamadamen A New Analytical Modeling for Fractional Telegraph Equation Arising in Electromagnetic 127 Theorem 2.2 Assume 𝑇(𝑣) is the Elzaki transform of the function 𝑓(𝑥, 𝑢). Thus 𝐸(𝐷𝑢 𝛼 𝑓(𝑥, 𝑢)) = 𝑇(𝑣) 𝑣 𝛼 − ∑ 𝑣 𝑖−𝛼+2𝑓 (𝑖)(𝑥, 0) 𝑛 − 1 < 𝛼 ≤ 𝑛 𝑛−1 𝑖=0 Proof: by Theorem 1 𝐸{𝐷𝛼 𝑓(𝑥, 𝑢), 𝑣} = 𝑣𝐿 {𝐷𝛼 𝑓(𝑥, 𝑢), 1 𝑣 }. Using equation (6), we obtain 𝐸{𝐷𝛼 𝑓(𝑥, 𝑢), 𝑣} = 𝑣 𝑣 𝛼 𝐺 ( 1 𝑣 ) − 𝑣 ∑ ( 1 𝑣 ) 𝛼−𝑖−1 𝑓 (𝑖)(𝑥, 0) 𝑛−1 𝑖=0 𝑣𝐺 ( 1 𝑣 ) 𝑣 𝛼 − ∑ 𝑣 𝑖−𝛼+2𝑓 (𝑖)(𝑥, 0) 𝑛 − 1 < 𝛼 ≤ 𝑛 𝑛−1 𝑖=0 = 𝑇(𝑣) 𝑣 𝛼 − ∑ 𝑣 𝑖−𝛼+2𝑓 (𝑖)(𝑥, 0). 𝑛−1 𝑖=0 3. Applications of HETM Recently, the Lagrange multiplier is introduced in new manners [38][39][40]. In this work, the Elzaki transform is used and multiply it by Lagrange multiplier in order to obtain the recurrence relation that is restricted in order to define the Lagrange multiplier. To avoid the convolution terms and integral evaluations, we use this technique. Due to the limitations of Elzaki transform on nonlinear parts, the HPM is used to decrease the computations. The innovative and modified scheme is constructed as follows: Taking the Elzaki transform of the proposed model and multiplying it via the Lagrange multiplier to obtain the recurrence relation that identifies the Lagrange multiplier, via variation approach. The Adomin polynomial is used to evaluate the nonlinear terms, and then the well-known HPM is used to find the series solution of the proposed problem. 𝑅𝑢 − 𝑁𝑢 − 𝑘 = 0. Taking the Elzaki transform, the following relation is obtained 𝐸[𝑅𝑢 − 𝑁𝑢 − 𝑘] = 0. Now, we take the Lagrange multiplier 𝜇(𝑣), 𝜇(𝑣){𝐸[𝑅𝑢 − 𝑁𝑢 − 𝑘]} = 0. Here, we can have the following recurrence relation 𝑈𝑗+1(𝑣) = 𝑈𝑗 (𝑣) + 𝜇(𝑣){𝐸[𝑅𝑢 − 𝑁𝑢 − 𝑘]}. (7) The recurrence relation represents the modified Elzaki variation; we apply the optimal condition using the following relation to introduce the Lagrange multiplier 𝜇(𝑣) 𝜌𝑈𝑗+1(𝑣) 𝜌𝑈𝑗 (𝑣) = 0. Here, the inverse Elzaki transform is applied on (7) to achieve the solution of equation (4) 𝑢𝑗+1(𝑣) = 𝑢𝑗 (𝑣) + 𝐸 −1[𝜇(𝑣){𝐸[𝑅𝑢𝑗 ] − 𝐸[𝐴𝑗 + 𝑘]}]. 𝑗 = 0,1,2,3, … where 𝐴𝑗 represents the Adomian polynomial as follows: 𝐴𝑗 = 1 𝑗! 𝑑𝑗 𝑑𝜏𝑗 (𝑁(∑ 𝑢𝑗 𝜏 𝑗 )).∞𝑗=0 (8) Jurnal Matematika MANTIK Vol. 8, No. 2, December 2022, pp. 124-138 128 Finally, to investigate the series approximate solution, the HPM is considered by equating the powers of the embedded parameter 𝑝. 4. Homotopy perturbation In this portion, we study the concept of HPM for the solution of our problem. Consider the following differential equation 𝑅𝑢 − 𝑁𝑢 = 𝑘, (9) where 𝑘 be a source term, 𝑅 is linear term and 𝑁 is the non-linear term, and 𝑢 the solution function. By the Homtopy theory 𝐻(𝑤, 𝑝), 𝐻(𝑤, 𝑝): 𝑅 × [0,1] → 𝑅 which satisfies the following: 𝐻(𝑤, 𝑝) = (1 − 𝑝)[𝑅(𝑤) − 𝑅(𝑤0)] + 𝑝[𝑅(𝑤) − 𝑁(𝑤) − 𝑘] = 0, Simple calculations, we obtain 𝑅(𝑤) − 𝑝𝑁(𝑤) = 𝑘, (10) The parameter 𝑝 ∈ [0,1], 𝑢0 is the initial term of (4), and 𝑤 is the homotopy function with 𝑅(𝑢0) = 𝑘. Since, 𝑤 can be written as: 𝑤 = lim 𝑝→1 (𝑤0 + 𝑝𝑤1 + 𝑝 2𝑤2 + ⋯ ). (11) Using (10) and (11), we have 𝑤0 + 𝑝𝑤1 + 𝑝 2𝑤2 + 𝑝 3𝑤3 … = 𝑘 + 𝑝𝑁(𝑤). Equating the powers of 𝑝 can be written as follows: 𝑝0: 𝑤0 = 𝑘, 𝑝1: 𝑤1 = 𝑁(𝑤0), 𝑝2: 𝑤2 = 𝑤1𝑁 ′(𝑤0), 𝑝3: 𝑤3 = 𝑁 ′(𝑤0) + 𝑤1 2𝑁′′(𝑤0) 2 , ⋮ Finally, as 𝑝 approach to 1, the following series solution is the solution of (4): 𝑢 = 𝑤0 + 𝑤1 + 𝑤2 + 𝑤3 … . (12) Indeed, the convergence of the solution (12) is studied in[41] [42]. 5. Applications The approximate patterns are employed to show the importance of the new method for solving the time fractional telegraph equation. Here, the following models of time fractional differential equations is given: Example 4.1 Consider the following linear telegraph equation of fractional order 𝜕2𝑧 𝜕𝑥2 = 𝜕2𝛼 𝑧 𝜕𝑡2𝛼 + 2 𝜕𝛼 𝑧 𝜕𝑡𝛼 + 𝑧, 𝑡 ≥ 0, 0 < 𝛼 ≤ 1. with the initial conditions 𝑧(𝑥, 0) = 𝑒 𝑥 , 𝑧𝑡 (𝑥, 0) = −2𝑒 𝑥. The exact solution of equation (13) is: 𝑧(𝑥, 𝑡) = 𝑒 𝑥−2𝑡 . Taking the Elzaki transform of equation (13) 𝐸 [ 𝜕2𝛼 𝑧 𝜕𝑡2𝛼 + 2 𝜕𝛼 𝑧 𝜕𝑡𝛼 + 𝑧 − 𝜕2𝑧 𝜕𝑥2 ] = 0. (13) Muhammad Amin Sadiq Murad, Mudhafar Hamed Mamadamen A New Analytical Modeling for Fractional Telegraph Equation Arising in Electromagnetic 129 Now, we multiply both sides of above equation by 𝜇(𝑣) 𝜇(𝑣)𝐸 [ 𝜕2𝛼 𝑧 𝜕𝑡2𝛼 + 2 𝜕𝛼 𝑧 𝜕𝑡𝛼 + 𝑧 − 𝜕2𝑧 𝜕𝑥2 ] = 0. The recurrence relation has the following form 𝑍𝑗+1(𝑥, 𝑣) = 𝑍𝑗 (𝑥, 𝑣) + 𝜇(𝑣)𝐸 [ 𝜕2𝛼𝑧 𝜕𝑡 2𝛼 + 2 𝜕𝛼𝑧 𝜕𝑡 𝛼 + 𝑧 − 𝜕2𝑧 𝜕𝑥2 ]. (14) Taking the variation of the above equation and using Elzaki property (5), we obtain 𝜌𝑍𝑗+1(𝑥, 𝑣) = 𝜌𝑍𝑗 (𝑥, 𝑣) + 𝜇(𝑣)𝜌 { 𝑍𝑗(𝑥,𝑣) 𝑣2𝛼 − 𝑣 2−2𝛼 �̂�𝑗 (𝑥, 0) − 𝑣 3−2𝛼 𝜕 𝛼�̂�𝑗(𝑥,0) 𝜕𝑡 + 𝐸 [2 𝜕𝛼�̂�𝑗 𝜕𝑡 𝛼 + �̂�𝑗 − 𝜕2�̂�𝑗 𝜕𝑥2 ]}. (15) Here, �̂�𝑗 = �̂�𝑗 (𝑥, 0) = �̂�𝑗 (𝑥, 0) are restricted variables, it means that 𝜌�̂�𝑗 (𝑥, 0) = 𝜌�̂�𝑗 (𝑥, 0) = 0 and since 𝑍𝑗+1(𝑥,0) 𝑍𝑗(𝑥,0) = 0. Substituting restricted variables in equation (15), gives 𝜌𝑍𝑗+1(𝑥, 𝑣) = 𝜌𝑍𝑗 (𝑥, 𝑣) + 1 𝑣 2𝛼 𝜇(𝑣)𝜌𝑍𝑗 (𝑥, 𝑣). Therefore, the Lagrange multiplier 𝜇(𝑣) = −𝑣2𝛼 . Substituting the Lagrange multiplier in equation (14), we obtain 𝑍𝑗+1(𝑥, 𝑣) = 𝑍𝑗 (𝑥, 𝑣) − 𝑣 2𝛼 𝐸 [ 𝜕2𝛼𝑧𝑗 𝜕𝑡 2𝛼 + 2 𝜕𝛼𝑧𝑗 𝜕𝑡 𝛼 + 𝑧 − 𝜕2𝑧𝑗 𝜕𝑥2 ]. Applying Elzaki inverse, we have 𝑧𝑗+1(𝑥, 𝑣) = 𝑧𝑗 (𝑥, 𝑣) − 𝐸 −1 [𝑣 2𝛼 𝐸 [ 𝜕2𝛼𝑧𝑗(𝑥,𝑡) 𝜕𝑡 2𝛼 + 2 𝜕𝛼𝑧𝑗(𝑥,𝑡) 𝜕𝑡 𝛼 + 𝑧𝑗 − 𝜕2𝑧𝑗(𝑥,𝑡) 𝜕𝑥2 ]] Since 𝜕2𝛼𝑧𝑗 𝜕𝑡 2𝛼 = 0, 𝑗 = 0,1,2, …, to obtain He’s polynomial the homotopy perturbation method is utilized 𝑧0 + 𝑝𝑧1 + 𝑝 2𝑧2 + 𝑝 3𝑧3 … = 𝑧𝑗 (𝑥, 𝑡) − 𝑝𝐸 −1 [𝑣 2𝛼 𝐸 [2 𝜕𝛼 𝑧𝑗 𝜕𝑡𝛼 + 𝑧𝑗 − 𝜕2𝑧𝑗 𝜕𝑥2 ]] = 𝑧𝑗 (𝑥, 𝑡) − 𝑝𝐸 −1 [𝑣 2𝛼 𝐸 [(2 𝜕𝛼 𝑧0 𝜕𝑡𝛼 + 𝑧0 − 𝜕2𝑧0 𝜕𝑥2 ) + 𝑝 (2 𝜕𝛼 𝑧1 𝜕𝑡𝛼 + 𝑧1 − 𝜕2𝑧1 𝜕𝑥2 ) + 𝑝2 (2 𝜕𝛼 𝑧2 𝜕𝑡𝛼 + 𝑧2 − 𝜕2𝑧2 𝜕𝑥2 ) + 𝑝3 (2 𝜕𝛼 𝑧3 𝜕𝑡𝛼 + 𝑧3 − 𝜕2𝑧3 𝜕𝑥 2 )]] Equating the highest powers of 𝑝 𝑝0 ∶ 𝑧0 = 𝑧0(𝑥, 𝑡) + 𝑡𝑧0𝑡 (𝑥, 𝑡) 𝑝1 ∶ 𝑧1 = −𝐸 −1 [𝑣 2𝛼 𝐸 [2 𝜕𝛼 𝑧0 𝜕𝑡𝛼 + 𝑧0 − 𝜕2𝑧0 𝜕𝑥2 ]] 𝑝2 ∶ 𝑧2 = −𝐸 −1 [𝑣 2𝛼 𝐸 [2 𝜕𝛼 𝑧1 𝜕𝑡𝛼 + 𝑧1 − 𝜕2𝑧1 𝜕𝑥2 ]] 𝑝3 ∶ 𝑧3 = −𝐸 −1 [𝑣 2𝛼 𝐸 [2 𝜕𝛼 𝑧2 𝜕𝑡𝛼 + 𝑧2 − 𝜕2𝑧2 𝜕𝑥2 ]] Therefore, we obtain 𝑧0 = 𝑒 𝑥 − 2𝑡𝑒 𝑥 . Jurnal Matematika MANTIK Vol. 8, No. 2, December 2022, pp. 124-138 130 Since by Theorem 2, we have 𝑧1 = −𝐸 −1[−𝑣𝛼+34𝑒 𝑥 ], 𝑧1 = 4𝑒 𝑥 𝑡𝛼+1 𝛤(𝛼 + 2) . Similarly, 𝑧2 = −8𝑒 𝑥 𝑡2𝛼+1 𝛤(2𝛼 + 2) , 𝑧3 = 16𝑒 𝑥 𝑡3𝛼+1 𝛤(3𝛼 + 2) , ⋯. One can expressed these results in a series such as: 𝑧 = 𝑧0 + 𝑧1 + 𝑧2 + 𝑧3 + ⋯. 𝑧 = 𝑒 𝑥 − 2𝑡𝑒 𝑥 + 4𝑒 𝑥 𝑡𝛼+1 𝛤(𝛼 + 2) − 8𝑒 𝑥 𝑡2𝛼+1 𝛤(2𝛼 + 2) + 16𝑒 𝑥 𝑡3𝛼+1 𝛤(3𝛼 + 2) − ⋯, when 𝛼 = 1, the HETM solution for equation (13) is 𝑧 = 𝑒 𝑥 − 2𝑡𝑒 𝑥 + 4𝑒 𝑥 𝑡2 2! − 8𝑒 𝑥 𝑡3 3! + 16𝑒 𝑥 𝑡4 4! − ⋯. 𝑧 = 𝑒 𝑥−2𝑡 . Thus, the obtained result using HETM can compute to the exact solution 𝑧 = 𝑒 𝑥−2𝑡 , when 𝛼 = 1 . Figure 1. (a) Exact solution and (b) HETM solution of 𝑧(𝑥, 𝑡) of equation (13) at 𝛼 = 1. The HETM solution of 𝑧(𝑥, 𝑡) of equation (13) at (c) 𝛼 = 0.8 and (d) 𝛼 = 0.2. Muhammad Amin Sadiq Murad, Mudhafar Hamed Mamadamen A New Analytical Modeling for Fractional Telegraph Equation Arising in Electromagnetic 131 Example 4.2 Consider the following time fractional telegraph equation 𝜕2𝑧 𝜕𝑥2 + 𝜕2𝑧 𝜕𝑦2 = 𝜕2𝛼 𝑧 𝜕𝑡2𝛼 + 3 𝜕𝛼 𝑧 𝜕𝑡𝛼 + 2𝑧, 𝑡 ≥ 0,0 < 𝛼 ≤ 1, (16) with the initial conditions 𝑧(𝑥, 𝑦, 0) = 𝑒 𝑥+𝑦 , 𝑧𝑡 (𝑥, 𝑦, 0) = −3𝑒 𝑥+𝑦. The exact solution of the equation (16) is 𝑧(𝑥, 𝑦, 𝑡) = 𝑒 𝑥+𝑦−3𝑡 . Taking the Elzaki transform of equation (16) 𝐸 [ 𝜕2𝛼 𝑧 𝜕𝑡2𝛼 + 3 𝜕𝛼 𝑧 𝜕𝑡𝛼 + 2𝑧 − 𝜕2𝑧 𝜕𝑥2 − 𝜕2𝑧 𝜕𝑦2 ] = 0. Now, we multiply both sides of above equation by 𝜇(𝑣) 𝜇(𝑣)𝐸 [ 𝜕2𝛼 𝑧 𝜕𝑡2𝛼 + 3 𝜕𝛼 𝑧 𝜕𝑡𝛼 + 2𝑧 − 𝜕2𝑧 𝜕𝑥2 − 𝜕2𝑧 𝜕𝑦2 ] = 0. The recurrence relation has the following form 𝑍𝑗+1(𝑥, 𝑣) = 𝑍𝑗 (𝑥, 𝑣) + 𝜇(𝑣)𝐸 [ 𝜕2𝛼𝑧𝑗 𝜕𝑡 2𝛼 + 3 𝜕𝛼𝑧𝑗 𝜕𝑡 𝛼 + 2𝑧𝑗 − 𝜕2𝑧𝑗 𝜕𝑥2 − 𝜕2𝑧𝑗 𝜕𝑦2 ]. (17) Taking the variation of the above equation and using Elzaki property (5), we obtain 𝜌𝑍𝑗+1(𝑥, 𝑦, 𝑣) = 𝜌𝑍𝑗 (𝑥, 𝑦, 𝑣) + 𝜇(𝑣)𝜌 { 𝑍𝑗(𝑥,𝑦,𝑣) 𝑣2𝛼 − 𝑣 2−2𝛼 �̂�𝑗 (𝑥, 𝑦, 0) − 𝑣 3−2𝛼 𝜕 𝛼�̂�𝑗(𝑥,𝑦,0) 𝜕𝑡 + 𝐸 [ 𝜕2𝛼�̂�𝑗(𝑥,𝑦,0) 𝜕𝑡 2𝛼 + 3 𝜕𝛼�̂�𝑗(𝑥,𝑦,0) 𝜕𝑡 𝛼 + 2�̂�𝑗 (𝑥, 𝑦, 0) − 𝜕2�̂�𝑗(𝑥,𝑦,0) 𝜕𝑥2 − 𝜕2�̂�𝑗(𝑥,𝑦,0) 𝜕𝑦2 ]}. (18) The variables �̂�𝑗 = �̂�𝑗 (𝑥, 𝑦, 0) = �̂�𝑗 (𝑥, 𝑦, 0) are restricted variable, since 𝜌�̂�𝑗 (𝑥, 𝑦, 0) = 𝜌�̂�𝑗 (𝑥, 𝑦, ,0) = 0 and �̂�𝑗+1(𝑥,𝑦,0) �̂�𝑗(𝑥,𝑦,0) = 0. Substituting restricted variables in equation (18), gives 𝜌𝑍𝑗+1(𝑥, 𝑦, 𝑣) = 𝜌𝑍𝑗 (𝑥, 𝑦, 𝑣) + 1 𝑣 2𝛼 𝜇(𝑣)𝜌𝑍𝑗 (𝑥, 𝑦, 𝑣). Therefore, the Lagrange multiplier 𝜇(𝑣) = −𝑣2𝛼 . Substituting the Lagrange multiplier in equation (17), we obtain 𝑍𝑗+1(𝑥, 𝑦, 𝑣) = 𝑍𝑗 (𝑥, 𝑦, 𝑣) − 𝑣 2𝛼 𝐸 [ 𝜕2𝛼𝑧𝑗 𝜕𝑡 2𝛼 + 3 𝜕𝛼𝑧𝑗 𝜕𝑡 𝛼 + 2𝑧𝑗 − 𝜕2𝑧𝑗 𝜕𝑥2 − 𝜕2𝑧𝑗 𝜕𝑦2 ]. Applying Elzaki inverse, we get 𝑧𝑗+1(𝑥, 𝑦, 𝑣) = 𝑧𝑗 (𝑥, 𝑦, 𝑣) − 𝐸 −1 [𝑣 2𝛼 𝐸 [ 𝜕2𝛼𝑧𝑗 𝜕𝑡 2𝛼 + 3 𝜕𝛼𝑧𝑗 𝜕𝑡 𝛼 + 2𝑧𝑗 − 𝜕2𝑧𝑗 𝜕𝑥2 − 𝜕2𝑧𝑗 𝜕𝑦2 ]] Since 𝜕2𝛼𝑧𝑗 𝜕𝑡 2𝛼 = 0, 𝑗 = 0,1,2, … to obtain He’s polynomial, we apply HPM 𝑧0 + 𝑝𝑧1 + 𝑝 2𝑧2 + 𝑝 3𝑧3 … = 𝑧𝑗 (𝑥, 𝑦, 𝑡) − 𝑝𝐸 −1 [𝑣 2𝛼 𝐸 [3 𝜕𝛼 𝑧𝑗 𝜕𝑡𝛼 + 2𝑧𝑗 − 𝜕2𝑧𝑗 𝜕𝑥2 − 𝜕2𝑧𝑗 𝜕𝑦2 ]] Jurnal Matematika MANTIK Vol. 8, No. 2, December 2022, pp. 124-138 132 = 𝑧𝑗 (𝑥, 𝑦, 𝑡) − 𝑝𝐸 −1 [𝑣 2𝛼 𝐸 [(3 𝜕𝛼 𝑧0 𝜕𝑡𝛼 + 2𝑧0 − 𝜕2𝑧0 𝜕𝑥2 − 𝜕2𝑧0 𝜕𝑦2 ) + 𝑝 (3 𝜕𝛼 𝑧1 𝜕𝑡𝛼 + 2𝑧1 − 𝜕2𝑧1 𝜕𝑥2 − 𝜕2𝑧1 𝜕𝑦2 ) + 𝑝2 (3 𝜕𝛼 𝑧2 𝜕𝑡𝛼 + 2𝑧2 − 𝜕2𝑧2 𝜕𝑥2 − 𝜕2𝑧2 𝜕𝑦2 ) + 𝑝3 (3 𝜕𝛼 𝑧3 𝜕𝑡𝛼 + 2𝑧3 − 𝜕2𝑧3 𝜕𝑥2 − 𝜕2𝑧3 𝜕𝑦2 )]]. Equating the highest powers of 𝑝 𝑝0 ∶ 𝑧0 = 𝑧0(𝑥, 𝑦, 𝑡) + 𝑡𝑧0𝑡 (𝑥, 𝑦, 𝑡) 𝑝1 ∶ 𝑧1 = −𝐸 −1 [𝑣 2𝛼 𝐸 [3 𝜕𝛼 𝑧0 𝜕𝑡𝛼 + 2𝑧0 − 𝜕2𝑧0 𝜕𝑥2 − 𝜕2𝑧0 𝜕𝑦2 ]] 𝑝2 ∶ 𝑧2 = −𝐸 −1 [𝑣 2𝛼 𝐸 [3 𝜕𝛼 𝑧1 𝜕𝑡𝛼 + 2𝑧1 − 𝜕2𝑧1 𝜕𝑥2 − 𝜕2𝑧1 𝜕𝑦2 ]] 𝑝3 ∶ 𝑧3 = −𝐸 −1 [𝑣 2𝛼 𝐸 [3 𝜕𝛼 𝑧2 𝜕𝑡𝛼 + 2𝑧2 − 𝜕2𝑧2 𝜕𝑥2 − 𝜕2𝑧2 𝜕𝑦2 ]] 𝑝4 ∶ 𝑧4 = −𝐸 −1 [𝑣 2𝛼 𝐸 [3 𝜕𝛼 𝑧3 𝜕𝑡𝛼 + 2𝑧3 − 𝜕2𝑧3 𝜕𝑥2 − 𝜕2𝑧3 𝜕𝑦2 ]]. Therefore, we obtain 𝑧0 = 𝑒 𝑥+𝑦 − 3𝑡𝑒 𝑥+𝑦 . Since by Theorem 2.2, we have 𝑧1 = −𝐸 −1[−𝑣𝛼+39𝑒 𝑥+𝑦 ], 𝑧1 = 9𝑒 𝑥+𝑦 𝑡𝛼+1 𝛤(𝛼 + 2) . Similarly, 𝑧2 = −27𝑒 𝑥+𝑦 𝑡2𝛼+1 𝛤(2𝛼 + 2) , 𝑧3 = 81𝑒 𝑥+𝑦 𝑡3𝛼+1 𝛤(3𝛼 + 2) , … Here, the HETM for equation (16) is 𝑧(𝑥, 𝑦, 𝑡) = 𝑧0 + 𝑧1 + 𝑧2 + 𝑧3 + ⋯. 𝑧(𝑥, 𝑦, 𝑡) = 𝑒 𝑥+𝑦 (1 − 3𝑡 + 9𝑡𝛼+1 𝛤(𝛼 + 2) − 27𝑡2𝛼+1 𝛤(2𝛼 + 2) + 81𝑡3𝛼+1 𝛤(3𝛼 + 2) − ⋯ ), when 𝛼 = 1, the HETM for equation (16) is 𝑧(𝑥, 𝑦, 𝑡) = 𝑒 𝑥+𝑦 − 3𝑡𝑒 𝑥+𝑦 + 9𝑒 𝑥+𝑦 𝑡2 2! − 27𝑒 𝑥+𝑦 𝑡3 3! + 81𝑒 𝑥+𝑦 𝑡4 4! − ⋯. 𝑧(𝑥, 𝑦, 𝑡) = 𝑒 𝑥+𝑦−3𝑡 . Thus, exact solution of model (15) is obtained when 𝛼 = 1 . Example 4.3 Consider the following time fractional telegraph equation 𝜕𝛼 𝑧(𝑥, 𝑡) 𝜕𝑡𝛼 − 𝜕𝑧(𝑥, 𝑡) 𝜕𝑡 = 𝜕2𝑧(𝑥, 𝑡) 𝜕𝑥2 − 𝑧2(𝑥, 𝑡) + 𝑥𝑧(𝑥, 𝑡)𝑧𝑥 (𝑥, 𝑡), 𝑡, 𝑥 ≥ 0, 1 < 𝛼 ≤ 2, (19) with the initial terms 𝑧(𝑥, 0) = 𝑥, 𝑧𝑡 (𝑥, 0) = 𝑥. Muhammad Amin Sadiq Murad, Mudhafar Hamed Mamadamen A New Analytical Modeling for Fractional Telegraph Equation Arising in Electromagnetic 133 Appling the Elzaki transform of model (19), we obtain 𝐸 [ 𝜕𝛼 𝑧(𝑥, 𝑡) 𝜕𝑡𝛼 − 𝜕2𝑧(𝑥, 𝑡) 𝜕𝑥2 − 𝜕𝑧(𝑥, 𝑡) 𝜕𝑡 + 𝑧2(𝑥, 𝑡) − 𝑥𝑧(𝑥, 𝑡)𝑧𝑥 (𝑥, 𝑡)] = 0. Now, we multiply both sides of above equation by 𝜇(𝑣) 𝜇(𝑣)𝐸 [ 𝜕𝛼 𝑧(𝑥, 𝑡) 𝜕𝑡𝛼 − 𝜕2𝑧(𝑥, 𝑡) 𝜕𝑥2 − 𝜕𝑧(𝑥, 𝑡) 𝜕𝑡 + 𝑧2(𝑥, 𝑡) − 𝑥𝑧(𝑥, 𝑡)𝑧𝑥 (𝑥, 𝑡)] = 0. The recurrence relation has the following form 𝑍𝑗+1(𝑥, 𝑣) = 𝑍𝑗 (𝑥, 𝑣) + 𝜇(𝑣)𝐸 [ 𝜕𝛼𝑧𝑗(𝑥,𝑡) 𝜕𝑡 𝛼 − 𝜕2𝑧𝑗(𝑥,𝑡) 𝜕𝑥2 − 𝜕𝑧𝑗(𝑥,𝑡) 𝜕𝑡 + 𝑧𝑗 2(𝑥, 𝑡) − 𝑥𝑧𝑗 (𝑥, 𝑡)𝑧𝑗 𝑥 (𝑥, 𝑡)]. (20) Taking the variation of the above equation and using Elzaki property (5), we obtain 𝜌𝑍𝑗+1(𝑥, 𝑣) = 𝜌𝑍𝑗 (𝑥, 𝑣) + 𝜇(𝑣)𝜌 { 𝑍𝑗(𝑥,𝑣) 𝑣2𝛼 − 𝑣 2−2𝛼 �̂�𝑗 (𝑥, 0) − 𝑣 3−2𝛼 𝜕 𝛼�̂�𝑗(𝑥,0) 𝜕𝑡 − 𝐸 [ 𝜕2�̂�𝑗(𝑥,𝑡) 𝜕𝑥2 + 𝜕�̂�𝑗(𝑥,𝑡) 𝜕𝑡 − �̂�𝑗 2(𝑥, 𝑡) + 𝑥�̂�𝑗 (𝑥, 𝑡)�̂�𝑗 𝑥 (𝑥, 𝑡)]}. (21) 𝜌𝑍𝑗+1(𝑥, 𝑣) = 𝜌𝑍𝑗 (𝑥, 𝑣) + 1 𝑣2𝛼 𝜇(𝑣)𝜌𝑍𝑗 (𝑥, 𝑣). The variables �̂�𝑗 = �̂�𝑗 (𝑥, 0) = �̂�𝑗 (𝑥, 0) are restricted variables, since 𝜌�̂�𝑗 (𝑥, 0) = 𝜌�̂�𝑗 (𝑥, 0) = 0 and 𝜌𝑍𝑗+1(𝑥,𝑣) 𝜌𝑍𝑗(𝑥,𝑣) = 0. Therefore, the Lagrange multiplier 𝜇(𝑣) = −𝑣2𝛼 . Substituting the Lagrange multiplier in (20), the following relation is acquired: 𝑍𝑗+1(𝑥, 𝑣) = 𝑍𝑗 (𝑥, 𝑣) − 𝑣 2𝛼 𝐸 [ 𝜕𝛼𝑧𝑗(𝑥,𝑡) 𝜕𝑡 𝛼 + 𝜕2𝑧𝑗(𝑥,𝑡) 𝜕𝑥2 + 𝜕𝑧𝑗(𝑥,𝑡) 𝜕𝑡 − 𝑧𝑗 2(𝑥, 𝑡) + 𝑥𝑧𝑗 (𝑥, 𝑡)𝑧𝑗 𝑥 (𝑥, 𝑡)]. Applying Elzaki inverse, we get 𝑧𝑗+1(𝑥, 𝑣) = 𝑧𝑗 (𝑥, 𝑣) − 𝐸 −1 [𝑣 2𝛼 𝐸 [ 𝜕𝛼𝑧𝑗(𝑥,𝑡) 𝜕𝑡 𝛼 + 𝜕2𝑧𝑗(𝑥,𝑡) 𝜕𝑥2 + 𝜕𝑧𝑗(𝑥,𝑡) 𝜕𝑡 − 𝑧𝑗 2(𝑥, 𝑡) + 𝑥𝑧𝑗 (𝑥, 𝑡)𝑧𝑗 𝑥 (𝑥, 𝑡)]]. Since 𝜕𝛼𝑧𝑗 𝜕𝑡 𝛼 = 0, 𝑗 = 0,1,2,3 … to get He’s polynomial, we apply HPM 𝑧0 + 𝑝𝑧1 + 𝑝 2𝑧2 + 𝑝 3𝑧3 … = 𝑧𝑗 (𝑥, 𝑡) − 𝑝𝐸 −1 [𝑣2𝛼 𝐸 [ 𝜕2𝑧𝑗(𝑥,𝑡) 𝜕𝑥2 + 𝜕𝑧𝑗(𝑥,𝑡) 𝜕𝑡 − 𝐴𝑗 + 𝑥𝐵𝑗 ]], (22) where 𝐴𝑗 and 𝐵𝑗are the Adomian polynomials of (𝑧0, 𝑧1, 𝑧2, 𝑧3 … ), we use (8) to calculate the Adomian polynomials: 𝐴0 = 𝑧0 2, 𝐵0 = 𝑧0𝑧0𝑥 , 𝐴1 = 2𝑧0𝑧1, 𝐵1 = 𝑧0𝑧1𝑥 + 𝑧0𝑥 𝑧1 , 𝐴2 = 2𝑧0𝑧2 + 𝑧1 2, 𝐵2 = 𝑧0𝑧2𝑥 + 𝑧1𝑧1𝑥 + 𝑧2𝑧0𝑥 , 𝐴3 = 2𝑧0𝑧3 + 2𝑧1𝑧2, 𝐵3 = 𝑧3𝑧0𝑥 + 𝑧2𝑧1𝑥 + 𝑧1𝑧2𝑥 + 𝑧0𝑧3 𝑥 , Table 1: The numerical and exact solutions at various values of 𝛼 and 𝑡 for equation (19), where 𝑥 = 0.5. Jurnal Matematika MANTIK Vol. 8, No. 2, December 2022, pp. 124-138 134 Table 1. demonstrate the comparison of exact and approximate solutions of (19) for different values of 𝑡 and 𝛼 using HETM. It is clear that the value 𝛼 = 2 using HETM gives almost the exact solution of model (19). 𝑡 Exact Solution 𝛼 = 2 𝛼 = 1.90 𝛼 = 1.80 0.0 0.5 0.5 0.5 0.5 0.5 0.8243606355 0.8243606355 0.8377559990 0.8531832410 1.0 1.359140914 1.359140914 1.398076434 1.439803212 1.5 2.240844535 2.240844535 2.312171661 2.385449643 2.0 3.694528050 3.694528049 3.803171996 3.911485870 Equating the highest powers of 𝑝, and substituting the Adomian polynomials in (22), leads 𝑝0 ∶ 𝑧0 = 𝑧0(𝑥, 𝑡) + 𝑡𝑧0𝑡 (𝑥, 𝑡) 𝑝1 ∶ 𝑧1 = −𝐸 −1 [𝑣 2𝛼 𝐸 [ 𝜕𝑧0 𝜕𝑡 + 𝜕2𝑧0 𝜕𝑥2 − 𝑧0 2 + 𝑥𝑧0𝑧0𝑥 ]] 𝑝2 ∶ 𝑧2 = −𝐸 −1 [𝑣 2𝛼 𝐸 [ 𝜕𝑧1 𝜕𝑡 + 𝜕2𝑧1 𝜕𝑥2 − 2𝑧0𝑧1 + 𝑥(𝑧0𝑧1𝑥 + 𝑧0𝑥 𝑧1)]] 𝑝3 ∶ 𝑧3 = −𝐸 −1 [𝑣 2𝛼 𝐸 [ 𝜕𝑧2 𝜕𝑡 + 𝜕2𝑧2 𝜕𝑥2 − 2𝑧0𝑧2 − 𝑧1 2 + 𝑥(𝑧0𝑧2𝑥 + 𝑧1𝑧1𝑥 + 𝑧2𝑧0𝑥 )]] Therefore, we obtain 𝑧0 = 𝑥(1 + 𝑡), 𝑧1 = 𝑥𝑡 𝛼 𝛤(𝛼+1) , 𝑧2 = 𝑥𝑡 𝛼+1 𝛤(𝛼+2) , 𝑧3 = 𝑥𝑡 𝛼+2 𝛤(𝛼+3) , …. Here, the HETM solution for equation (19) is 𝑧 = 𝑧0 + 𝑧1 + 𝑧2 + 𝑧3 + ⋯. 𝑧 = 𝑥 (1 + 𝑡 + 𝑡𝛼 𝛤(𝛼 + 1) + 𝑡𝛼+1 𝛤(𝛼 + 2) + 𝑡𝛼+2 𝛤(𝛼 + 3) + ⋯ ), when 𝛼 = 2, the HETM solution for equation (19) is 𝑧 = 𝑥 (1 + 𝑡 + 𝑡2 2! + 𝑡3 3! + 𝑡4 4! + ⋯ ). 𝑧 = 𝑥𝑒 𝑡 . Thus, the exact solution of equation (19) is obtained when 𝛼 = 2 . Figure 2. (a) The error plot of 𝑧(𝑥, 𝑡) of equation (19) at 𝛼 = 2. Muhammad Amin Sadiq Murad, Mudhafar Hamed Mamadamen A New Analytical Modeling for Fractional Telegraph Equation Arising in Electromagnetic 135 In this paper, several models of time fractional telegraph equations are studied using a novel numerical approach. The present outcomes are compared with the analytic solutions via tables and illustrative graphs. Table 1. illustrates the comparison between the approximate solutions acquired via the proposed technique for various orders of fractional derivative 𝛼 with the exact solution. In Figure 1. graph (a) the exact solution is given, graph (b) the solution of HETM for 𝛼 = 2 is given, and graphs (c) and (d) the solutions of HETM for 𝛼 = 0.8 and 𝛼 = 0.2 are given, respectively. Finally, the error plot of equation (19) is given in Figure 2. As a result, it can be observed that there is an excellent agreement between the present results and the exact solution. 6. Conclusion In this work, a novel computational method called Elzaki integral transform combined with a new technique of He’s variation iteration technique to investigate the solution of linear and nonlinear telegraph equations of fractional orders. The Caputo sense is used to describe the fractional derivatives. This method is implemented on the several models of telegraph equations; the exact and approximate solutions are obtained for each model. The advantage of the proposed technique is that for defining the Lagrange multiplier, there is no need to integration or convolution theorem in recurrence relation. Because of the limitations of Elzaki transform on nonlinear parts, the HPM is used to reduce the computations. Finally, the present results show the accuracy of the novel computational method according to the obtained results. In future, the proposed method can be used to investigate the solutions of the differential equations. References [1] H. Khan, R. Shah, P. Kumam, D. Baleanu, and M. Arif, “An efficient analytical technique, for the solution of fractional-order telegraph equations,” Mathematics, vol. 7, no. 5, pp. 1–19, 2019, doi: 10.3390/math7050426. [2] M. A. Abdou, “Adomian decomposition method for solving the telegraph equation in charged particle transport,” J. Quant. Spectrosc. Radiat. Transf., vol. 95, no. 3, pp. 407–414, 2005, doi: https://doi.org/10.1016/j.jqsrt.2004.08.045. [3] A. Yıldırım, “He’s homotopy perturbation method for solving the space- and time- fractional telegraph equations,” Int. J. Comput. Math., vol. 87, no. 13, pp. 2998– 3006, Oct. 2010, doi: 10.1080/00207160902874653. [4] F. A. Alawad, E. A. Yousif, and A. I. Arbab, “A new technique of Laplace variational iteration method for solving space-time fractional telegraph equations,” Int. J. Differ. Equations, vol. 2013, 2013, doi: 10.1155/2013/256593. [5] H. Al-badrani, S. Saleh, H. O. Bakodah, and M. Al-Mazmumy, “Numerical solution for nonlinear telegraph equation by modified Adomian decomposition method,” Nonlinear Anal. Differ. Equations, vol. 4, no. 5, pp. 243–257, 2016, doi: 10.12988/nade.2016.6418. [6] V. K. Srivastava, M. K. Awasthi, R. K. Chaurasia, and M. Tamsir, “The telegraph equation and its solution by reduced differential transform method,” Model. Simul. Eng., vol. 2013, 2013, doi: 10.1155/2013/746351. [7] A. Sevimlican, “An approximation to solution of space and time fractional telegraph equations by he’s variational iteration method,” Math. Probl. Eng., vol. 2010, 2010, doi: 10.1155/2010/290631. Jurnal Matematika MANTIK Vol. 8, No. 2, December 2022, pp. 124-138 136 [8] A. Saadatmandi and M. Dehghan, “Numerical solution of hyperbolic telegraph equation using the Chebyshev tau method,” Numer. Methods Partial Differ. Equ., vol. 26, no. 1, pp. 239–252, 2010, doi: doi:10.1002/num.20442. [9] A. R. P. Rau, M. Inokuti, and D. A. Douthat, “Variational treatment of electron degradation and yields of initial molecular species,” Phys. Rev. A, vol. 18, no. 3, pp. 971–988, Sep. 1978, doi: 10.1103/PhysRevA.18.971. [10] M. Inokuti, H. Sekine, and T. Mura, “General Use of the Lagrange Multiplier in Nonlinear Mathematical Physics,” in Variational Methods in the Mechanics of Solids, Elsevier, 1980, pp. 156–162. [11] J. H. He, “Variational iteration method - A kind of non-linear analytical technique: Some examples,” Int. J. Non. Linear. Mech., vol. 34, no. 4, pp. 699–708, 1999, doi: 10.1016/s0020-7462(98)00048-1. [12] J. Biazar and H. Ghazvini, “He’s variational iteration method for fourth-order parabolic equations,” Comput. Math. with Appl., vol. 54, no. 7–8, pp. 1047–1054, 2007, doi: 10.1016/j.camwa.2006.12.049. [13] M. Akbarzade and J. Langari, “Application of variational iteration method to partial differential equation systems,” Int. J. Math. Anal., vol. 5, no. 17–20, pp. 863–870, 2011. [14] F. Geng and Y. Lin, “Application of the variational iteration method to inverse heat source problems,” Comput. Math. with Appl., vol. 58, no. 11–12, pp. 2098–2102, 2009, doi: 10.1016/j.camwa.2009.03.002. [15] A. Saadatmandi and M. Dehghan, “Variational iteration method for solving a generalized pantograph equation,” Comput. Math. with Appl., vol. 58, no. 11–12, pp. 2190–2196, 2009, doi: 10.1016/j.camwa.2009.03.017. [16] M. A. S. Murad, “Property Claim Services by Compound Poisson Process And Inhomogeneous Levy Process,” Sci. J. Univ. Zakho, vol. 6, no. 1, pp. 32–34, 2018. [17] M. Javidi and B. Ahmad, “Numerical solution of fourth-order time-fractional partial differential equations with variable coefficients,” J. Appl. Anal. Comput., vol. 5, no. 1, pp. 52–63, 2015, doi: 10.11948/2015005. [18] D. H. Shou, “The homotopy perturbation method for nonlinear oscillators,” Comput. Math. with Appl., vol. 58, no. 11–12, pp. 2456–2459, 2009, doi: 10.1016/j.camwa.2009.03.034. [19] J. Biazar, B. Ghanbari, M. G. Porshokouhi, and M. G. Porshokouhi, “He’s homotopy perturbation method: A strongly promising method for solving non- linear systems of the mixed Volterra–Fredholm integral equations,” Comput. Math. with Appl., vol. 61, no. 4, pp. 1016–1023, 2011, doi: https://doi.org/10.1016/j.camwa.2010.12.051. [20] J. Biazar and H. Ghazvini, “Homotopy perturbation method for solving hyperbolic partial differential equations,” Comput. Math. with Appl., vol. 56, no. 2, pp. 453– 458, 2008, doi: https://doi.org/10.1016/j.camwa.2007.10.032. [21] J. Biazar, F. Badpeima, and F. Azimi, “Application of the homotopy perturbation method to Zakharov–Kuznetsov equations,” Comput. Math. with Appl., vol. 58, no. 11, pp. 2391–2394, 2009, doi: https://doi.org/10.1016/j.camwa.2009.03.102. [22] T. M. Elzaki and J. Biazar, “Homotopy perturbation method and Elzaki transform for solving system of nonlinear partial differential equations,” World Appl. Sci. J., vol. 24, no. 7, pp. 944–948, 2013, doi: 10.5829/idosi.wasj.2013.24.07.1041. Muhammad Amin Sadiq Murad, Mudhafar Hamed Mamadamen A New Analytical Modeling for Fractional Telegraph Equation Arising in Electromagnetic 137 [23] M. A. S. Murad, F. K. Hamasalh, and H. F. Ismael, “Numerical study of stagnation point flow of Casson-Carreau fluid over a continuous moving sheet,” AIMS Mathematics, vol. 8, no. December 2022, pp. 7005–7020, 2023, doi: 10.3934/math.2023353. [24] M. A. S. Murad and F. K. Hamasalh, “Computational Technique for the Modeling on MHD Boundary Layer Flow Unsteady Stretching Sheet by B-Spline Function,” in 2022 International Conference on Computer Science and Software Engineering (CSASE), 2022, pp. 236–240. [25] A. C. Loyinmi and T. K. Akinfe, “Exact solutions to the family of Fisher’s reaction‐ diffusion equation using Elzaki homotopy transformation perturbation method,” Eng. Reports, vol. 2, no. 2, pp. 1–32, 2020, doi: 10.1002/eng2.12084. [26] J. Ul Rahman, D. Lu, M. Suleman, J. H. He, and M. Ramzan, “HE-ELZAKI METHOD for SPATIAL DIFFUSION of BIOLOGICAL POPULATION,” Fractals, vol. 27, no. 5, 2019, doi: 10.1142/S0218348X19500695. [27] N. Anjum, M. Suleman, D. Lu, J. H. He, and M. Ramzan, “Numerical iteration for nonlinear oscillators by Elzaki transform,” J. Low Freq. Noise Vib. Act. Control, 2019, doi: 10.1177/1461348419873470. [28] T. M. Elzaki, “The New Integral Transform ’ ’ ELzaki Transform ’ ’,” vol. 7, no. 1, pp. 57–64, 2011. [29] E. M. A. Hilal, “Elzaki and Sumudu Transforms for Solving Some,” vol. 8, no. 2, pp. 167–173, 2012. [30] M. A. S. Murad, “Modified integral equation combined with the decomposition method for time fractional differential equations with variable coefficients,” Appl. Math. J. Chinese Univ., vol. 37, no. 3, pp. 404–414, 2022. [31] D. Ziane and M. H. Cherif, “Resolution of Nonlinear Partial Di ¤ erential Equations by Elzaki Transform Decomposition Method Laboratory of mathematics and its applications ( LAMAP ),” vol. 5, pp. 17–30, 2015. [32] O. E. Ige, R. A. Oderinu, and T. M. Elzaki, “Adomian polynomial and Elzaki transform method for solving sine-gordon equations,” IAENG Int. J. Appl. Math., vol. 49, no. 3, pp. 1–7, 2019. [33] D. H. Malo, M. A. S. Murad, R. Y. Masiha, and S. T. Abdulazez. , “A New Computational Method Based on Integral Transform for Solving Linear and Nonlinear Fractional Systems,” J. Mat. MANTIK, vol. 7, no. 1, pp. 9–19, 2021. [34] R. M. Jena and S. Chakraverty, “Solving time-fractional Navier–Stokes equations using homotopy perturbation Elzaki transform,” SN Appl. Sci., vol. 1, no. 1, pp. 1– 13, 2019, doi: 10.1007/s42452-018-0016-9. [35] M. Hamed, S. Taha, and M. A. S. Murad, “ Modified Computational Method Based on Integral Transform for Solving Fractional Zakharov-Kuznetsov Equations ,” Matrix Science Mathematic, vol. 7, no. 1, pp. 1–6, 2023, doi: 10.26480/msmk.01.2023.01.06. [36] R. V Slonevskii and R. R. Stolyarchuk, “Rational-fractional methods for solving stiff systems of differential equations,” J. Math. Sci., vol. 150, no. 5, pp. 2434–2438, 2008, doi: 10.1007/s10958-008-0141-x. [37] A. Prakash and V. Verma, “Numerical method for fractional model of newell- whitehead-segel equation,” Front. Phys., vol. 7, no. FEB, pp. 1–10, 2019, doi: 10.3389/fphy.2019.00015. Jurnal Matematika MANTIK Vol. 8, No. 2, December 2022, pp. 124-138 138 [38] H. Kumar Mishra and A. K. Nagar, “He-Laplace method for linear and nonlinear partial differential equations,” J. Appl. Math., vol. 2012, 2012, doi: 10.1155/2012/180315. [39] Z. J. Liu, M. Y. Adamu, E. Suleiman, and J. H. He, “Hybridization of homotopy perturbation method and laplace transformation for the partial differential equations,” Therm. Sci., vol. 21, no. 4, pp. 1843–1846, 2017, doi: 10.2298/TSCI160715078L. [40] M. Nadeem and F. Li, “Modified Laplace Variational Iteration Method for Analytical Approach of Klein–Gordon and Sine–Gordon Equations,” Iran. J. Sci. Technol. Trans. A Sci., vol. 43, no. 4, pp. 1933–1940, 2019, doi: 10.1007/s40995- 018-0667-9. [41] J. Biazar and H. Aminikhah, “Study of convergence of homotopy perturbation method for systems of partial differential equations,” Comput. Math. with Appl., vol. 58, no. 11, pp. 2221–2230, 2009, doi: https://doi.org/10.1016/j.camwa.2009.03.030. [42] M. Turkyilmazoglu, “Convergence of the homotopy perturbation method,” Int. J. Nonlinear Sci. Numer. Simul., vol. 12, no. 1–8, pp. 9–14, 2011, doi: 10.1515/IJNSNS.2011.02.y