EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 17, No. 3, 2024, 1497-1515 ISSN 1307-5543 – ejpam.com Published by New York Business Global An Efficient Numerical Approach Based On the Adomian Chebyshev Decomposition Method for Two-Point Boundary Value Problems Amani AL-Refaidi2, Nawal ALZaid1, Huda Bakodah1,∗, Mariam AL-Mazmumy1 1 Department of Mathematics and statistics, Faculty of Science, University of Jeddah, P.O. Box 80327, Jeddah, Saudi Arabia. 2 Department of Mathematics, AL-Qunfudhah University college, Umm Al-Qura University, Makkah, Saudi Arabia. Abstract. The current manuscript devises an efficient numerical method for solving two-point nonhomogeneous Boundary Value Problems (BVPs) with Dirichlet conditions. The method is based on the application of the celebrated Adomian Decomposition Method (ADM) and, the Chebyshev polynomials. This method which refers to ”Adomian Chebyshev Decomposition Method” (ACDM) is further proved to be a robust numerical method as the associated nonhomogeneous terms are successfully reinstated with a reliable Chebyshev series. Lastly, a comparative study between the acquired numerical results and the existing exact solutions of the test problems has been established to demonstrate the salient features of the devised method. 2020 Mathematics Subject Classifications: 34A30,34A34,34B15 Key Words and Phrases: Boundary Value Problems (BVPs), Adomian Decomposition Method (ADM), Chebyshev polynomials, Adomian Chebyshev Decomposition Method (ACDM), Dirichlet conditions 1. Introduction The celebrated Adomian decomposition method (ADM) is a well-known semi-analytical method that has been attested by many researchers to be an effective approach for solv- ing wide classes of differential, integral and, integro-differential equations. The method reveals the solution as a rapidly convergent series that tends to accurately converge to the exact analytical solution [2, 3, 7, 8, 11, 24]. Additionally, the method is so flexible to be applied to the governing functional equations directly without the need of either ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v17i3.5232 Email addresses: ahrfede@uqu.edu.sa (A. Al-Refaidi), naalzaid@uj.edu.sa (N. Al-Zaid), hobakodah@uj.edu.sa (H. Bakodah), mhalmazmumy@uj.edu.sa (M. Al-Mazmumy) https://www.ejpam.com 1497 © 2024 EJPAM All rights reserved. H. Bakodah et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 1497-1515 1498 restricting the method/solution to a particular domain, or by making use of certain sim- plifications like perturbation, discretization, or linearization [23] . Furthermore, like any other mathematical method in the literature, various researchers have equally proposed different extensions and modifications to the ADM, thereby improving the convergence rate of the method and, also, extending its applicability, [6, 10, 12, 13, 20]. In addition, the utilization of Green’s function approach on the BVPs for treating ordinary differen- tial equations (ODEs) has been overcome with the invention of the ADM[5, 21]; as the acquisition of approximate analytic solutions are attained and greatly facilitated. There are numerous efficient methods based on the ADM for solving BVPs of ODEs such as the double decomposition method [3, 4] and, the Duan-Rach modified decomposition method [16] to mention a few. However, in the current manuscript, we shall devise an efficient numerical method based on the application of the ADM and, Chebyshev polynomials to solve a class of two-point nonhomogeneous BVPs with Dirichlet boundary conditions. This method that called ACDM makes use of the traditional ADM together with the Chebyshev orthogonal polynomials to solve BVPs. More, the method first appeared in [19] to solve only certain initial value problems and; thus, the method is extended in the present study in order to be able to cover a wide range of applications. Additionally, a comparative study between the acquired numerical results by the devised method and the existing exact solutions of the test problems will be established to demonstrate certain salient features of the devised method. Lastly, we arrange the manuscript in the following way: in sections 2 and 3, we present the methods based on the traditional ADM and ACDM for solving BVPs with Dirichlet boundary conditions, respectively. Section 4 gives the numerical application of the devised method in Section 3; while Section 5 gives some concluding notes. 2. Adomian Decomposition Method (ADM) for BVPs Let us consider the generalized second-order nonlinear version of the nonhomogeneous ODE expressed in operator form as follows Lu(x) +Ru(x) +Nu(x) = g(x), (1) together with the following Dirichlet boundary conditions u (α1) = β1 and u (α2) = β2, (2) where β1 and β2 are real constants, L is the highest linear operator and, R is an operator with degree less than of L;N is the nonlinear operator, while the function g(x) on the other side of the above equation is a prescribed source term, which is a given continuous function. What’s more, if the operator L is assumed to be invertible and, further apply its corre- sponding inverse operator L−1 to all the parts of Eq. (1), we get after solving for Lu(x) the following expression for u(x) u(x) = Φ(x) + L−1(g(x))− L−1(Ru(x))− L−1(Nu(x)), (3) H. Bakodah et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 1497-1515 1499 where Φ(x) satisfies LΦ(x) = 0. In particular, since we are considering a generalized second-order nonlinear nonhomogeneous ODE, let us consider the inverse operator of L to be a two-fold definite integration of the form L−1(.) = ∫ x α1 ∫ x α1 (.)dxdx. (4) Appling L−1 defined above to all the parts of Eq. (1) and using the boundary condition u (α1) = β1 yields u(x) = u (α1) + z (x− α1) + L−1g(x)− L−1Ru− L−1Nu, (5) where z = u′ (α1). Next, by decomposing the solution u and the nonlinear term Nu to series forms, say u = ∑∞ n=0 un and, Nu = ∑∞ n=0An, where the components un(x), n ≥ 0, are to be determined in a recursive manner, and An ’s are the Adomian polynomials that take care of the nonlinear terms and, recurrently determined via the following formula [2, 3, 7, 8, 11, 24] An = 1 n! dn dλn [ N ( n∑ i=0 λiui )] λ=0 , n = 0, 1, 2, . . . (6) and thereafter substitute them into the above equation yields the following ∞∑ n=0 un = u (α1) + z (x− α1) + L−1g(x)− L−1R ( ∞∑ n=0 un ) − L−1 ( ∞∑ n=0 An ) , (7) More so, the ADM identifies the zeroth component u0(x) with the terms emanating from the nonhomogeneous term and boundary conditions, and the rest follows recurrently as given in the following recurrence relation{ u0(x) = u (α1) + z (x− α1) + L−1g(x), un+1(x) = −L−1Run − L−1An, n ≥ 0. (8) The recurrence relation or procedure could be carried out to calculate as many components as we like. Furthermore, using the second boundary condition u (α2) = β2 in the obtained series solution and comparing coefficients of like powers of x leads to the determination of z. Moreover, in many BVPs of exact solutions, the obtained series provides the solution in a closed-form. 3. Adomian Chebyshev Decomposition Method (ACDM) for BVPs Here, we present the methodology as the mixture of the ADM and, the Chebyshev poly- nomials. In numerical analysis, it is often important to find the polynomials Pn, of specified H. Bakodah et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 1497-1515 1500 maximum degree n, such that, for a function f(x) on the interval [a, b], |f(x)− Pn(x)| has as small maximum value over [a, b] as possible. Usually, this polynomials Pn is rather difficult to find, but a very good approximation to Pn can be found with relative easily. This approximation takes the form of a linear combination of a special group of polynomials that are named after the prolific mathematician Pafnuty Lvovich Chebyshev (1821-1894). Definition 1. [14] (1). The Chebyshev polynomials of the first kind are defined as Tn(x) = cosnθ, x = cos θ, −1 ≤ x ≤ 1, n = 0, 1, 2, . . . (9) (2). The Chebyshev polynomials of the second kind are defined as Un(x) = sin[(n+ 1)θ] sin θ , x = cos θ, −1 ≤ x ≤ 1, n = 0, 1, 2, . . . (10) It may be instantly obvious from the above definitions that Chebyshev polynomials are indeed polynomials. Therefore, note that the trigonometric identity, cos[(n+ 1)θ] + cos[(n− 1)θ] = 2 cos θ cosnθ, (11) supplies a recurrence relation for the functions as, Tn(x) = cosnθ, (12) Tn+1(x) + Tn−1(x) = 2xTn(x), (13) or equally expressed as Tn+1(x) = 2xTn(x)− Tn−1(x). (14) Since T0(x) = cos 0 = 1 and T1(x) = cos θ = x, Eq. (14) can be used to generate any number of the Tn(x) in their specific polynomial forms as T2(x) = 2x2 − 1, T3(x) = 4x3 − 3x, T4(x) = 8x4 − 8x2 + 1, T5(x) = 16x5 − 20x3 + 5x, ... (15) Besides, many mathematical techniques have been devised based on the Chebyshev poly- nomials to treat different types of functional equations; more especially, with regards to its discrete orthogonality relations which were found to have considerable advantages. For instance, Hoseini [19] presented a promising modification of the ADM through the appli- cation of the Chebyshev polynomials to solve initial value problems. Moreover, we shall expand the work done in [19] by performing approximations using Chebyshev polynomials with fractional function as in the Padé method [9], but except by replacing each term of H. Bakodah et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 1497-1515 1501 x in Padé approximation with the Chebyshev polynomials of degree k. Furthermore, to derive the ACDM based on the ADM and Chebyshev polynomials, we first rewrite the function g(x) in Eq. (8) in terms of the Chebyshev series as follows g(x) = m∑ i=0 aiTi(x), (16) where Ti(x) are the first kind of orthogonal Chebyshev polynomials, and the coefficients ak are determined from the orthogonal polynomials as follows{ a0 = 1 π ∫ 1 −1 f(x)T0√ 1−x2 dx, ak = 2 π ∫ 1 −1 f(x)Tk(x)√ 1−x2 dx, k ≥ 1. (17) Therefore, from Eq. (8), we, thus obtain the following ACDM recurrent relation as follows{ u0(x) = u (α1) + z (x− α1) + L−1 (a0T0(x) + a1T1(x) + · · ·+ amTm(x)) , un+1(x) = −L−1Run − L−1An, n ≥ 0. (18) Finally, in what follows, we will be demonstrating the devised ACDM scheme given in Eq. (18) to obtain an approximate solution via φn = ∑n−1 n=0 un. Remark 1. [14] The technique for using the Chebyshev polynomial is extended to a general closed interval [a, b] through the change of the following variables t = 1 2 [(b− a)x+ a+ b]. (19) 4. Numerical applications In this section, certain BVPs have been are considered as test problems to demon- strate the efficiency of the devised ACDM methodology. It is worthy to mention that all computations are performed by using Maple 18 version. Example 1. Consider the linear nonhomogeneous BVP [22] u′′ = u+ 2ex, u(0) = 0, u(1) = e, (20) that admits the following exact analytical solution u(x) = xex. Thus, according to the ADM, the BVP admits the following{ u0(x) = u(0) + zx+ L−1g(x), un+1(x) = L−1un, n ≥ 0. H. Bakodah et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 1497-1515 1502 where L(.) = d2 dx2 (.), L −1(.) = ∫ x 0 ∫ x 0 (.)dxdx , and, g(x) = 2ex. We now make use of the Chebyshev expansion for the function g(x) from Eq. (16) say, g(x) ∼= ∑10 i=0 aiTi(2x − 1), where the coefficients a0 and ai are computed from Eq. (17) as a0 = 1 π ∫ 1 −1 g(0.5 + 0.5x)T0(x)√ 1− x2 dx, ai = 2 π ∫ 1 −1 g(0.5 + 0.5x)Ti(x)√ 1− x2 dx, i = 1, 2, . . . 10. Thus, g(x) becomes g(x) ∼= 2.0000000000000387 + 1.99999999999064x+ 1.0000000003722x2 +0.3333333275x3 + 0.0833333792x4 + 0.01666645453x5 + · · · Therefore, having determined g(x), the ACDM solution is recurrently given from Eq. (18) as u0 = zx+ 1.0000000000000193x2 + 0.33333333333177x3 + 0.083333333643x4 + 0.016666666377x5 + · · · , u1 = 0.166666667zx3 + 0.0833333333x4 + 0.0166666666x5 + 0.0027777777x6+ 0.0003968253899x7 + · · · , ... In the same way, we compute u2, u3 and u4, then we have φ5 = ∑4 i=0 ui. Now, to determine z, we substitute the boundary condition at x = 1 in φ5 to get φ5 = 1.543080630 + 1.175201168z. More, φ5 requires the boundary condition φ5(1) = e, such after solving this relation gives z = 1.0000000253. Thus, since the constant z is determined, the solution in a series form follows instantly. The matching of φ5 with the exact solution is accurate as demonstrated in Table 1. In Figure 1, we depict the curves of φ5 and the exact analytical solution u(x) = xex in the approximate region 0 ≤ x ≤ 1. H. Bakodah et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 1497-1515 1503 Table 1: Absolute error between the exact and ACDM solutions in Example 1. x Exact solution ACDM solution Absolute error 0.0 0 0 0 0.1 1.10517092× 10−1 1.10517094× 10−1 2.53649074× 10−9 0.2 2.44280552× 10−1 2.44280557× 10−1 5.09836706× 10−9 0.3 4.04957642× 10−1 4.04957650× 10−1 7.71122380× 10−9 0.4 5.96729879× 10−1 5.96729889× 10−1 1.04002264× 10−8 0.5 8.24360635× 10−1 8.24360649× 10−1 1.31822171× 10−8 0.6 1.09327128 1.09327130 1.60211319× 10−8 0.7 1.40962690 1.4096269 1.86512214× 10−8 0.8 1.78043274 1.78043276 2.00216909× 10−8 0.9 2.21364280 2.21364282 1.68194966× 10−8 1.0 2.71828183 2.71828183 1.× 10−29 Figure 1: Comparison between the exact and ACDM solutions in Example 1. Example 2. Consider the nonlinear nonhomogeneous BVP [17] u′′ = u2 + 2π2 cos 2πx− sin4 πx, u(0) = u(1) = 0, (21) having the following exact analytical solution u(x) = sin2 πx. Again, based on the ADM, the BVP have the following{ u0(x) = u(0) + zx+ L−1g(x), un+1(x) = L−1An, n ≥ 0. where Adomian polynomials An of a nonlinear term u2 are determined from Eq. (6) as H. Bakodah et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 1497-1515 1504 follows: A0 = u20, A1 = 2u0u1, A2 = 2u0u2 + u21, A3 = 2u0u3 + 2u1u2, ... In similar manner, g(x) becomes g(x) ∼= 19.739432398− 0.063110987x− 386.709633571x2 − 52.590635088x3 +1664.947764846x4 − 2526.987198109x5 + · · · Therefore, having determined g(x) above, the ACDM solution is recurrently given from Eq. (18), thus obtain φ5 = ∑4 i=0 ui, then we find z = 4.213845191× 10−6. Therefore, since the constant value of z is determined, the series solution of the problem follows instantly. The matching of φ5 with the exact solution is accurate as demonstrated in Table 2. In Figure 2, we depict the curves of φ5 and the exact analytical solution u(x) = sin2 πx in the approximate region 0 ≤ x ≤ 1. Table 2: Absolute error between the exact and ACDM solutions in Example 2 x Exact solution ACDM solution Absolute error 0.0 0 0 0 0.1 9.54915028× 10−2 9.54918613× 10−2 3.58489171× 10−7 0.2 3.45491503× 10−1 3.45492103× 10−1 6.00678252× 10−7 0.3 6.54508497× 10−1 6.54509467× 10−1 9.70293389× 10−7 0.4 9.04508497× 10−1 9.04510189× 10−1 1.69139683× 10−6 0.5 1.00000000 1.00000133 1.33401846× 10−6 0.6 9.04508497× 10−1 9.04510994× 10−1 2.49639628× 10−6 0.7 6.54508497× 10−1 6.54511059× 10−1 2.56152588× 10−6 0.8 3.45491503× 10−1 3.45494307× 10−1 2.80459804× 10−6 0.9 9.54915028× 10−2 9.54939626× 10−2 2.45981055× 10−6 1.0 −1.30000000× 10−26 2.47124121× 10−61 1.30000000× 10−26 H. Bakodah et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 1497-1515 1505 Figure 2: Comparison between the exact and ACDM solutions in Example 2 Example 3. Consider the following nonhomogeneous BVP [18] u′′′(x) = u(x)− 3ex, u(0) = 1, u(1) = u′(0) = 0, (22) that admits the following exact analytical solution u(x) = (1− x)ex. Therefore, the ADM of the BVP is recurrently given as followsu0(x) = u(0) + xu′(0) + tx2 2 + L−1g(x), un+1(x) = L−1un, n ≥ 0, where L(.) = d3(.) dxx , g(x) = −3ex and, L−1(.) = ∫ x 0 ∫ x 0 ∫ x 0 (.)dxdxdx. As proceeding before, we determine g(x) as follows g(x) ∼= −3.0000000000000581− 2.99999999998x− 1.500000000558x2− 0.49999999133x3 + · · · In similar manner, we get t = −0.9999999999985. Therefore, since the constant value of t is determined, the series solution of the problem follows instantly. The matching of φ5 with the exact solution is remarkable as reported in Table 3. In Figure 3, we depict the curves of φ5 and the exact analytical solution u(x) = (1− x)ex in the approximate region 0 ≤ x ≤ 1. H. Bakodah et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 1497-1515 1506 Table 3: Absolute error between the exact and ACDM solutions in Example 3 x Exact solution ACDM solution Absolute error 0.0 1 1 0 0.1 9.94653826× 10−1 9.94653826× 10−1 7.49228630× 10−15 0.2 9.77122207× 10−1 9.77122207× 10−1 2.99634613× 10−14 0.3 9.44901165× 10−1 9.44901165× 10−1 6.74469668× 10−14 0.4 8.95094819× 10−1 8.95094819× 10−1 1.19977986× 10−13 0.5 8.24360635× 10−1 8.24360635× 10−1 1.87622933× 10−13 0.6 7.28847520× 10−1 7.2887520× 10−1 2.70272115× 10−13 0.7 6.04125812× 10−1 6.04125812× 10−1 3.65511599× 10−13 0.8 4.45108186× 10−1 4.45108186× 10−1 4.56646544× 10−13 0.9 2.45960311× 10−1 2.45960311× 10−1 4.57186783× 10−13 1.0 0 1× 10−30 1× 10−30 Figure 3: Comparison between the exact and ACDM solutions in Example 3 Example 4. Consider the following nonlinear nonhomogeneous BVP [1] u(iv)(x) = u2(x)− x10 + 4x9 − 4x8 − 4x7 + 8x6 − 4x4 + 120x− 48, u(0) = u′(0) = 0, u(1) = u′(1) = 1, (23) having the following exact analytical solution u(x) = x5 − 2x4 + 2x2. H. Bakodah et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 1497-1515 1507 Accordingly, the ADM gives the following recurrent relation to the BVP u0(x) = u(0) + xu′(0) + tx2 2! + kx3 3! + L−1g(x), un+1(x) = L−1An, n ≥ 0. where L(.) = d4(.) dx4 , g(x) = −x10 + 4x9 − 4x8 − 4x7 + 8x6 − 4x4 + 120x− 48 and L−1(.) =∫ x 0 ∫ x 0 ∫ x 0 ∫ x 0 (.)dxdxdxdx. In similar manner, g(x) becomes g(x) ∼= −48 + 120x− 1× 10−29x2 − 4x4 + 8x6 − 4x7 + · · · . Now, we determine t = 3.9999999999, and k = 3.319466361× 10−17. Therefore, since the constant values of t and k are determined, the series solution of the problem follows instantly. The matching of φ5 with the exact solution is amazing as reported in Table 4. In Figure 4, we depict the curves of φ5 and that of the exact analytical solution u(x) = x5 − 2x4 + 2x2 in the approximate region 0 ≤ x ≤ 1. Table 4: Absolute error between the exact and ACDM solutions in Example 1 x Exact solution ACDM solution Absolute error 0.0 0 0 0 0.1 1.98100000× 10−2 1.9810000× 10−2 4.77923286× 10−20 0.2 7.71200000× 10−2 7.71200000× 10−2 1.69039567× 10−19 0.3 1.66230000× 10−1 1.66230000× 10−1 3.30547636× 10−19 0.4 2.79040000× 10−1 2.79040000× 10−1 4.99126063× 10−19 0.5 4.06250000× 10−1 4.06250000× 10−1 6.41596782× 10−19 0.6 5.38560000× 10−1 5.38560000× 10−1 7.24810853× 10−19 0.7 6.67870000× 10−1 6.67870000× 10−1 7.15675166× 10−19 0.8 7.88480000× 10−1 7.88480000× 10−1 5.81377176× 10−19 0.9 8.98290000× 10−1 8.98290000× 10−1 2.96356361× 10−19 1.0 1 1 2.00000000× 10−29 H. Bakodah et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 1497-1515 1508 Figure 4: Comparison between the exact and ACDM solutions in Example 4 Example 5. Consider the linear nonhomogeneous system of second-order BVPs [15]{ u′′(x) + (2x− 1)u′(x) + cos(πx)v′(x) = g1(x), u(0) = u(1) = 0, 0 ≤ x ≤ 1, v′′(x) + xu(x) = g2(x), v(0) = v(1) = 0, 0 ≤ x ≤ 1, (24) where g1(x) and g2(x) are given by g1(x) = −π2 sin(πx) + (2x − 1)π cos(πx) + (2x− 1) cos(πx), and g2(x) = 2 + x sin(πx). In addition, the following exact analytical solution set satisfies the system u(x) = sin(πx) and, v(x) = x2 − x. Consequently, the ADM recurrent relation for the system is given by u0(x) = u(0) + z1x+ L−1g1(x), v0(x) = v(0) + z2x+ L−1g2(x), un+1(x) = −L−1 ((2x− 1)u′n)− L−1 (cos(πx)v′n) , n ≥ 0, vn+1(x) = −L−1 (xun) , n ≥ 0. we use the Chebyshev expansion to determine the series forms of g1(x) and g2(x) from Eq. (16) say, g1(x) ∼= ∑10 i=0 aiTi(2x − 1), and g2(x) ∼= ∑10 i=0 biTi(2x − 1), where the coefficients a0, b0, ai and bi are computed from Eq. (17) as a0 = 1 π ∫ 1 −1 g1(0.5 + 0.5x)T0(x)√ 1− x2 dx, b0 = 1 π ∫ 1 −1 g2(0.5 + 0.5x)T0(x)√ 1− x2 dx ai = 2 π ∫ 1 −1 g1(0.5 + 0.5x)Ti(x)√ 1− x2 dx, bi = 2 π ∫ 1 −1 g2(0.5 + 0.5x)Ti(x)√ 1− x2 dx i = 1, 2, . . . 10. H. Bakodah et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 1497-1515 1509 Additionally, g1(x) and g2(x) are explicitly found from Eq. (16) as g1(x) ∼= −4.141592658− 22.723090021x+ 20.437876252x2 + 10.12859625x3 + · · · g2(x) ∼= 1.999999988 + 0.000002814x+ 3.141480108x2 + 0.001753695x3 + · · · Now, we determine z1 = 3.141584852, and z2 = −1.0000214289. Therefore, since the constant values of z1 and z2 are determined above, the series solution of the problem follows instantly. The matching of ψ5 = ∑4 i=0 vi with the exact solution is commendable as reported in Table 5. In Figure 5, we depict the curves of ψ5 and the exact analytical solution v(x) = x2 − x in the approximate region 0 ≤ x ≤ 1. Table 5: Absolute error between the exact and ACDM solutions v(x) in Example 5 x ACDM solution ψ5(x) Exact solution v(x) Absolute error 0.0 0 0 0 0.1 −9.00021428× 10−2 −9.00000000× 10−2 2.14282294× 10−6 0.2 −1.60004284× 10−1 −1.60000000× 10−1 4.28395716× 10−6 0.3 −2.10006407× 10−1 −2.10000000× 10−1 6.40680363× 10−6 0.4 −2.40008430× 10−1 −2.40000000× 10−1 8.43004588× 10−6 0.5 −2.50010128× 10−1 −2.50000000× 10−1 1.01279366× 10−5 0.6 −2.40011187× 10−1 −2.4000000× 10−1 1.10872057× 10−5 0.7 −2.10010776× 10−1 −2.10000000× 10−1 1.07759673× 10−5 0.8 −1.60008755× 10−1 −1.60000000× 10−1 8.75509042× 10−6 0.9 −9.00049736× 10−2 −9.00000000× 10−2 4.97362094× 10−6 1.0 −3.45366895× 10−28 0 3.45366895× 10−28 Figure 5: Comparison between the exact and ACDM solutions in Example 5 for v(x) H. Bakodah et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 1497-1515 1510 The matching of φ5 = ∑4 i=1 ui with the exact solution is commendable as reported in Table 6. In Figure 6, we depict the curves of φ5 and the exact analytical solution u(x) = sin(πx) in the approximate region 0 ≤ x ≤ 1. Table 6: Absolute error between the exact and ACDM solutions u(x) in Example 5 x ACDM solution φ5(x) Exact solution u(x) Absolute error 0.0 1.00000000× 10−27 0 1.00000000× 10−27 0.1 3.09016283× 10−1 3.09016994× 10−1 7.11807212× 10−7 0.2 5.87783841× 10−1 5.87785252× 10−1 1.41152139× 10−6 0.3 8.09014318× 10−1 8.09016994× 10−1 2.67611588× 10−6 0.4 9.51051322× 10−1 9.51056516× 10−1 5.19434595× 10−6 0.5 9.99991110× 10−1 1 8.89040303× 10−6 0.6 9.51043993× 10−1 9.51056516× 10−1 1.2523000× 10−5 0.7 8.09002815× 10−1 8.09016994× 10−1 1.41795077× 10−5 0.8 5.87772869× 10−1 5.87785252× 10−1 1.23837067× 10−5 0.9 3.09009889× 10−1 3.09016994× 10−1 7.10572450× 10−6 1.0 −3.03006339× 10−27 −4.97115803× 10−31 3.02956627× 10−27 Figure 6: Comparison between the exact and ACDM solutions in Example 5 for u(x) Example 6. Consider the nonlinear nonhomogeneous system of second-order BVPs [15]{ u′′(x) + xu′(x) + cos(πx)v′(x) = g1(x), u(0) = u(1) = 0, 0 ≤ x ≤ 1, v′′(x) + xu′(x) + x ( v′(x) )2 = g2(x), v(0) = v(1) = 0, 0 ≤ x ≤ 1, (25) H. Bakodah et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 1497-1515 1511 where g1(x) and g2(x) are given by g1(x) = sin(x)+ ( x2 − x+ 2 ) cos(x)+(1− 2x) cos(πx), g2(x) = −2 + x sin(πx) + ( x2 − x ) cos(x) + x(1− 2x)2. Additionally, the above system admits the following exact analytical solution u(x) = (x− 1) sin(x) and v(x) = x− x2. As preceded, the recurrent relation via the ADM takes the following form u0(x) = u(0) + z1x+ L−1g1(x), v0(x) = v(0) + z2x+ L−1g2(x), un+1(x) = −L−1xu′n − L−1 cos(πx)v′n, n ≥ 0, vn+1(x) = −L−1xu′n − L−1xAn, n ≥ 0, From Eq. (6), the nonlinear term Nv = (v′)2 is also expressed through the following Adomian polynomials An are given as A0 = (v′0) 2, A1 = 2(v′0)(v ′ 1), A2 = 2(v′0)(v ′ 2) + (v′1) 2, A3 = 2(v′0)(v ′ 3) + 2(v′1)(v ′ 2), ... In similar manner, using the relation given in Eq. (16), we equally obtain g1(x) ∼= 3.000000001− 2.000000476x− 4.934779553x2 + 10.20251612x3 + · · · , g2(x) ∼= −2.0− 1.5896928× 10−10x− 2.0x2 + 4.500000096x3 + · · · So, we find z1 = −1.000001152, z2 = 0.999830865. Therefore, since the constant values of z1 and z2 are determined, the series solution of the problem follows instantly. The matching of ψ5 = ∑4 i=0 vi with the exact solution is demonstrated in Table 7. In Figure 7 we depict the curves of ψ5 and the exact analytical solution v(x) = x − x2 in the approximate region 0 ≤ x ≤ 1. Table 7: Absolute error between the exact and ACDM solutions v(x) in Example 6. x ACDM solution ψ5(x) Exact solution v(x) Absolute error 0.0 1.24003340× 10−22 0 1.24003340× 10−22 0.1 8.99831359× 10−2 9.00000000× 10−2 1.68640966× 10−5 0.2 1.59966511× 10−1 1.6000000× 10−1 3.34886510× 10−5 0.3 2.09950234× 10−1 2.1000000× 10−1 4.97663491× 10−5 0.4 2.39934409× 10−1 2.4000000× 10−1 6.55905833× 10−5 0.5 2.49919498× 10−1 2.5000000× 10−1 8.05024359× 10−5 0.6 2.39906904× 10−1 2.4000000× 10−1 9.30960974× 10−5 0.7 2.09900091× 10−1 2.1000000× 10−1 9.99091708× 10−5 0.8 1.59906373× 10−1 1.6000000× 10−1 9.36273128× 10−5 0.9 8.99368847× 10−2 9.000000× 10−2 6.31153078× 10−5 1.0 1.29287648× 10−21 0 1.29287648× 10−21 H. Bakodah et al. / Eur. J. Pure Appl. Math, 17 (3) (2024), 1497-1515 1512 Figure 7: Comparison between the exact and ACDM solutions in Example 6 for v(x) The matching of φ5 with the exact solution is commendable as reported in Table 8. In Figure 8, we depict the curves of φ5 = ∑4 i=0 ui and the exact analytical solution u(x) = (x− 1) sin(x) in the approximate region 0 ≤ x ≤ 1. Table 8: Absolute error between the exact and ACDM solutions u(x) in Example 6. x ACDM solution φ5(x) Exact solution u(x) Absolute error 0.0 −2.15883226× 10−15 0 2.15883226× 10−15 0.1 −8.98493539× 10−2 −8.98500750× 10−2 7.21035532× 10−7 0.2 −1.58932458× 10−1 −1.558935465× 10−1 3.00633437× 10−6 0.3 −2.06857574× 10−1 −2.06864145× 10−1 6.57114337× 10−6 0.4 −2.33639930× 10−1 −2.3365100× 10−1 1.10754214× 10−5 0.5 −2.39696515× 10−1 −2.39712769× 10−1 1.62538649× 10−5 0.6 −2.25835101× 10−1 −2.25856989× 10−1 2.18878914× 10−5 0.7 −1.93237792× 10−1 −1.93265306× 10−1 2.7514182910−5 0.8 −1.43439660× 10−1 −1.43471218× 10−1 3.15577978× 10−5 0.9 −7.83043755× 10−2 −7.83326910× 10−2 2.83154484× 10−5 1.0 −1.48468813× 10−14 0 1.48468813× 10−14 REFERENCES 1513 Figure 8: Comparison between the exact and ACDM solutions in Example 6 for u(x) . 5. 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