EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 16, No. 3, 2023, 1491-1507 ISSN 1307-5543 – ejpam.com Published by New York Business Global An Iterative Approach to Solve Volterra Nonlinear Integral Equations Rania Saadeh Department of Mathematics, Zarqa University, Zarqa 13110, Jordan Abstract. In this study, we provide the Aboodh decomposition method, a novel analytical tech- nique. The fundamental definitions and theorems of the suggested approach are provided and analyzed. This new method is a novel mixture of the Aboodh transform and the Adomian decom- position method. The new method is used to solve nonlinear integro-differential equations (IDEs), and the solutions are given as quickly expanding series of terms. We compute the maximum ab- solute error and provide some figures to compare the resulting approximative solutions with the exact ones in order to demonstrate the method’s applicability and efficiency. 2020 Mathematics Subject Classifications: 44A05,45D05,49M27 Key Words and Phrases: Aboodh transform, Decomposition method, Integral equations, Non- linear integro-differential equations 1. Introduction Several scientific and engineering problems include integral equations. Volterra or Fredholm integral equations can be used to solve a wide variety of initial and boundary value problems. The potential theory made the greatest contribution to the develop- ment of integral equations. The development of integral equations was further facilitated by mathematical physics models of diffraction issues, astrophysics, quantum mechanics scattering, conformal mapping, and water waves [4, 5, 7, 9, 26]. Moreover, the study of nonlinear IDEs has appeared in many fields of science, because of the great number of applications that could describe, such as chemical kinetics, queuing theory, and others [12, 21, 27, 28, 34]. Thus researchers have developed many techniques to handle these problems such as He’s homotopy perturbation method [37], variation iteration method [14], least square method [8], decomposition method [13] and others. Decomposition method is one of the most powerful methods to solve nonlinear differ- ential and integral equations, it presents approximate analytical series solutions of the target problems. Adomian decomposition method was presented by Adomian in [6, 7] to solve integral equations, then it was developed by Wazwaz to solve Volterra IDEs [39]. DOI: https://doi.org/10.29020/nybg.ejpam.v16i3.4791 Email address: rsaadeh@zu.edu.jo https://www.ejpam.com 1491 © 2023 EJPAM All rights reserved. R. Saadeh / Eur. J. Pure Appl. Math, 16 (3) (2023), 1491-1507 1492 Then the method was used by many researchers to solve various kinds of problems [15– 19, 22, 31, 32]. Integral transforms have played important roles in solving integral equations, such as Laplace transform [38], ARA transform [36], formable transform [35] and others [10, 20]. One of the most important transforms in literature is Aboodh transform, which was in- troduced in 2013 [1], and it has great applications in mathematics. Aboodh transform is defined by the following improper integral: A [φ (τ)] = 1 v ∫ ∞ 0 e−vτφ (τ) dτ, v > 0. This transform has a great attention from mathematicians, because of its applicability to solve different types of problems, also it could be combined easily with other iteration methods to solve nonlinear problems [2, 3, 11]. The main goal of this article, is to introduce a new combination between the Aboodh transform and the Adomian decomposition method, namely the Aboodh -decomposition method (ADM). The proposed method is utilized to establish analytical series solutions of nonlinear Volterra IDEs, these approximate solutions converge rapidly to the exact ones. for the nonlinear VIE. The novelty of this approach is the powerful combination between the decomposition method and Aboodh transform for the first time. Moreover, the high speed of convergence of the approximate analytical solutions obtained by ADM to the exact solutions, make it an effective method to solve nonlinear IDEs in comparison to other numerical methods. This research investigates the solution of the nonlinear Volterra IDE of the form φ(n) (τ) = ψ (τ) + ∫ τ 0 k(τ − u)H(φ (u))du, where the kernel k(τ − v) and ψ(τ) are real-valued functions, and H(φ (u)) is a nonlinear function of φ (v), such as φ3 (v), coshφ (v), sinhφ (v). The main contribution of this work is to present a new analytical approach for solving nonlinear IDEs with simple and easy steps, the method basically depends on applying the Aboodh transform the using the technique of ADM to handle the nonlinear terms. The method is new and simple with less computatios than other methods. This paper is structured as follows, in Section 2, we introduce the definition of Aboodh transform and some basic properties of it, and we illustrate the main idea of the Adomian decomposition method. In Section 3, the ADM is presented to handle nonlinear Volterra IDEs. To show the applicability of the method, we solve some numerical examples on IDEs. Finally, the conclusion of this article is introduced in Section 5. 2. Basic preliminaries of ADM 2.1. Aboodh integral transform In this section, we present the basic definitions and properties of Aboodh transform. R. Saadeh / Eur. J. Pure Appl. Math, 16 (3) (2023), 1491-1507 1493 Definition 1. [1] Let φ (τ) be a piecewise continuous function defined on (0,∞). Then Aboodh transform for φ (τ) is denoted and defined by A [φ (τ)] = Φ (v) = 1 v ∫ ∞ 0 e−vτφ (τ) dτ, τ > 0. The inverse Aboodh transform for Φ (v) denoted and defined by A−1 [Φ (v)] = φ (τ) = 1 2πi ∫ c+i∞ c−i∞ vevτΦ (v) dv. Theorem 1. (Existence Condition)[1] Ifφ (τ) is a piecewise continuous function on [0,∞) and satisfies the condition |φ (τ)| ≤ Neατ , for some N > 0. Then, Aboodh transform A[φ (τ)] exists for Re (v) > α. Proof. Using the definition of Aboodh transform, we obtain |Φ (v) | = ∣∣∣∣1v ∫ ∞ 0 e−vτφ (τ) dτ ∣∣∣∣ ≤ 1 v ∫ ∞ 0 e−vτ |φ (τ)| dτ ≤ 1 v ∫ ∞ 0 e−vτNeατdτ = 1 v N ∫ ∞ 0 e−τ(v−α)dτ = N v (v − α) , Re (v) > α > 0. Hence, Aboodh integral transform exists for Re (v) > α > 0. Now, we mention some properties of Aboodh transform to the basic functions. Suppose that Φ1 (v) = A[φ1 (τ)] and Φ2 (v) = A[φ2 (τ)] and α, β ∈ R, then • A [αφ1 (τ) + βφ2 (τ)] = αΦ1 (v) + βΦ2 (v). • A−1 [αΦ1 (v) + βΦ2 (v)] = αφ1 (τ) + βφ2 (τ). Now the following table (Table 1) introduces some values of Aboodh transform to some elementary functions, for more details, see [1]. Table 1: Aboodh transform for some functions. φ(τ) A [φ (τ)] 1 1/v2 τa Γ(a+1) va+2 , a > −1 eaτ 1 v(v−a) , s > a sin (aτ) a v(v2+a2 cos (aτ) 1 v2+a2 sinh (aτ) a v(v2−a2 cosh (aτ) 1 v2−a2 φ′ (τ) uΦ (v)− 1 vφ(0) φ(n) (τ) Φ (v)− ∑n−1 j=0 v n−j−2φ(j)(0) (φ ∗ ψ)(τ) vA [φ(τ)]A [ψ(τ)] R. Saadeh / Eur. J. Pure Appl. Math, 16 (3) (2023), 1491-1507 1494 2.2. Adomian Decomposition Method The Adomian decomposition method [7], which has many applications in engineering, physics, and applied mathematics, is a very effective technique for solving many classes of nonlinear partial and ordinary differential equations. The core idea behind the Adomian decomposition technique is to decompose the nonlinear term in the equation into a sum of component. These parts add up to a highly accurate representation of the solution. We explain the steps of the method as: • Suppose that the solution of the target problem has the following series representa- tion φ (τ) = ∞∑ n=0 φn (τ) = φ0 (τ) + φ1 (τ) + . . . . • Establish a recursive relation of the nonlinear term of the discussed equation, then substitute the value of the series solution in the equation. • Simplify the resulting equation and solve it for the series components recursively. 3. Solving nonlinear Volterra IDEs by ADM In this part of the study, we operate Aboodh transform to the target IDE, then apply the decomposition method, which is the main idea of the ADM. Moreover, we suppose that the given kernel in the equation has a difference form, that could be presented as: k (x− τ), for examples, cos (x− τ), (x− τ)2, ex−τ . Now, consider the following nonlinear Volterra IDE: φ(n) (τ) = ψ (τ) + ∫ τ 0 k(τ − u)H(φ (u))du, (1) Subject to the initial conditions (ICs) φ(i) (0) = δi, i = 0, 1, . . . , n− 1. (2) To get the solution of equation (1) by ADM, we operate Aboodh transform to equation (1) A [ φ(n) (τ) ] = A [ψ (τ)] +A [∫ τ 0 k (τ − v)H(φ (v))dv ] . The differential property and the convolution property stated in Table 1 of Aboodh trans- form imply that equation (1) can be simplified to vn−1A [φ (τ)]− vn−2δ0 − vn−3δ1 − . . .− 1 v δn−1 = A [ψ (τ)] + vA [k (τ − v)]A [H (φ (τ))] . (3) R. Saadeh / Eur. J. Pure Appl. Math, 16 (3) (2023), 1491-1507 1495 Hence, substituting the ICs (2) in (3) and simplifying equation (3), we obtain A [φ (τ)] = 1 v δ0 + 1 v2 δ1 + . . .+ 1 vn δn−1 + 1 vn−1 A [ψ (τ)] + 1 vn−2 A [k (τ − v)]A [H (φ (τ))] . (4) Now, utilizing the Adomian decomposition method to treat the nonlinear functionH (φ (τ)), we have to present φ (τ) as an infinite series with the components: φ (τ) = ∞∑ i=0 φi(τ) = φ0 (τ) + φ1 (τ) + . . . , (5) where the components φi(τ), τ = 0, 1, . . ., are determined by the recurrence relation and the nonlinear term H (φ (τ)) can be expressed as H (φ (τ)) = ∞∑ i=0 Ai(τ), (6) where Ai (τ), i = 0, 1, 2, . . . are defined as Ai = 1 i! di dλi H  i∑ j=0 λjφj ∣∣∣∣∣∣ λ=0  , i = 0, 1, 2, · · · . (7) The Ai’s are called the Adomian polynomials for the nonlinear function H(φ (τ)), that can be determined by A0 = H (φ0) , A1 = φ1H ′ (φ0) , A2 = φ2H ′ (φ0) + 1 2! φ2 1H ′′ (φ0) , (8) A3 = φ3H ′ (φ0) + φ1φ2H ′′ (φ0) + 1 3! φ3 1H ′′′ (φ0) , A4 = φ4H ′ (φ0) + ( 1 2! φ2 2 + φ1φ3 ) H ′′ (φ0) + 1 2! φ2 1φ2H ′′′ (φ0) + 1 4! φ4 1H (4) (φ0) . Thus, substituting equations (5) and (6) in equation (4), we get A [ ∞∑ i=0 φi(τ) ] = 1 v δ0 + 1 v2 δ1 + . . .+ 1 vn δn−1 (0) + 1 vn−1 A [ψ (τ)] + 1 vn−2 A [k (τ − v)]A [ ∞∑ i=0 Ai(τ) ] . (9) R. Saadeh / Eur. J. Pure Appl. Math, 16 (3) (2023), 1491-1507 1496 The recursive relation from Adomian decomposition method implies A [φ0 (τ)] = 1 v δ0 + 1 v2 δ1 + . . .+ 1 vn δn−1 (0) + 1 vn−1 A [ψ (τ)] . (10) From equation (9), one can get A [φn+1 (τ)] = 1 vn−2 A [k (τ − v)] A [An(τ)] . (11) Operating the inverse Aboodh transform to the equations in (10) and (11) recursively, one can obtain the values of the components φ0 (τ) , φ1 (τ) , · · · . The solution of the Volterra IDE (1) is φ (τ) = φ0 (τ) + φ1 (τ) + . . . . Remark 1. A necessary condition for equation (11) to be well defined is that lim v→∞ 1 vn−2 A [k (τ)] = 0. The presented method is effective in establishing approximate solutions of nonlinear Volterra IDEs. To test the validity of the method, we discuss some applications and compute the maximum absolute error, defined as AbsErr = max |φexact − φapp|, which is given in some interval. 4. Applications In this section, we apply ADM to solve some applications of Volterra IDEs, and com- pute the maximum absolute error to show the efficiency of our results. Problem 1. Consider the following nonlinear Volterra integral equation: φ (τ) = 2τ − τ4 12 + 1 4 ∫ τ 0 (τ − u)φ2 (u) du. (12) Solution The exact solution of equation (12) is φ (τ) = 2τ . To get the solution by ADM, we apply Aboodh transform to equation (12) to get Φ (v) = A [ 2τ − τ4 12 ] + 1 4 vA [τ ]A [ φ2 (τ) ] = 2 v3 − 5! 12 v6 + 1 4 v 1 v3 A [ φ2 (τ) ] . (13) Substituting the value of the series solution Φ (v) and the Adomian components for φ2 (u), we obtain A [φ0 (τ)] = 2 v3 − 5! 12v6 , R. Saadeh / Eur. J. Pure Appl. Math, 16 (3) (2023), 1491-1507 1497 A [φn+1 (τ)] = 1 4v2 A [An (τ)] , n ≥ 0. For the nonlinear term φ2 (v), it can be decomposed using the formula in equation (7), one can obtain the following components A0 = φ2 0, A1 = 2φ0φ1, A2 = φ2 1 + 2φ0φ2, (14) A3 = 2φ1φ2 + 2φ0φ3, A4 = φ2 2 + 2φ1φ3 + 2φ0φ4. Making comparisons in the iterative form of equation (7) and applying the inverse Aboodh transform, we obtain φ0 (τ) = 2τ − τ4 12 , φ1 (τ) = τ4 12 − τ7 126 + τ10 51840 , φ2 (τ) = τ7 504 − τ10 181440 + 127 τ13 56609280 − τ16 298598400 , φ3 (τ) = τ4 12 − τ7 504 + τ10 2792 − 19τ13 14152320 + 71τ16 2264371200 − 7893 τ19 575787643000000 . Thus, the approximate solution can be expressed as φ (τ) = φ0 (τ) + φ1 (τ) + φ2 (τ) + φ3 (τ) + . . . = 2τ + τ4 12 − τ7 126 − τ10 362880 + 51τ13 56609280 + . . . . Table 2, below presents the values of the exact solution and approximate solution of Problem 1, and to test the efficiency we compute the absolute error. Table 2: The exact and approximate solution of equation (12), and the absolute error. Nodes Exact Solution Approximate Solution Absolute Error 0.0 0.0 0.0000000000 0.0000000000 0.1 0.2 0.2000083325 0.0000083325 0.2 0.4 0.4001332317 0.0001332317 0.3 0.6 0.6006732643 0.0006732643 0.4 0.8 0.8021203322 0.0021203322 0.5 1.0 1.0051463480 0.0051463480 0.6 1.2 1.2105779450 0.0105779450 0.7 1.4 1.4193552730 0.0193552730 0.8 1.6 1.6328034650 0.0328034650 0.9 1.8 1.8508857190 0.0508857190 1.0 2.0 1.8833526250 0.1166473750 R. Saadeh / Eur. J. Pure Appl. Math, 16 (3) (2023), 1491-1507 1498 In the following figure below, we sketch the exact and approximate solutions in Figure 1 below. In Figure 2, we sketch the absolute error of the exact and approximate solution of Problem 1. Figure 1: The exact and approximate solutions of the Problem 1. Figure 2: The absolute error of the exact and approximate solutions of equation 12. Problem 2. Consider the following nonlinear Volterra integral equation: φ (τ) = τ + ∫ τ 0 φ2 (u) du. (15) Solution. The exact solution of equation (15) is φ (τ) = tan τ . Applying Aboodh transform to equation (15), we get Φ (v) = 1 v2 + 1 v A [ φ2 (τ) ] . (16) Thus, by similar arguments to Problem 1, one can obtain φ0 (τ) = τ, φ1 (τ) = τ3 3 , R. Saadeh / Eur. J. Pure Appl. Math, 16 (3) (2023), 1491-1507 1499 φ2 (τ) = 2τ5 15 , φ3 (τ) = 17τ7 315 . Thus, the approximate solution can be expressed as φ (τ) = τ + τ3 3 + 2τ5 15 + 17τ7 315 + . . . . Table 3 below presents the values of the exact solution and approximate solution of Prob- lem 2, and to test the efficiency we compute the absolute error below. Table 3: The exact and approximate solutions of equation (15) and the absolute error. Nodes Exact Solution Approximate Solution Absolute Error 0.0 0.00000000000 0.0000000000 0.0000000000 0.1 0.1002940335 0.1003346721 0.0000406386 0.2 0.2026262629 0.2027100241 0.0000837612 0.3 0.3092040035 0.3093358029 0.0001317994 0.4 0.4226035289 0.4227870883 0.0001835594 0.5 0.5460413117 0.5462549603 0.0002136486 0.6 0.6837824776 0.6838787656 0.0000962880 0.7 0.8418070516 0.8411871844 0.0006198672 0.8 1.0289756740 1.0256752970 0.0033003770 0.9 1.2592215210 1.2475448490 0.0116766720 1.0 1.5560303730 1.5206349210 0.03539545198 In the following figure below, we sketch the exact and approximate solutions of Problem 2 in Figure 3 below. Figure 3: The exact and approximate solutions of Problem 2. R. Saadeh / Eur. J. Pure Appl. Math, 16 (3) (2023), 1491-1507 1500 Figure 4: The absolute error of the exact and approximate solutions of equation 15. Problem 3. Consider the following nonlinear Volterra IDE of the form φ′ (τ) = 3 2 eτ − 1 2 e3τ + ∫ τ 0 eu−τφ3 (τ) dτ. (17) φ (0) = 1. (18) Solution. Applying Aboodh transform to equation (17), we get Φ (v) = 1 v + 3 2v(v − 1) − 1 2v (v − 3) + v 1 v (v − 1) A [ φ3 (τ) ] . In an equivalent form, we have Φ (v) = 1 v + 3 2v(v − 1) − 1 2v (v − 3) + 1 v − 1 A [ φ3 (τ) ] . Now, we have A [φ0 (τ)] = 1 v + 3 2v(v − 1) − 1 2v (v − 3) , A [φn+1 (τ)] = 1 v − 1 A [An(τ)] , n ≥ 0. (19) The Adomian polynomials An(τ) of φ 3 (τ), can be determined as A0 = φ3 0, A1 = 3φ2 0φ1, A2 = 3φ2 0φ2 + 3φ0φ 3 1, A3 = 3φ2 0φ3 + 6φ0φ1φ2 + φ3 1. R. Saadeh / Eur. J. Pure Appl. Math, 16 (3) (2023), 1491-1507 1501 Taking the inverse Aboodh transform to the functions (19) and use the given recursive relation, one can obtain φ0 (τ) = 1 + τ − 1 2 τ3 − τ4 2 − 13 40 τ5 + . . . , φ1 (τ) = 1 2 τ2 + 2 3 τ3 + 5 12 τ4 + 7 120 τ5 + . . . , φ2 (τ) = 1 8 τ4 + 11 40 τ5 + . . . . Hence, the approximate series solution of Problem 3 is φ (τ) = 1 + τ + τ2 2! + τ3 3! + τ4 4! + . . . , which converges to the exact solution φ (τ) = eτ . Table 4 below, presents the values of the exact solution and approximate solution of Problem 3, and to test the efficiency we compute the absolute error. Table 4: The exact and approximate solution of Problem 3, and the absolute error. Nodes Exact Solution Approximate Solution Absolute Error 0.0 1 1 0 0.1 1.1051709181 1.1051709181 2.2204460493× 10−16 0.2 1.2214027582 1.2214027582 0 0.3 1.3498588076 1.3498588076 2.2204460493× 10−16 0.4 1.4918246976 1.4918246976 2.2204460492× 10−16 0.5 1.6487212707 1.6487212707 8.8817841970× 10−16 0.6 1.8221188004 1.8221188004 9.5479180118× 10−15 0.7 2.0137527075 2.0137527075 8.1268325403× 10−14 0.8 2.2255409285 2.2255409285 5.3290705182× 10−13 0.9 2.4596031112 2.4596031112 2.7911006839× 10−12 1.0 2.7182818285 2.7182818284 1.228617207× 10−11 In the following figure below, we sketch the exact and approximate solutions in Figure 5 below. We also sketch the absolute error of the exact and approximate solutions of Problem 3. R. Saadeh / Eur. J. Pure Appl. Math, 16 (3) (2023), 1491-1507 1502 Figure 5: The exact and approximate solutions of Problem 3. Figure 6: The absolute error of the exact and approximate solutions of Problem 3. Problem 4. Consider the following nonlinear Volterra IDE of the form φ′ (τ) = −2 sin τ − 2τ 3 cos τ + ∫ τ 0 cos (u− τ)φ2 (τ) dτ, (20) φ (0) = 1. (21) Solution. Applying the same procedure from the previous examples, we can obtain φ0 (τ) = 1− τ − τ2 + 1 2 τ3 + 1 12 τ4 − 11 120 τ5 + . . . , φ1 (τ) = 1 2 τ2 − 1 3 τ3 − 1 8 τ4 + 1 6 τ5 + . . . , φ2 (τ) = 1 12 τ4 − 1 12 τ5 + . . . . Thus, the approximate solution of (20) and (21) can be expressed as φ (τ) = ( 1− τ2 2! + τ4 4! + . . . ) − ( τ − τ3 3! − τ5 5! + . . . ) , R. Saadeh / Eur. J. Pure Appl. Math, 16 (3) (2023), 1491-1507 1503 which converge to the exact solution φ (τ) = cos τ − sin τ . Table 5 below presents the values of the exact solution and approximate solution of Prob- lem 4, and to test the efficiency we compute the absolute error. Table 5: The exact and ARA-DM solutions of Problem 4, and the absolute error. Nodes Exact Solution Approximate Solution Absolute Error 0.0 1 1 0 0.1 0.8951707486 0.8951709167 0.0000001680 0.2 0.7813972470 0.7814026667 0.0000054196 0.3 0.6598162825 0.65985775 0.0000414675 0.4 0.5316426517 0.5318186667 0.00017601497 0.5 0.3981570233 0.3986979167 0.0005408933 0.6 0.2606931415 0.262048 0.0013548589 0.7 0.12062450 0.1235714167 0.0029469166 0.8 -0.0206493816 -0.0148693333 0.0057800482 0.9 -0.1617169414 -0.151241750 0.0104751914 1.0 -0.3011686789 -0.2833333333 0.0178353456 In the following figure below, we sketch the exact and approximate solutions of Problem 4 in Figure 7 below. Figure 7: The exact and approximate solutions of the nonlinear Problem 4. REFERENCES 1504 Figure 8: The absolute error of the exact and approximate solutions of Problem 4. 5. Conclusion The purpose of this study is to provide a new efficient method for solving nonlinear Volterra IDEs. We presented approximate solutions of a family of nonlinear IDE in a form of infinite series solutions using the ADM, that is a combines Aboodh transform with the decomposition technique. Some examples of Volterra IDEs are discussed to verify the validity and applicability of the proposed method. As a result, it turned out that the ADM is an effective and simple method for solving nonlinear IDEs. In the future, we will modify the method [29, 30] and solve nonlinear fractional integral equations of several types [23–25, 33]. Funding statement This research received no external funding. Acknowledgements The author express their gratitude to the dear referees, who wish to remain anonymous, and the editor for their helpful suggestions, which improved the final version of this paper. Conflict of interest The author declares no conflict of interest. References [1] K S Aboodh. The New Integral Transform’Aboodh Transform. Global Journal of Pure and Applied Mathematics, 9(1):35–43, 2013. 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