EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 4, Article Number 6982 ISSN 1307-5543 – ejpam.com Published by New York Business Global Qualitative Study on Semi-Analytical Methods for Solving Nonlinear Time-Fractional partial differential equations Alaa Mohammad Alhammad1, Abdulkafi Mohammed Saeed1,∗ 1 Department of Mathematics, College of Science, Qassim University, Buraydah 51452, Saudi Arabia Abstract. This paper focuses on finding Semi-Analytical solutions of nonlinear partial differential equations of fractional order by using four techniques such as Sumudu decomposition, Natural decomposition, Adomian decomposition and modified Laplace variational iteration methods. The fractional derivatives are described in the Caputo sense. In these methods, the solution manifests as a convergent series with conveniently computable components. Numerical results show that the four approaches are easy to implement and accurate when applied to partial differential equations of fractional order, although there are some distinct differences between the methods studied, which depend on the nature of the equations and the conditions associated with them. 2020 Mathematics Subject Classifications: 35B45, 35D30, 35D35 Key Words and Phrases: Nonlinear partial differential equations, Adomian decomposition method, Modified Variational Iteration Laplace Transform Method 1. Introduction Fractional calculus is a field of mathematics study that grows out of the traditional definitions of calculus integral and derivative operators in much the same way frictional are an outgrowth of exponents with integer values. Fractional calculus has been part of mathematics and science literature for more than three centuries. It has been used to model physical and engineering processes that are found to be best described by fractional differential equations [1–4]. Recent research has studied the applications of fractional cal- culus in several domains, including fractional-order epidemic models [5], fractional-order Optimal Control Problem [6]. Most phenomena in nature are described by nonlinear differential equations. Therefore, scientists in different branches of science try to solve them. But because of the nonlinear part of these groups of equations, finding an exact solution is not easy [7]. Approximation and numerical techniques must be used. The Ado- mian decomposition method [8–11], the Sumudu decomposition method [12], the natural ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.6982 Email addresses: 451214285@qu.edu.sa (A. M. Alhammad), abdulkafi.ahmed@qu.edu.sa (A. M. Saeed) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) A. M. Alhammad, A. M. Saeed / Eur. J. Pure Appl. Math, 18 (4) (2025), 6982 2 of 22 decomposition method [13] and modified Laplace variational iteration method [14] are relatively new approaches to provide an analytical approximation to linear and nonlinear problems, and they are particularly valuable as tools for scientists and applied mathe- maticians. For nonlinear models, the previous methods have shown dependable results and give analytical approximations solutions that converge very rapidly. This paper is organized as follows: Section 2 provides a brief overview of Basic Concept of Fraction Calculus. Section 3 presents the description of proposed Transformations Methods and the analysis of these methods given by Section 4. Section 5 includes the numerical experiments and the comparison among these methods. The paper ends with conclusion and final remarks. 2. Basic Concept of Fractional Calculus This section is devoted to a description of the basic ideas of fractional calculus, which are essential to comprehending complex systems, and explores the fundamental meanings of integrals and fractional derivatives, as well as their different forms and use. Definition 1. [8] A real function f(x), x > 0, is siaid to be in the space Cµ, µ ∈ R if there exists a real number p(> µ), such that f(x) = xpf1(x) ∈ C[0,∞), and it is said to be in the space Cm µ iff fm ∈ Cµ,m ∈ N . Definition 2. [8] The Riemann-Liouville fractional integral operator of order α ≥ µ, of function f ∈ Cµ, µ ≥ −1, is defined as Jαf(x) = { 1 Γ(α) ∫ x a (x− t)α−1f(t)dt, α > 0, x > 0, f(x), α = 0. We mention only the following Properties of the operator Jα. For function f ∈ Cµ, µ ≥ −1, α, β ≥ 0 and γ > −1 (i) JαJβf(x) = Jα+βf(x), (ii) JαJβf(x) = JβJαf(x), (iii) Jαxγ = Γ(γ+1) Γ(α+γ+1)x α+γ. The Riemann-Liouville derivative exhibits certain drawbacks for modeling real-world occurrences using fractional differential equations, as noted in [8, 12]. Consequently, we will now provide a modified fractional differentiation operator Dα. presented by Caputo in his study of viscoelasticity theory [15]. Definition 3. [12] For m to be the smallest integer that exceeds α, the Caputo fractional derivatives of order α > 0 is defined as: Dαf(x, t) = dαf(x, t) dtα = { 1 Γ(m−α) ∫ t 0 (t− τ)m−α−1 d mf(x,τ) dτm , m− 1 < α ≤ m dmf(x,t) dtm , α = m,∈ N A. M. Alhammad, A. M. Saeed / Eur. J. Pure Appl. Math, 18 (4) (2025), 6982 3 of 22 One may refer to the cited sources for the mathematical characteristics of fractional derivatives and integrals. In Section 5, we demonstrate the application of semi-analytical methods for solving known and commonly encountered categories of fractional partial differential equations (FPDEs), such as nonlinear time-fractional advection partial differential equation and nonlinear time-fractional hyperbolic partial differential equation. 3. The proposed Transformations Methods There are several integral transformations available for solving fractional differential equations. The Laplace transform is the most extensively utilized. The core idea behind the Laplace transforms is mopping of a time-domain function f(t) into an s-domain function F (s) using an integral transformation. By this, differentiation and integration operations in the time domain are made equivalent to multiplication and division by s in the Laplace domain [16, 17]. In the Sumudu transform, the differentiation and integration operations in the t-domain are made equivalent to division and multipli- cation by u in a u-domain. This makes it possible to treat the variable u and transformed function f(u) as replicas of t and f(t), respectively. It is even possible to express them in the same engineering units as t and f(t) so that the consistency of units in a differential equation describing a physical process can be maintained even after the transformation [16]. The Natural transform is derived from the Fourier integral, and it converges to the Laplace transform and the Sumudu transform [18]. Definition 4. [19] The Laplace transform of f(t): L[f(t)] = F (s) = ∫ ∞ 0 e−stf(t)dt Definition 5. [12] The Sumudu transform over the following set of functions A = { f(t) ∣∣∣∣∃M, τ1, τ2 > 0, |f(t)| < Me |t| tj , if t ∈ (−1)j × [0,∞) } , is defined as, for u ∈ (τ1, τ2), we have S[f(t)] = G(u) = ∫ ∞ 0 f(ut)e−tdt = ∫ ∞ 0 1 u e− t u f(t)dt. Definition 6. [18] The Natural transform over the following set of functions A = { f(t) ∣∣∣∣∃M, τ1, τ2 > 0, |f(t)| < Me t tj , if t ∈ (−1)j × [0,∞) } , is defined as N[f(t)] = R(s, u) = ∫ ∞ 0 e−stf(ut)dt, s > 0, u > 0. A. M. Alhammad, A. M. Saeed / Eur. J. Pure Appl. Math, 18 (4) (2025), 6982 4 of 22 Remark 1. If S[f(t)] = G(u), L[f(t)] = F (s) and N[f(t)] = R(s, u), then the relationship between Sumudu and Laplace transforms [20]: The transformations are connected via a change of variables: • G(u) = 1 uF ( 1 u ) . Equivalently: • F (s) = 1 sG ( 1 s ) , where s = 1 u and u = 1 s . Relationship Between Sumudu and Natural Transforms [18]: R(s, u) = 1 s G (u s ) . Relationship Between Natural and Laplace Transforms [18]: R(s, u) = 1 u F ( s u ) . Definition 7. [19] The Laplace transform of the Caputo fractional derivative is defined as L [Dαf(x)] = sαL[f(x)]− m−1∑ k=0 sα−k−1fk(0), m− 1 < α ≤ m Definition 8. [12] The Sumudo transform of the Caputo fractional derivative is defined as S [Dα t f(x, t)] = S[f(x, t)] uα − m−1∑ k=0 fk(x, 0) uα−k , m− 1 < α ≤ m Definition 9. [21] The Natural transform of the Caputo fractional derivative is defined as N [Dαf(x)] = sα uα N[f(x)]− m−1∑ k=0 sα−(k+1) uα−k fk(0), m− 1 < α ≤ m 4. Analysis of Methods The aim of this section is to discuss the use of the transform algorithm for nonlinear partial fractional differential equations. We consider the following time-fractional partial differential equation [8]: Dα t u(x, t) = f (u, ux, uxx) + g(x, t), m− 1 < α ≤ m, (1) Where Dα t = dα dtα is the Caputo fractional derivative of order α,m ∈ N, f is a nonlinear function and g is the source function. The initial and boundary conditions associated with (1) are of the form u(x, 0) = h(x), 0 < α ≤ 1 u(x, t) → 0 as |x| → ∞, t > 0, (2) A. M. Alhammad, A. M. Saeed / Eur. J. Pure Appl. Math, 18 (4) (2025), 6982 5 of 22 and u(x, 0) = h(x), du(x, 0) dt = k(x), 1 < α ≤ 2 u(x, t) → 0 as |x| → ∞, t > 0. (3) The decomposition method requires that the nonlinear fractional differential Eq. (1) be expressed in terms of operator form as Dα t u(x, t) + Lu(x, t) +Nu(x, t) = g(x, t), x > 0, (4) Where L is a linear operator, which might include other fractional derivatives of order less than α, N is a nonlinear operator that might also include other fractional derivatives of order less than α, g(x, t) and Dα t are defined as in Eq. (1). 4.1. Adomian Decomposition and Transform-Based Variants Although the Adomian decomposition method, the Sumudu decomposition method, and the natural decomposition method apply different integral transforms in their formu- lations, the obtained decomposition components and numerical results are identical for all the considered problems. Adomian Decomposition Method: Applying the operator Jα, the inverse of the operator Dα t , to both sides of Eq. (4) yields [8]: u(x, t) = m−1∑ k=0 dku dtk ( x, 0+ ) tk k! + Jαg(x, t)− Jα[Lu(x, t) +Nu(x, t)]. (5) The Adomian decomposition method suggests that the solution u(x, t) be decomposed into an infinite series of components. u(x, t) = ∞∑ n=0 un(x, t). (6) And the nonlinear function in Eq. (5) is decomposed as follows: Nu = ∞∑ n=0 An, (7) Here An are the so-called Adomian polynomials. The general form of the formula for An Adomian polynomials is An = 1 n! [ dn dλn N ( n∑ k=0 λkuk )] λ=0 . (8) The Adomian polynomial An can be calculated for all forms of nonlinearity according to specific algorithms constructed by Adomian [9]. A. M. Alhammad, A. M. Saeed / Eur. J. Pure Appl. Math, 18 (4) (2025), 6982 6 of 22 Substituting the series of decompositions (6) and (7) into both sides of (5) gives. ∞∑ n=0 un(x, t) = m−1∑ k=0 dku dtk ( x, 0+ ) tk k! + Jαg(x, t)− Jα [ L ( ∞∑ n=0 un(x, t) ) + ∞∑ n=0 An ] . (9) From this equation, the iterates are determined by the following recursive way. u0(x, t) = m−1∑ k=0 dku dtk ( x, 0+ ) tk k! + Jαg(x, t), u1(x, t) = −Jα [Lu0 +A0] , u2(x, t) = −Jα [Lu1 +A1] , ... un+1(x, t) = −Jα [Lun +An] . (10) Finally, we approximate the solution u(x, t) by the truncated series. ϕN (x, t) = N−1∑ n=0 un(x, t) and lim N→∞ ϕN (x, t) = u(x, t). (11) Sumudu Decomposition Method: Applying the Sumudu transform on both sides of Eq. (4) yields [12] S [Dα t u(x, t)] = S[g(x, t)− Lu(x, t)−Nu(x, t)]. (12) Using the differentiation property of the Sumudu transform for Caputo, we get u−αS[u(x, t)]− m−1∑ k=0 u−(α−k)uk(x.0) = S[g(x, t)− Lu(x, t)−Nu(x, t)]. (13) Simplify, we get. u(x, t) = m−1∑ k=0 u(k)fk(x) + S−1 (uαS[g(x, t)])− S−1 (uαS[Lu(x, t) + Nu(x, t)]) . (14) Substituting the series of decompositions (6) and (7) into both sides of (5) gives, ∞∑ n=0 un(x, t) = m−1∑ k=0 u(k)fk(x) + S−1 (uαS[g(x, t)]) − S−1 [ uαS ( L ( ∞∑ n=0 un(x, t) ) + ∞∑ n=0 An )] , (15) where An defined in Eq. (8). A. M. Alhammad, A. M. Saeed / Eur. J. Pure Appl. Math, 18 (4) (2025), 6982 7 of 22 On comparing both sides of the Eq. (15), we get u0(x, t) = m−1∑ k=0 u(k)fk(x) + S−1 (uαS[g(x, t)]) , u1(x, t) = −S−1 (uαS [Lu0 +A0]) , u2(x, t) = −S−1 (uαS [Lu1 +A1]) , ... un+1(x, t) = −S−1 (uαS [Lun +An]) , n ≥ 1. (16) Finally, we approximate the solution u(x, t) by the truncated series. ϕN (x, t) = N−1∑ n=0 un(x, t) and lim N→∞ ϕN (x, t) = u(x, t). (17) Natural Decomposition Method: We apply the Natural transform on both sides of Eq. (4), which yields [13]. N [Dα t u(x, t)] = N[g(x, t)− Lu(x, t)−Nu(x, t)]. (18) Using the differentiation property of the Natural transform for Caputo, we get sα uα N[u(x, t)]− m−1∑ k=0 sα−(k+1) uα−k uk(x, 0) = N[g(x, t)− Lu(x, t)−Nu(x, t)]. (19) Simplify, we get u(x, t) = m−1∑ k=0 s−(k+1) u−k uk(x, 0) + N−1 ( uα sα N[g(x, t)] ) − N−1 ( uα sα N[Lu(x, t) +Nu(x, t)] ) . (20) Substitution the decomposition series (6) and (7) into both sides of (20) gives. ∞∑ n=0 un(x, t) = m−1∑ k=0 s−(k+1) u−k uk(x, 0) + N−1 ( uα sα N[g(x, t)] ) − N−1 [ uα sα N ( L ( ∞∑ n=0 un(x, t) ) + ∞∑ n=0 An )] . (21) Where An defined in Eq. (8). From this equation, the iterates are determined by the following recursive way u0(x, t) = m−1∑ k=0 s−(k+1) u−k uk(x, 0) + N−1 ( uα sα N[g(x, t)] ) , A. M. Alhammad, A. M. Saeed / Eur. J. Pure Appl. Math, 18 (4) (2025), 6982 8 of 22 u1(x, t) = −N−1 ( uα sα N [Lu0 +A0] ) , u2(x, t) = −N−1 ( uα sα N [Lu1 +A1] ) , (22) ... un+1(x, t) = −N−1 ( uα sα N [Lun +An] ) , n ≥ 1. Finally, we approximate the analytical solution u(x, t) by truncated series: ϕN (x, t) = N−1∑ n=0 un(x, t) and lim N→∞ ϕN (x, t) = u(x, t). 4.2. Modified Laplace Variational Iteration Method The new tactic of modified Laplace variation iteration technique is applied as follows. We apply Laplace transform on both sides of Eq. (4) yields [14]. L [Dα t u(x, t)] = L[g(x, t)− Lu(x, t)−Nu(x, t)]. (23) Using the differentiation property of Laplace transform for Caputo, we get sαL[u(x, t)]− m−1∑ k=0 s(α−k−1)uk(x, 0) = L[g(x, t)−Nu(x, t)], (24) u(x, s) = h(x) s + k(x) s2 + L sα [g(x, t)]− L sα [Nu(x, t)], (25) u(x, t) = f(x, t)− L−1 [ L sα [Nu(x, t)] ] . (26) Differentiating the results obtained above with respect to α, we then get the value of the general Lagrange multiplier, for the correction functional iterative formula to equal one. du(x, t) dt = df(x, t) dt − d dt L−1 [ L sα [Nu(x, t)] ] . (27) Therefore, equation (27) can be put above in the following formula: un+1(x, t) = un(x, t) + ∫ t 0 λ [ dun(x, y) dy − df(x, y) dy + d dy L−1 [ L sα [Nun(x, y)] ]] dy. (28) The general Lagrange multiplier for equation (28) can be identified optimally via variation theory to get 1 + λ|y=t = 0, λ|y=t = −1. A. M. Alhammad, A. M. Saeed / Eur. J. Pure Appl. Math, 18 (4) (2025), 6982 9 of 22 Substituting λ = −1 into equation (28) we get the iterative formula n = 1, 2, · · · follows: un+1(x, t) = un(x, t)− ∫ t 0 [ dun(x, y) dy − df(x, y) dy + d dy L−1 [ L sα [Nun(x, y)] ]] dy. (29) Equation (29) is the new modified function of the Laplace transform and variational it- eration method. Start with the initial iteration u(x, t) = u(x, 0). The exact solution is present as the sequent approximation un+1(x, t), n = 1, 2, · · · . In other words, u(x, t) = lim n→∞ un(x, t). 5. Numerical Experiments and Discussion of Results In this section, we shall illustrate the four techniques by several examples. These examples are somewhat artificial in the sense that the exact answer, for the special cases α = 1, α = 2 is known in advance, and the initial and boundary conditions are directly taken from this answer. Nonetheless, such an approach is needed to evaluate the accuracy of the analytical techniques and to examine the effect of varying the order of the time- fractional derivative on the behavior of the solution. All the results are calculated by using the software MATLAB. Example 1. Consider the nonlinear time-fractional advection partial differential equation. Dαu(x, t) + u(x, t)ux(x, t) = x+ xt2, t > 0, x ∈ R, 0 < α ≤ 1, (30) subject to the initial condition u(x, 0) = 0. (31) The values of α = 1 is the only case for which we know the exact solution u(x, t) = xt. To solve the problem using the decomposition method [8], we simply substitute (30) and the initial conditions (31) into (10), to obtain the following recurrence relation. u0(x, t) = u(x, 0) + Jα ( x+ xt2 ) = x ( tα Γ(α+ 1) + 2tα+2 Γ(α+ 3) ) , un+1(x, t) = −Jα [An] , n ≥ 0. (32) Where An are the Adomian polynomials for the nonlinear function N = u(x, t)ux(x, t). The few components of Adomian polynomials, are given by [9]: A0 = u0u0x, A1 = u0xu1 + u0u1x, A2 = u0xu2 + u1xu1 + u2xu0, (33) and we can continue the calculations to find A3 and so on by the same manner. A. M. Alhammad, A. M. Saeed / Eur. J. Pure Appl. Math, 18 (4) (2025), 6982 10 of 22 In view of (32), the first few components of the decomposition series are derived as follows u0(x, t) =x ( tα Γ(α+ 1) + 2tα+2 Γ(α+ 3) ) , (34) u1(x, t) =− x [ Γ(2α+ 1)t3α Γ(α+ 1)2Γ(3α+ 1) + 4Γ(2α+ 3)t3α+2 Γ(α+ 1)Γ(α+ 3)Γ(3α+ 3) + 4Γ(2α+ 5)t3α+4 Γ(α+ 3)2Γ(3α+ 5) ] , (35) u2(x, t) =2x [ Γ(2α+ 1)Γ(4α+ 1)t5α Γ(α+ 1)3Γ(3α+ 1)Γ(5α+ 1) + 8Γ(2α+ 5)Γ(4α+ 7)t5α+6 Γ(α+ 3)3Γ(3α+ 5)Γ(5α+ 7) + . . . ] . (36) And soon, in this manner the rest of components of the decomposition series can be obtained. The first three terms of the decomposition series (6) are given by u(x, t) = x [ tα Γ(α+ 1) + 2tα+2 Γ(α+ 3) − Γ(2α+ 1)t3α Γ(α+ 1)2Γ(3α+ 1) − 4Γ(2α+ 3)t3α+2 Γ(α+ 1)Γ(α+ 3)Γ(3α+ 3) − 4Γ(2α+ 5)t3α+4 Γ(α+ 3)2Γ(3α+ 5) + Γ(2α+ 1)Γ(4α+ 1)t5α Γ(α+ 1)3Γ(3α+ 1)Γ(5α+ 1) + · · · ] . (37) To solve the problem using the Sumudu decomposition method, we simply substitute (30) and the initial conditions (31) into (16), to obtain the following recurrence relation u0(x, t) = m−1∑ k=0 u(k)fk(x) + S−1 ( uαS [ x+ xt2 ]) = x ( tα Γ(α+ 1) + 2tα+2 Γ(α+ 3) ) , un+1(x, t) = −S−1 (uαS [An]) , n ≥ 1. (38) The few components of Adomian polynomials show in (33). In view of (38), the first few components of the decomposition series are derived as in equations (35) and (36) and so on, in this manner the rest of components of the Sumudu decomposition series can be obtained. The first three terms of the decomposition series are given by u(x, t)as written in equation (37). To solve the problem using the Natural decomposition method, we simply substitute (30) and the initial conditions (31) into (22), to obtain the following recurrence relation. u0(x, t) = m−1∑ k=0 s−(k+1) u−k uk(x, 0) + N−1 ( uα sα N[g(x, t)] ) = x ( tα Γ(α+ 1) + 2tα+2 Γ(α+ 3) ) , un+1(x, t) = −N−1 ( uα sα N [An] ) , n ≥ 1. (39) In view of (39), the first few components of the decomposition series are derived as in equations (35) and (36) and so on, in this manner the rest of components of the Natural A. M. Alhammad, A. M. Saeed / Eur. J. Pure Appl. Math, 18 (4) (2025), 6982 11 of 22 decomposition series can be obtained. The first three terms of the decomposition series are given by u(x, t) as written in equation (37). To solve the problem using the Modified Laplace Variational Iteration Method [14], by enforcement of the Laplace transform to the sides of equation (30) and using the initial conditions (31), we get u(x, s) = x s1+α + 2x s3+α − ( 1 sα L (u(x, t)ux(x, t)) ) . (40) Taking the inverse Laplace transform to the sides of equation (40), we obtain u(x, t) = xtα Γ(1 + α) + 2xtα+2 Γ(α+ 3) − L−1 ( 1 sα L (u(x, t)ux(x, t)) ) . (41) Now the new tactic of the modified Laplace variational iteration technique is instituted on (27) from equation (41), and we get du(x, t) dt = d dt ( xtα Γ(1 + α) ) + d dt ( 2xtα+2 Γ(α+ 3) ) − d dt ( L−1 ( 1 sα L (u(x, t)ux(x, t)) )) . (42) We simply substitute equation (42) and the initial condition equation (31) into equation (29), by the new modified function, we find u0(x, t) =u(x, 0) = 0 u1(x, t) = xtα Γ(1 + α) + 2xtα+2 Γ(α+ 3) u2(x, t) = xtα Γ(1 + α) + 2xtα+2 Γ(α+ 3) − ( Γ(1 + 2α)xt3α Γ(1 + α)2Γ(3α+ 1) ) − 4xΓ(2α+ 3)t3α+2 Γ(1 + α)Γ(α+ 3)Γ(3α+ 3) − 4xΓ(2α+ 5)t3α+4 Γ(α+ 3)2Γ(3α+ 5) . Tables (1, 2, 3) show the approximate solutions for Eq. (30) obtained for different val- ues of a using the decomposition method, Sumudu decomposition method, Natural de- composition method, Adomian decomposition method, and Modified Laplace Variational Iteration. It is to be noted that only three terms of this decomposition series and the Modified Laplace Variational Iteration were used in evaluating the approximate solutions for obtaining the results. The accuracy can be improved by computing more terms of the approximate solution. A. M. Alhammad, A. M. Saeed / Eur. J. Pure Appl. Math, 18 (4) (2025), 6982 12 of 22 Table 1: Numerical values when α = 0.5 for Eq. (30). t x UADM UNDM − USDM UMLVIM 0.2 0.25 0.1128438450 0.1128438450 0.1067989518 0.5 0.2256876900 0.2256876900 0.2135979036 0.75 0.3385315349 0.3385315349 0.3203968554 1 0.4513753799 0.4513753799 0.4271958072 0.4 0.25 0.1640101327 0.1640101327 0.1257268756 0.5 0.3280202655 0.3280202655 0.2514537512 0.75 0.4920303982 0.4920303982 0.3771806267 1 0.6560405310 0.6560405310 0.5029075023 0.6 0.25 0.2440592839 0.2440592839 0.1176010458 0.5 0.4881185678 0.4881185678 0.2352020917 0.75 0.7321778516 0.7321778516 0.3528031375 1 0.9762371355 0.9762371355 0.4704041833 Table 2: Numerical values when α = 0.75 for Eq. (30). t x UADM UNDM − USDM UMLVIM 0.2 0.25 0.0787870110 0.0787870110 0.0784882208 0.5 0.1575740220 0.1575740220 0.1569764416 0.75 0.2363610330 0.2363610330 0.2354646624 1 0.3151480440 0.3151480440 0.3139528832 0.4 0.25 0.1289407026 0.1289407026 0.1245800056 0.5 0.2578814051 0.2578814051 0.2491600112 0.75 0.3868221077 0.3868221077 0.3737400168 1 0.5157628102 0.5157628102 0.4983200224 0.6 0.25 0.1772526142 0.1772526142 0.1544935814 0.5 0.3545052283 0.3545052283 0.3089871628 0.75 0.5317578425 0.5317578425 0.4634807443 1 0.7090104566 0.7090104566 0.6179743257 Table 3: Numerical values when α = 1 for Eq. (30). t x UADM UNDM − USDM UMLVIM UExact 0.2 0.25 0.0500001744 0.0500001744 0.0499892825 0.0500000000 0.5 0.1000003488 0.1000003488 0.0999785651 0.1000000000 0.75 0.1500005233 0.1500005233 0.1499678476 0.1500000000 1 0.2000006977 0.2000006977 0.1999571302 0.2000000000 0.4 0.25 0.1000229939 0.1000229939 0.0996521651 0.1000000000 0.5 0.2000459878 0.2000459878 0.1993043302 0.2000000000 0.75 0.3000689818 0.3000689818 0.2989564952 0.3000000000 1 0.4000919757 0.4000919757 0.3986086603 0.4000000000 0.6 0.25 0.1504123340 0.1504123340 0.1472969143 0.1500000000 0.5 0.3008246680 0.3008246680 0.2945938286 0.3000000000 0.75 0.4512370020 0.4512370020 0.4418907429 0.4500000000 1 0.6016493361 0.6016493361 0.5891876571 0.6000000000 A. M. Alhammad, A. M. Saeed / Eur. J. Pure Appl. Math, 18 (4) (2025), 6982 13 of 22 Table (4) shows the absolute error between the exact and approximate solutions for Eq. (30) produced using Adomian Decomposition Method and Modified Laplace Variational Iteration Method. The results are computed after applying three iterations of each method for various values. Table 4: The absolute error for α = 1 for Eq. (30). t x UADM UMLVIM 0.2 0.25 0.0000001744 0.0000107175 0.5 0.0000003488 0.0000214349 0.75 0.0000005233 0.0000321524 1 0.0000006977 0.0000428698 0.4 0.25 0.0000229939 0.0003478349 0.5 0.0000459878 0.0006956698 0.75 0.0000689818 0.0010435048 1 0.0000919757 0.0013913397 0.6 0.25 0.0004123340 0.0027030857 0.5 0.0008246680 0.0054061714 0.75 0.0012370020 0.0081092571 1 0.0016493361 0.0108123429 Figure 1: Comparison of ADM and MLVIM solution for the first three approximations α = 0.5, t = 0.2, with x = 0 : 1, for equation (30). A. M. Alhammad, A. M. Saeed / Eur. J. Pure Appl. Math, 18 (4) (2025), 6982 14 of 22 Figure 2: Comparison of ADM and MLVIM solution for the first three approximations α = 1, t = 0.6, with x = 0 : 1, for equation (30). Figure 3: Comparison of the absolute error between the ADM and MLVIM solution for the first three approxi- mations α = 1, t = 0.6, with x = 0 : 1, for Eq. (30). Example 2. Consider the nonlinear time-fractional hyperbolic partial differential equation. Dαu(x, t) = d dx ( u(x, t) u(x, t) dx ) , t > 0, x ∈ R, 1 < α ≤ 2, (43) A. M. Alhammad, A. M. Saeed / Eur. J. Pure Appl. Math, 18 (4) (2025), 6982 15 of 22 subject to the initial condition u(x, 0) = x2, ut(x, 0) = −2x2. (44) The values of α = 2 is the only case for which we know the exact solution u(x, t) = ( x t+ 1 )2 . To solve the problem using the decomposition method [8], we simply substitute (43) and the initial conditions (44) into (10), to obtain the following recurrence relation. u0(x, t) = u(x, 0) + Jα ( x+ xt2 ) = x ( tα Γ(α+ 1) + 2tα+2 Γ(α+ 3) ) , un+1(x, t) = Jα [An] , n ≥ 0, (45) Where An are the Adomian polynomials for the nonlinear function N = u(x, t)ux(x, t), The few components of Adomian polynomials, are given by [9] A0 = u0u0x A1 = u0xu1 + u0u1x, A2 = u0xu2 + u1xu1 + u2xu0, (46) and we can continue the calculations to find A3 and so on by the same manner . In view of (45), the first few components of the decomposition series are derived as follows u0(x, t) = x2(1− 2t), u1(x, t) =6x2 ( tα Γ(α+ 1) − 4tα+1 Γ(α+ 2) + 8tα+2 Γ(α+ 3) ) , (47) u2(x, t) =72x2 ( t2α Γ(2α+ 1) − 4t2α+1 Γ(2α+ 2) + 8t2α+2 Γ(2α+ 3) − 2Γ(α+ 2)t2α+1 Γ(α+ 1)Γ(2α+ 2) , + 8Γ(α+ 3)t2α+2 Γ(α+ 2)Γ(2α+ 3) − 16Γ(α+ 4)t2α+3 Γ(α+ 3)Γ(2α+ 4) ) . (48) And soon, in this manner the rest of components of the decomposition series can be obtained. The first three terms of the decomposition series (6) are given by. u(x, t) =x2(1− 2t) + 6x2 ( tα Γ(α+ 1) − 4tα+1 Γ(α+ 2) + 8tα+2 Γ(α+ 3) ) + 72x2 ( t2α Γ(2α+ 1) − 4t2α+1 Γ(2α+ 2) + · · · ) . (49) To solve the problem using the Sumudu decomposition method [12], we simply substitute (43) and the initial conditions (44) into (16), to obtain the following recurrence relation u0(x, t) = x2(1− 2t) un+1(x, t) = S−1 (uαS [An]) , n ≥ 1. (50) A. M. Alhammad, A. M. Saeed / Eur. J. Pure Appl. Math, 18 (4) (2025), 6982 16 of 22 In view of (50), the first few components of the decomposition series are derived as follows u0(x, t) =x2(1− 2t), where u1(x, t) and u2(x, t) as written in equations (47) and (48) and soon, in this manner the rest of components of the decomposition series can be obtained. The first three terms of the decomposition series (6) are given by u(x, t) = Eq.(49). To solve the problem using the Natural decomposition method, we simply substitute (43) and the initial conditions (44) into (22), to obtain the following recurrence relation u0(x, t) = x2(1− 2t), un+1(x, t) = N−1 ( uα sα N [An] ) , n ≥ 1. (51) In view of (51), the first few components of the decomposition series are derived as follows u0(x, t) =x2(1− 2t), where u1(x, t) and u2(x, t) as written in equations (47) and (48) and soon, in this manner the rest of components of the decomposition series can be obtained. The first three terms of the decomposition series (6) are given by u(x, t) as written in equation (49). To solve the problem using the Modified Laplace Variational Iteration Method [14], by enforcement of the Laplace transform to the sides of equation (43) and using the initial conditions (44), we get sαL[u(x, t)]− sα−1x2 + 2sα−2x2 = L ( d dx ( u(x, t) u(x, t) dx )) , L[u(x, t)] = x2 s − 2x2 s2 + s−αL ( d dx ( u(x, t) u(x, t) dx )) , u(x, s) = x2 s − 2x2 s2 + ( 1 sα L (u(x, t)ux(x, t)) ) . (52) Taking the inverse Laplace transform to the sides of equation (52), we obtain u(x, t) = x2 − 2x2t+ L−1 ( s−αL ( d dx ( u(x, t) u(x, t) dx ))) . (53) Now the new tactic of the modified Laplace variational iteration technique is instituted on (27) from equation (53), and we get du(x, t) dt = d dt ( x2 − 2x2t ) + d dt ( L−1 ( 1 sα L (u(x, t)ux(x, t)) )) . (54) A. M. Alhammad, A. M. Saeed / Eur. J. Pure Appl. Math, 18 (4) (2025), 6982 17 of 22 We simply substitute equation (54) and the initial condition equation (44) into equation (29); by the new modified function, we find u0(x, t) =x2(1− 2t), u1(x, t) =x2(1− 2t) + 6x2tα Γ(α+ 1) − 24x2tα+1 Γ(α+ 2) + 48x2tα+2 Γ(α+ 3) , u2(x, t) =x2(1− 2t) + 6x2 ( tα Γ(α+ 1) − 3tα+1 Γ(α+ 2) + 4tα+2 Γ(α+ 3) + 12t2α Γ(2α+ 1) · · · ) Tables (5, 6, 7) show the approximate solutions for Eq. (43) obtained for different values of a using the decomposition method Sumudu decomposition method, Natural decomposition method, Adomian decomposition method and Modified Laplace Variational Iteration. It is to be noted that only three terms of this decomposition series and the Modified Laplace Variational Iteration were used in evaluating the approximate solutions for Table (5, 6, 7). The Accuracy can be improved by computing more terms of the approximate solution Table 5: Numerical values when α = 1.5 for Eq. (43). t x UADM UNDM − USDM UMLVIM 0.2 0.25 0.0592832468 0.0592832468 0.0596734742 0.5 0.2371329870 0.2371329870 0.2386938966 0.75 0.5335492208 0.5335492208 0.5370612674 1 0.9485319481 0.9485319481 0.9547755866 0.4 0.25 0.0654118621 0.0654118621 0.0708846937 0.5 0.2616474484 0.2616474484 0.2835387748 0.75 0.5887067589 0.5887067589 0.6379622433 1 1.0465897936 1.0465897936 1.1341550992 0.6 0.25 0.0631775739 0.0631775739 0.0846789648 0.5 0.2527102956 0.2527102956 0.3387158594 0.75 0.5685981652 0.5685981652 0.7621106836 1 1.0108411825 1.0108411825 1.3548634375 A. M. Alhammad, A. M. Saeed / Eur. J. Pure Appl. Math, 18 (4) (2025), 6982 18 of 22 Table 6: Numerical values when α = 1.75 for Eq. (43). t x UADM UNDM − USDM UMLVIM 0.2 0.25 0.0487012403 0.0487012403 0.0487471818 0.5 0.1948049610 0.1948049610 0.1949887272 0.75 0.4383111623 0.4383111623 0.4387246362 1 0.7792198441 0.7792198441 0.7799549088 0.4 0.25 0.0437480416 0.0437480416 0.0448869280 0.5 0.1749921662 0.1749921662 0.1795477122 0.75 0.3937323741 0.3937323741 0.4039823524 1 0.6999686650 0.6999686650 0.7181908488 0.6 0.25 0.0381836436 0.0381836436 0.0445168272 0.5 0.1527345746 0.1527345746 0.1780673088 0.75 0.3436527928 0.3436527928 0.4006514448 1 0.610938298 0.610938298 0.7122692353 Table 7: Numerical values when α = 2 for Eq. (43). t x UADM UNDM − USDM UMLVIM UExact 0.2 0.25 0.0433950857 0.0433950857 0.0433999819 0.0434027778 0.5 0.1735803429 0.1735803429 0.1735999276 0.1736111111 0.75 0.3905557714 0.3905557714 0.3905998371 0.3906250000 1 0.6943213714 0.6943213714 0.6943997104 0.6944444444 0.4 0.25 0.0315669714 0.0315669714 0.0317794680 0.0318877551 0.5 0.1262678857 0.1262678857 0.1271178720 0.1275510204 0.75 0.2841027429 0.2841027429 0.2860152121 0.2869897959 1 0.5050715429 0.5050715429 0.5084714881 0.5102040816 0.6 0.25 0.0220044571 0.0220044571 0.0236648775 0.0244140625 0.5 0.0880178286 0.0880178286 0.0946595101 0.0976562500 0.75 0.1980401143 0.1980401143 0.2129838978 0.2197265625 1 0.3520713143 0.3520713143 0.3786380405 0.3906250000 Table (8) shows the absolute error between the exact and approximate solutions for Eq. (43) produced using Adomian Decomposition Method and Modified Laplace Variational Iteration Method. The results are computed after applying three iterations of each method for various values. A. M. Alhammad, A. M. Saeed / Eur. J. Pure Appl. Math, 18 (4) (2025), 6982 19 of 22 Table 8: The absolute error for α = 2 for Eq. (43). t x UADM UMLVIM 0.2 0.25 0.0000076921 0.0000027959 0.5 0.0000307682 0.0000111835 0.75 0.0000692286 0.0000251629 1 0.0001230730 0.0000447341 0.4 0.25 0.0003207837 0.0001082871 0.5 0.0012831347 0.0004331484 0.75 0.0028870530 0.0009745838 1 0.0051325387 0.0017325935 0.6 0.25 0.0024096054 0.0007491850 0.5 0.0096384214 0.0029967399 0.75 0.0216864482 0.0067426647 1 0.0385536857 0.0119869595 Figure 4: Comparison of ADM and MLVIM solution for the first three approximations α = 1.5, t = 0.4, with x = 0 : 1, for equation (43). A. M. Alhammad, A. M. Saeed / Eur. J. Pure Appl. Math, 18 (4) (2025), 6982 20 of 22 Figure 5: Comparison of ADM and MLVIM solution for the first three approximations α = 2, t = 0.6, with x = 0 : 1, for equation (43). Figure 6: Comparison of the absolute error between the ADM and MLVIM solution for the first three approxi- mations α = 2, t = 0.6, with x = 0 : 1, for Eq. (43). 6. Conclusions The main objective of this article was to investigate an accurate approximate solutions for nonlinear partial differential equations of fractional order. This goal was achieved by applying the modified Laplace variational iteration method and three variants of the Adomian Decomposition Method (ADM): the standard ADM, ADM with the Sumudu transform, and the natural ADM. All these methods yield solutions expressed as con- vergent series that are straightforward to compute. There are two key observations to highlight. First, all three variants of the Adomian Decomposition Method (ADM)—the standard ADM, ADM with the Sumudu transform, and the natural ADM—yielded iden- tical numerical solutions and convergence rates. This demonstrates their equivalence in A. M. Alhammad, A. M. Saeed / Eur. J. Pure Appl. Math, 18 (4) (2025), 6982 21 of 22 both accuracy and efficiency for the cases studied. Therefore, the choice among them can primarily depend on ease of implementation, although differences may emerge for more complex problems. Second, in Example 1, the decomposition method provided a faster-converging approximate solution than the modified Laplace variational iteration method. Conversely, in Example 2, the modified Laplace variational iteration method showed faster convergence toward the exact solution than the decomposition method. Hence, the accuracy and performance of these methods depend on the specific nonlinear fractional differential equation under consideration. Both approaches can thus serve as effective alternatives for solving fractional partial differential equations. These findings are consistent with previously reported results [8, 14]. 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