EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 15, No. 4, 2022, 1917-1936 ISSN 1307-5543 – ejpam.com Published by New York Business Global The possible solutions for the two KdV-type equations using a semi-analytical Kamal-iteration method Amal Jasim Mohammed1,∗, Sohaib Talal Al-Ramadhani1, Rabeea Mohammed Hani Darghoth2 1 College of Education For Pure Sciences, Mathematics Department, University of Mosul, Iraq 2 College of Basic Education, Mathematics Department, University of Mosul, Iraq Abstract. In this paper, we used a semi-analytical method that combines the Kamal transform and a modified iteration method to solve the nonlinear homogeneous and nonhomogeneous modified Kerteweg-de Vries and the fifth-order Kerteweg-de Vries equations. The modified iteration method has been used to with a condition that guarantees fast convergence of the approximate solution. The analysis and convergence of the combined method have been discussed. Furthermore, the numerical simulations are presented to illustrate the effectiveness of the proposed semi-analytical method. 2020 Mathematics Subject Classifications: 35Q53, 35Q35,41A58, 74H15, 74J35 Key Words and Phrases: KdV, mKdV, FKdV equations, Modified Iteration method, Kamal transform 1. Introduction The Kerteweg-de Vries equation (KdV eq.) is a nonlinear integrable system that has various applications in fluid mechanics, hydromagnetics plasma, electrical transmission lines, and others, see for example [8]. Thus, numerical and semi-analytical methods are designed to solve the nonlinear partial differential equations (PDEs), where; several mecha- nism methods were employed to study and find out a suitable solution for the PDEs. Then, some of these methods are analytical and others are semi-analytical (numerical with ana- lytic technique) and numerical, for instance, homotopy analysis, differential transformation methods, and many others. These methods are used in different branches of sciences such as; physics, chemistry, and engineering which is why the numerical and semi-analytical methods have become more common in use than pure analytical methods [1, 7, 10, 16]. For example, the integral transforms; by Laplace, Fourier, Hilbert, and Sumida, have ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v15i4.4599 Email addresses: a.j.moha7@uomosul.edu.iq. (A.J. Mohammed), s.alramadhani@uomosul.edu.iq. (S.T. Al-Ramadhani), r.darghoth@uomosul.edu.iq. (R.M.H. Darghoth) https://www.ejpam.com 1917 © 2022 EJPAM All rights reserved. A.J. Mohammed, S.T. Al-Ramadhani, R.M.H. Darghoth / Eur. J. Pure Appl. Math, 15 (4) (2022), 1917-1936 1918 been smoothly used to solve linear homogeneous or nonhomogeneous ordinary and par- tial differential equations (PDEs) [4, 25]. For instance, heat, wave, telegraph equations, advection problems. Kamal transform (KT) is used to solve various types of differential and integral equations, see [2, 5, 15, 17, 26]. However, all these transforms cannot be used directly if they are employed for the nonlinear homogeneous or nonhomogeneous ordinary and PDEs, because there are no transforms for the nonlinear part. For that reason, some researchers have used the integral transform with the help of decomposition, variational methods and others to solve this problem [6, 23, 24]. The problem in this project is based on the nonlinear partial KdV eq. [3, 9, 11, 28] of the form (∂t − 6 w(x, t) ∂x + ∂xx)w(x, t) = 0, −∞ < x < ∞ t > 0, (1) where, the initial condition w(x, 0) = F (x), F (x) is a known function. A differentiable solution of eq. (1) is w = w(x, t) which is travelling wave solution tends to zero as |x| → ∞. This kind of equation is classified with the group of integrable equations such as; the nonlinear Schrödinger, Sine-Gordon, Kadomtsev-Petviashvili and Bogoyavlensky- Konopelchenko equations [12, 13, 19, 27]. In addition, the family of the nonlinear PDEs of the KdV eq. comes from the theory of dispersive wave motions. The solutions of these equations have been used to describe several physical phenomena like; ”widespread class, elastic scattering property” and others (see [26]). Here, we are interested in employing the modified iteration method (MI) [21], which was used for the homogeneous and nonhomogeneous nonlinear ordinary differential equa- tions (ODEs). In this paper, we used a combination of the KT [5, 17] and MI [21] method which we denote for as K-MI. Using this method, we have explored the acceptable solu- tion for two types of the family of KdV eq.: the modified KdV eq. (mKdV eq.) and the fifth order of KdV eq. (FKdV eq.). The outline of this paper is arranged as follows. The mathematical model has been presented in Section 2, where, we displayed theKT and some of its related theorems. Next, we have shown the combined method K-MI and applied it to the general description of nonlinear PDEs. The limit of iteration of this method has been discussed. In section 3 we have applied the K-MI method to several examples to show the success of the method in reaching accuracy of the results. Section 4 is related to comparing the classical iterations method to the modified one, both combined with the Kamal transform. Finally, our discussion and conclusion are given in section 5 and 6 respectively. 2. Mathematical Method 2.1. Kamal transform This transformation is one of the integral transforms that is employed to find out the solution to differential equations [22]. We have presented some of its properties. We have denoted the integral transform of the Kamal by K and it is specified as: K [S(t)] = ∫ ∞ 0 S(t) e− t υ t = G(υ), (2) A.J. Mohammed, S.T. Al-Ramadhani, R.M.H. Darghoth / Eur. J. Pure Appl. Math, 15 (4) (2022), 1917-1936 1919 where, l1 ≤ υ ≤ l2. This transform defined for functions on the set A A = { S(t) : ∃ M, l1, l2 > 0, |S(t)| < M e |t| lj , if t ∈ (−1)l × [0,∞) } (3) Theorem 1. Assume G(υ) is the KT of S(t) then [17]: K [ S(m)(t) ] = υ(−m)G(υ)− m−1∑ k=0 υk−m+1S(k)(0), for m ≥ 0, (4) where (S(m)) is the mth derivative. For m = 1, K [S′(t)] = υ(−1)G(υ)− S(0). Theorem 2. Let a be a constant and a ≥ 0 then [17]: K [a] = a υ. (5) Theorem 3. For an integer number m ≥ 0, the K [tm] is [17]: K [tm] = m! vm+1 (6) For more information about the properties of this transformation see [5, 22] and many others. In the next section, we use the above theorems to calculate the appropriate solution for the nonlinear PDEs with the help of the MI method. 2.2. Kamal-Modified Iteration method for the KdV equation types Consider the nonlinear PDEs of the form L̇[w(x, t)] +M [w(x, t)] +G(x, t) = 0, (7) where, L̇ is the time dependent operator, M is containing only the dependants of the variable x and w(x, 0) = F (x) is the initial condition (IC) of eq. (7). Here, F (x) and G(x, t) are known functions. First, we apply KT to eq. (7) K[L[w(x, t)]] +K[M [w(x, t)]] +K[G(x, t)] = 0, (8) next, we need to use theorems (1 and 2) and the IC in eq. (8) as W (x, υ) = υ w(x, 0)− υ K[M [w(x, t)] +G(x, t)]. (9) To find w(x, t), we need to take the inverse of Kamal transform (K−1) of eq. (9), which yields w(x, t) = w(x, 0)−K−1[υK[M [w(x, t)] +G(x, t)]]. (10) Second, we use the MI method on eq. (10) to deduce the solution of eq. (7). Let w(x, 0) = w0(x, t), be the first approximate solution then, eq. (10) becomes wn+1(x, t) = w0(x, 0)−K−1[υK[M [wn(x, t)] +G(x, t)]]. (11) A.J. Mohammed, S.T. Al-Ramadhani, R.M.H. Darghoth / Eur. J. Pure Appl. Math, 15 (4) (2022), 1917-1936 1920 Remark 1. During the calculation of classical iteration method, some terms are created, we need to cancel the un necessary terms which make noise to our calculation (see section (4). For that reason, we have defined the condition of the MI method as M [wn(x, t)] = Qn(x, t) +O(tn+1), n = 0, 1, 2, . . . (12) where, O(tn+1) are higher order terms to be neglected. Therefore, eq. (11) becomes: wn+1(x, t) = w0(x, 0)−K−1[υK[Qn(x, t) +G(x, t)]]. (13) Suppose that G(x, t) is smooth enough, so it has Taylor’s expansion in the variable t (about t = 0) G(x, t) = Gn(x, t) +O(tn+1), (14) where, Gn = n+1∑ k=0 Skt k = S0 + S1t+ S2t 2 + · · ·+ Sn+1t n+1, (15) Sk = 1 k! [ k tk G(x, t) ] , so, eq. (13) becomes wn+1(x, t) = w0(x, 0)−K−1[υK[Qn +Gn]]. (16) On the other hand, eq. (16) can be written as wn+1(x, t) = w0(x, 0)−K−1[υK[Qn−1+Gn−1]]−K−1[υK[(Qn−Qn−1)+(Gn−Gn−1)]], (17) set n = n− 1 in (16), then we have used it in eq. (17), yields, wn+1(x, t) = wn(x, 0)−K−1[υK[(Qn −Qn−1) + (Gn −Gn−1)]], (18) here, Q−1 = 0, and G−1 = 0. Now, let T [wn(x, t)] is define as T [wn(x, t)] = −K−1[υK[(Qn −Qn−1) + (Gn −Gn−1)]], (19) if we assume ηn+1 = T [η0 + η1 + η2 + · · ·+ ηn], such that η0 =w0, η1 =T [η0], η2 =T [η0 + η1] = T [w1], ... ηn+1 =T [η0 + η1 + η2 + · · ·+ ηn], (20) A.J. Mohammed, S.T. Al-Ramadhani, R.M.H. Darghoth / Eur. J. Pure Appl. Math, 15 (4) (2022), 1917-1936 1921 then, eq. (18) will be wn+1(x, t) = wn(x, 0) + T [wn(x, t)], (21) where, w1 =w0 + T [w0] = η0 + η1, w2 =w1 + T [w1] = η0 + η1 + T [η0 + η1], =η0 + η1 + η2, ... wn =η0 + η1 + η2 + · · ·+ ηn. (22) Therefore, we have obtained an approximate solution wn+1(x, t) as a series solution. Hence, w(x, t) is the sum of the iterative solutions, i.e: w(x, t) = lim n→∞ wn(x, t) = ∞∑ k=0 ηk. (23) 3. Numerical Examples In this section we clearly show the effectiveness of the K-MI method to solve the certain PDEs mKdV eq. and FKdV eq. The technique of semi-analytical (numerical with analytic ) K-MI method is used on the homogeneous and nonhomogeneous mKdV eq. and FKdV eq. This method started by the initial condition w(x, 0) as a zero iteration solution w0(x, t), then, we applied the KT and MI method on the PDE. Example 1. Consider the third order homogeneous mKdV eq. [26] wt(x, t) + 6 w2 wx(x, t) + w3x(x, t) = 0, (24) using the IC w(x, 0) = 2 λ eλx 1 + e2λx . Compering eq. (24) to eq. (7), It is clear that ẇ = ∂w(x, t) ∂t , M [w(x, t)] = 6 w2 wx(x, t) + w3x(x, t), and G(x, t) = 0, (25) using the K-MI method and following the procedure in section (2.2), we obtain M [wn(x, t)] = 6 w2 n w{n,x}(x, t) + w{n,3x}(x, t) = Qn(x, t) +O(tn+1), (26) the first iteration w1(x, t) of the homogeneous mKdV eq. can be found from substituting w0(x, t) in eq. (22). When n = 0, η0 = w0 = 2 λ eλx 1 + e2λx , (27) A.J. Mohammed, S.T. Al-Ramadhani, R.M.H. Darghoth / Eur. J. Pure Appl. Math, 15 (4) (2022), 1917-1936 1922 to find Q0, substitute eq. (27) in eq. (26) yields Q0 =M [w0(x, t)] = 6 w2 0 w{0,x}(x, t) + w{0,3x}(x, t) +O(t1), =− 2 λ4eλx ( −1 + e2λx ) (1 + e2λx) 2 . (28) Here, Q0 in (28) is time independent, so, substitute the value of Q0 in eq. (19) to find η1 η1 =T [w0] = −K−1[υK[(Q0 −Q−1)]], where Q−1 = 0, =−K−1[υK[−2 λ4eλx ( −1 + e2λx ) (1 + e2λx) 2 ]], since −2 λ4eλx(−1+e2λx) (1+e2λx) 2 is time independent, which means it is constant with respect to t, then K transform of a constant is v (Theorem 2) =K−1[υ(2 λ4eλx ( −1 + e2λx ) (1 + e2λx) 2 υ)], Theorem 2, =K−1[υ2(2 λ4eλx ( −1 + e2λx ) (1 + e2λx) 2 )], Theorem 3, and K−1[v2] is t (Theorem 3), so we have η1 = −2 λ4eλx ( 1− e2λx ) (1 + e2λx) 2 t. (29) Then, the first iteration w1 is w1(x, t) =η0 + η1 = 2 λ eλx 1 + e2λx − 2 λ4eλx ( 1− e2λx ) (1 + e2λx) 2 t. (30) When n = 1, we have substituted w1 in eq. (26) to find Q1. In this case, we neglected the coefficients of t with indices grater than or equal 2 (O(t2)). Here, Q1 is t independent, Q1 =6 w2 1 w{1,x}(x, t) + w{1,3x}(x, t) +O(t2), =− Aλ3 ( −2 e10λx − 17 e8λx − 28 e6λx + e12λx − 17 e4λx − 2 e2λx + 1 ) t B7 − A ( 4 e10λx + 5 e8λx + e12λx − 5 e4λx − 4 e2λx − 1 ) B7 + A ( e2λx − 1 ) B2 , (31) where, A = 2λ4 eλx and B = (1 + e2λx), and using equation (19), we have η2 =T [w1] = −K−1[υK[(Q1 −Q0)]], A.J. Mohammed, S.T. Al-Ramadhani, R.M.H. Darghoth / Eur. J. Pure Appl. Math, 15 (4) (2022), 1917-1936 1923 = λ7 ( e5λx − 6 e3λx + eλx ) B3 t2. (32) So, the second iteration w2 is w2(x, t) =η0 + η1 + η2 = 2 λ eλx B − A ( 1− e2λx ) B2 t+ A λ3 ( e4λx − 6 e2λx + 1 ) B3 t2 2 . (33) When n → ∞, the nth iteration (approximate solution) will be w(x, t) = lim n→∞ n∑ k=0 ηk w(x, t) = 2 λ eλx B − A ( 1− e2λx ) B2 t+ A λ3 ( e4λx − 6 e2λx + 1 ) B3 t2 2 − A λ5 ( e6λx − 23 e4λx + 23 e2λx − 1 ) B4 t3 3 + . . . (34) the series solution (34) of the mKdV eq. is the Taylor’s expansion in the variable t of the function w(x, t) = √ c sech( √ c(x− c t)), c > 0, (35) which is exactly the same as the one obtained in [26]. Figure (1) shows the solution deduced from the convergent Taylor’s series in eq. (35). Figure 1: The exact solution of eq. (24) induced from K-MI method eq. (35) when c = 4. This is soliton type of solution. A.J. Mohammed, S.T. Al-Ramadhani, R.M.H. Darghoth / Eur. J. Pure Appl. Math, 15 (4) (2022), 1917-1936 1924 Example 2. Consider the third order nonhomogeneous mKdV eq. [20] wt(x, t)− w2 wx(x, t) + w3x(x, t) = x − 3 t2x− ( x t− t3x )2 ( t − t3 ) , (36) using the IC w(x, 0) = 0. It is clear that, when we compare eq. (36) to eq. (7), we have ẇ = ∂w(x, t) ∂t , M [w(x, t)] = w3x(x, t)− w2 wx(x, t), (37) and G(x, t) = −3 t2x+ x− ( −t3x+ tx )2 (−t3 + t ) . (38) Similarly as in example (1), we have used K-MI method and following the procedure in section (2.2), we have M [wn(x, t)] = w{n,3x}(x, t)− w2 {n,x}(x, t) = Qn(x, t) +O(tn+1), (39) Gn(x, t) = −t9x2 + 3 t7x2 − 3 t5x2 + t3x2 + 3 t2x− x+O(tn+1), (40) here, the function Gn is a polynomial of degree (9) in the variable t. When n = 0, then η0 = w0 = 0, using (39) and (40) to find Q0(x, t) = 0, G0(x, t) = −x+−t9x2 + 3 t7x2 − 3 t5x2 + t3x2 + 3 t2x︸ ︷︷ ︸ neglect , (41) Hence, substituting Q0 and G0 in eq (19), we have η1 =T [η0 = w0] =−K−1[υK[(Q0 −Q−1) + (G0 −G−1)]], G−1 = 0 = xt, (42) therefore, the first iteration w1 is w1(x, t) =η0 + η1 = xt. (43) When n = 1, then, Q1(x, t) = 0, and G1(x, t) = −x+ xt3︸︷︷︸ neglect , so, η2 =T [w1] = −K−1[υK[(Q1 −Q0) + (G1 −G0)]] = 0, (44) therefore, the second iteration w2 is w2(x, t) =η0 + η1 + η2 = x t. (45) A.J. Mohammed, S.T. Al-Ramadhani, R.M.H. Darghoth / Eur. J. Pure Appl. Math, 15 (4) (2022), 1917-1936 1925 When n = 2 Q2(x, t) = −x2t3︸ ︷︷ ︸ neglect , G2(x, t) = −x+ 3 x t2 +−t9x2 + 3 t7x2 − 3 t5x2 + t3x2︸ ︷︷ ︸ neglect . (46) So, Q2 = 0 and G2 = 3 x t2 − x η3 =T [w2] = −K−1[υK[(Q2 −Q1) + (G2 −G1)]] = −x t3, (47) therefore, the third iteration w3 is w3(x, t) =η0 + η1 + η2 + η3 = x t− x t3. (48) Finally, when n → ∞, the series solution (48) of the nonhomogeneous mKdV equation (36) has the sum: w(x, t) = lim n→∞ n∑ k=0 ηk =0 + xt+ 0− x t3 − 1 4 x t4(x− 1) + 1 4 x t4(x− 1) + · · · = x t− x t3. (49) which is the exact solution. The w(x, t) in eq. (49) is a polynomial of degree (3) function. Equation (36) was solved in [20] and Figure (2) display the exact solution which concluded from the Taylor’s series in eq. (49). Figure 2: The exact solution of the nonhomogeneous mKdV equation (36) is a polynomial type. In the next example, we apply the K-MI method on the nonlinear homogeneous and nonhomogeneous FKdV eq. A.J. Mohammed, S.T. Al-Ramadhani, R.M.H. Darghoth / Eur. J. Pure Appl. Math, 15 (4) (2022), 1917-1936 1926 Example 3. Consider the nonlinear homogeneous FKdV eq. wt(x, t) + w2 wxx(x, t) + wxw3x(x, t)− 20w2 w3x(x, t) + w5x(x, t) = 0, (50) with the IC w(x, 0) = 1 x , x ̸= 0. That is clear from equation (50), we have ẇ = ∂w(x, t) ∂t , M [w(x, t)] = w2 wxx(x, t)+wxw3x(x, t)−20w2 w3x(x, t)+w5x(x, t), andG(x, t) = 0, (51) following the same structures as above examples, the Qn is M [wn(x, t)] =w2 n w{n,xx}(x, t) + w{n,x}w{n,3x}(x, t)− 20w2 nw{n,3x}(x, t)+ w{n,5x}(x, t) = Qn(x, t) +O(tn+1), (52) when n = 0, η0 = w0 = 1 x , Q0(x, t) = −1 x2 , (53) then, η1 =T [w0]−K−1[υK[(Q0 −Q−1)]] = t x2 (54) therefore, the first iteration w1 is w1(x, t) =η0 + η1 = 1 x + t x2 . (55) When n = 1, Q1(x, t) = (6x+ 480) t3 x9 + ( 2x2 + 1080x ) t2 x9︸ ︷︷ ︸ neglect −2 t x3 − 1 x2 , (56) so, Q1 = −2 t x3 − 1 x2 , then η2 =T [w1] = −K−1[υK[(Q1 −Q0)]] = t2 x3 , (57) therefore, the second iteration w2 is w2(x, t) =η0 + η1 + η2 = 1 x + t x2 + t2 x3 . (58) When n = 2 Q2(x, t) = − 1 x2 − 2 t x3 − 3 t2 x4 , (59) A.J. Mohammed, S.T. Al-Ramadhani, R.M.H. Darghoth / Eur. J. Pure Appl. Math, 15 (4) (2022), 1917-1936 1927 so, η3 =T [w2] = −K−1[υK[(Q2 −Q1)]] = t3 x4 . (60) Then, the third iteration w3 is w3(x, t) =η0 + η1 + η2 + η3 = 1 x + t x2 + t2 x3 + t3 x4 . (61) Finally, when n → ∞, the series solution (61) of the homogeneous FKdV eq. (50) w(x, t) = lim n→∞ n∑ k=0 ηk = 1 x + t x2 + t2 x3 + t3 x4 + . . . , (62) when we take 1 x as a common factor from eq. (62), we have w(x, t) = 1 x ( 1 + t x + t x2 + . . . ) , (63) the second part of eq.(63) is the Taylor’s expansion of, w(x, t) = 1 x(1− t x) , then, the Taylor’s expansion of the exact solution is w(x, t) = 1 x− t . (64) Equation (50) was solved in [14, 20], we have reached the same result with the K-IM method. According to Taylor’s series, Figure (3) display the exact solution of the homo- geneous FKdV eq. (50). The function in eq. (64) has singularities at (t = x). A.J. Mohammed, S.T. Al-Ramadhani, R.M.H. Darghoth / Eur. J. Pure Appl. Math, 15 (4) (2022), 1917-1936 1928 Figure 3: The singular solution of the homogeneous FKdV eq. (50). The solution is growing up rapidly on time. In addition, the solution when |x| → ∞ is decay to zero. Example 4. Consider the nonlinear nonhomogeneous FKdV eq. wt(x, t) + w wx(x, t) + w w3x(x, t) + w5x(x, t) = 2 cos(x+ t), (65) with the IC w(x, 0) = sin(x). Compering (65) to eq. (7), this in turn gives ẇ = ∂w(x, t) ∂t , M [w(x, t)] = w wx(x, t)+w w3x(x, t)+w5x(x, t), and G(x, t) = 2 cos(x+t). (66) Similarly, as above examples, we have M [wn(x, t)] =wn w{n,x}(x, t) + wnw{n,3x}(x, t) + w{n,5x}(x, t) = Qn(x, t) +O(tn+1), (67) and G(x, t) = 2 cos(x+ t), (68) when n = 0, η0 = w0 = sin(x), Q0(x, t) = cos(x). (69) Using the Taylor’s expansion in the function G0(x, t), for e −t = 1− t+ t2 2! − t3 3! + t4 4! + . . . , then, G(x, t) = cos (x)− sin (x) t− cos (x) 2 t2 + sin (x) 6 t3 + cos (x) 24 t4 − sin (x) 120 t5 + . . . , A.J. Mohammed, S.T. Al-Ramadhani, R.M.H. Darghoth / Eur. J. Pure Appl. Math, 15 (4) (2022), 1917-1936 1929 G0(x, t) = cos(x). (70) Substituting Q0 and G0 in eq (19), we have η1 =T [w0] = −K−1[υK[(Q0 −Q−1) + (G0 −G−1)]] = cos(x)t, (71) therefore, the first iteration w1 is, w1(x, t) =η0 + η1 = sin(x) + cos(x) t. (72) When n = 1, Q1(x, t) = cos(x)− sin(x) t, (73) and G1 = −2 cos (x) + 2 sin (x) t after cancelling the neglected terms. So, η2 =T [w1] = −K−1[υK[(Q1 −Q0) + (G1 −G0)]] = −1 2 sin (x) t2, (74) therefore, the second iteration w2 is, w2(x, t) =η0 + η1 + η2 = sin (x) + cos (x) t− 1 2 sin (x) t2. (75) When n = 2 Q2(x, t) = cos (x)− sin (x) t− 1/2 cos (x) t2, (76) and G2(x, t) = −2 cos (x) + 2 sin (x) t+ cos (x) t2, (77) so, η3 =T [w2] = −K−1[υK[(Q2 −Q1) + (G2 −G1)]] = −1 6 cos (x) t3, (78) then, the third iteration w3 is, w3(x, t) = sin (x) + cos (x) t− 1 2 sin (x) t2 − 1 6 cos (x) t3. (79) Finally, when n → ∞, the series solution (79) of the homogeneous FKdV equation (65) w(x, t) = lim n→∞ n∑ k=0 ηk =η0 + η1 + η2 = sin (x) + cos (x) t− 1 2 sin (x) t2 − 1 6 cos (x) t3 + . . . , =(sin(x)− 1 2 sin (x) t2 + 1 24 sin (x) t4 + . . . )+ (cos (x) t− 1 6 cos (x) t3 + 1 120 cos (x) t5 + . . . ) (80) A.J. Mohammed, S.T. Al-Ramadhani, R.M.H. Darghoth / Eur. J. Pure Appl. Math, 15 (4) (2022), 1917-1936 1930 the first part of eq. (80) represents the Taylor’s expansion of sin(x) cos(t) where sin(x) is a common factor, and the second part represents the Taylor’s expansion of cos(x) sin(t), then the Taylor’s expansion of the exact solution w(x, t) = sin(x) cos(t) + cos(x) sin(t) = sin(x+ t), (81) which is the periodic exact solution. Figure (4) shows the exact solution deduce from the convergent of Taylor’s series in eq. (81). Figure 4: The periodic solution of the nonhomogeneous FKdV eq. (65). 4. The effects of noise terms on Kamal transform with classical iteration method Here, the K-CI method is applied on the homogenous FKdV eq. and the result does not converge well. In previous examples, the homogenous FKdV eq. (50) was solved by the K-MI method. So, we present the following example to prove that the K-CI method is not a good choice to solve the KdV eq. types. Example 5. Consider the homogeneous FKdV eq. wt(x, t) + w2 wxx(x, t) + wxw3x(x, t)− 20w2 w3x(x, t) + w5x(x, t) = 0, (82) with given IC w(x, 0) = 1 x , taking KT and it’s inverse on eq. (82), lead to w(x, t) = 1 x −K−1[υK[w2 wxx(x, t) + wxw3x(x, t)− 20w2 w3x(x, t) + w5x(x, t)]], (83) A.J. Mohammed, S.T. Al-Ramadhani, R.M.H. Darghoth / Eur. J. Pure Appl. Math, 15 (4) (2022), 1917-1936 1931 next, applying the classical iteration method on eq. (83) wn+1(x, t) = 1 x − K−1[υK[w2 n w{n,xx}(x, t) + w{n,x}w{n,3x}(x, t)− 20w2 n w{n,3x}(x, t) + w{n,5x}(x, t)]], (84) where, w0(x, t) = 1 x . Then when n = 0, in eq. (84) the first iteration will be w1(x, t) = 1 x − K−1[υK[w2 0 w{0,xx}(x, t) + w0,x}w{0,3x}(x, t)− 20w2 0 w{0,3x}(x, t) + w{0,5x}(x, t)]], w1(x, t) = 1 x + t x2 , (85) and when n = 1, yields w2(x, t) = x−1 + t x2 + t2 x3 − 3 2 t4 x8 − 2 3 t3 x7 − 120 t4 x9 − 360 t3 x8 . (86) Figure (5) display the sketch of w2(x, t) by eq. (86) when n = 1, we can see clearly it Figure 5: the plot of w2(x, t) given by (86) when n = 1. is far from our expectation, so we need to find the next iteration to see whether the next iteration will be better. When n = 2, the third iteration is w3(x, t) = 1621620 t3x26 + 1621620 t2x27 + 1621620 tx28 + 1621620x29 − 2779920 t7x19− 8108100 t6x20 − 8432424 t5x21 − 3513510 t4x22 − 277992000 t7x18 − 778377600 t6x19− − 1868106240 t5x20 − 2821618800 t4x21 + 25945920 t9x14 + 65540475 t8x15 + 3397680 t7x16− − 20900880 t6x17 − 11459448 t5x18 − 2702700 t4x19 + 7005398400 t9x13 + 23724300600 t8x14+ A.J. Mohammed, S.T. Al-Ramadhani, R.M.H. Darghoth / Eur. J. Pure Appl. Math, 15 (4) (2022), 1917-1936 1932 33016183200 t7x15 + 11578366800 t6x16 − 38776177440 t5x17 − 13945932000 t4x18+ 479999520000 t9x12 + 2270916648000 t8x13 + 4433416416000 t7x14 + 5876750880000 t6x15− 1727998272000 t5x16 − 11734042320000 t4x17 − 51744420 t11x9 − 100135035 t10x10+ 116105990 t9x11 + 129639510 t8x12 + 28622880 t7x13 − 19185238800 t11x8 − . . . , (87) Figure (6) shows the sketch of w3(x, t) when n = 2. In addition, as n → ∞, the wn+1 has Figure 6: The Figure shows the un stability of the approximate solution due to the higher order noise terms in the iteration in eq. (87) when n = 2. unnecessary (noise) terms which makes the approximate solution looks divergent from the exact solution. This case will happen when we try to use the CI method instead of the MI method for the above examples (1,2) and (4). 5. Discussion Kamal transform has been used recently in a combination with another method to solve nonlinear differential equations [18, 22]. In this work, we use KT in combination with the IT method to solve the KdV eq. Our analysis shows that using the KT and the classical iteration method can lead to divergent series of approximate solutions due to the high order terms induced during the iterations causing high noise and error in the calculation as shown in section (4). However, modifying the iteration method by removing the noise terms and combining them with the KT leads to better approximation results since the series of the solutions converges fast to the accurate solution as shown in the examples in section (3). The semi-analytical K-MI method is used to solve the homogeneous and nonhomogeneous mKdV eq. and FKdV eq. This method begins with the initial condition w(x, 0) as the first approximate solution w0(x, t), then, we apply the KT and MI method to construct the other iterations. Each iteration is an approximate solution of the PDE. The approximate solution is a polynomial in the variable t of a degree equal to the order of the iteration, hence, it is not bounded as |t| → ∞. However, in many cases, one can REFERENCES 1933 deduce an infinite series by taking the limit of the iteration. When the deduced series is comparable to the Taylor expansion of a known function, then the exact solution is inferred from our semi-analytic method, as we have shown in our examples. In example (1), the exact solution of the homogeneous KdV equation is shown in Figure (1). The solution is called soliton which is a traveling wave that maintains its amplitude for a long distance while vanishing as |x| → ∞. In example (2), the exact solution of the nonhomogeneous mKdV equation is an odd polynomial that is not bounded as shown in Figure (2). In example (3) the solution of the homogeneous FKdV equation has a discontinuity at x = t while decaying to zero as |x| → ∞, hence, the equation in example (3) admits a singular solution. The solution of the nonhomogenous FKdV eq. in example (4) is a periodic sine wave as shown in Figure (4). Figure (5), illustrates the approximate solution of equation (87) obtained after 2 iterations (n = 0, 1) using the CI method. Both approximate solutions w1 and w2 are polynomials in the variable t, therefore, they are unbounded along time progress. However, the error of the approximate solution of the classical iterations method is higher than the error of the modified iterations method. The error induced by the polynomial terms of degree higher than the iteration order n = 3 which cause high noise in the calculation, see Figure(6). Removing the noise terms in the modified iterations method leads to better convergence towards the exact solution, see Figure(3). 6. 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