EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 3, Article Number 6422 ISSN 1307-5543 – ejpam.com Published by New York Business Global Abundant Solitary Solutions for the Fractional Unidirectional Wave Model Using in Oceanography, Coastal engineering, and Meteorology Wael W. Mohammed1,∗, Monirah W. Alshammary1, Ahmed E. Matouk2,3, Naveed Iqbal1 1 Department of Mathematics, College of Science, University of Ha’il, Ha’il 2440, Saudi Arabia 2 Department of Mathematics, College of Science Al-Zulfi, Majmaah University, Al-Majmaah 11952, Saudi Arabia 3 College of Engineering, Majmaah University, Al-Majmaah 11952, Saudi Arabia Abstract. In this paper, we consider the unidirectional wave model (UWM) with beta-derivative operator (BDO). This model simplifies the complexity of wave interactions by providing a one- dimensional approach to understanding wave behavior, particularly under conditions where wave directionality plays a crucial role. Its applications are vital in different area, such as oceanography, coastal engineering, and environmental science, contributing significantly to our understanding of wave dynamics and their impact on coastal and marine environments. Therefore, it is crucial to find the solutions for this model. By applying the F-expansion method, we can obtain abundant solutions including periodic, bright, kink, anti-kink, singular, and dark-bright solitons. Further- more, the graphs of the solutions are displayed using the MATLAB software to demonstrate how the beta-derivative operator affects the obtained solution. 2020 Mathematics Subject Classifications: 35Q51, 35Q80 Key Words and Phrases: Beta-derivative, F-expansion method, Exact solutions, nonlinear evolution equations 1. Introduction Nonlinear evolution equations (NLEEs) provide a powerful mathematical framework for studying complex systems with memory effects and non-linear interactions, leading to novel insights and applications in engineering and science [1–5]. The non-linear na- ture of these equations makes them suitable for modeling a wide range of phenomena in engineering, biology, chemistry, and physics. As a result, it is important to solve these NLEEs. Recently, numerous powerful approaches for discovering exact solutions to ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.6422 Email addresses: w.mohammed@uoh.edu.sa (W. W. Mohammed), n.iqbal@uoh.edu.sa (N. Iqbal) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) W. W. Mohammed et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6422 2 of 13 NLEEs have been introduced, such as, sine-Gordon expansion technique [6], F-expansion method [7], modified extended tanh-function method [8], Jacobi elliptic function expan- sion [9], exp-function method [10], (G′/G)-expansion [11, 12], sine-cosine method [13], exp(−ϕ(ς))-expansion method [14], Sumudu perturbation transform method [15], unified Riccati equation expansion technique [16, 17] and etc. One of these equations is the unidirectional wave model (UWM). UWM is an essential tool in understanding and predicting the behavior of waves in various natural phenomena. This model simplifies the complex interactions of waves by assuming that they propagate in only one direction, making it easier to analyze and interpret wave motion. One of the major fields where the unidirectional wave model is crucial in oceanography. This model helps scientists and researchers in studying the dynamics of ocean waves, including their formation, propagation, and interaction with other components of the ocean system. By using this model, oceanographers can accurately predict wave height, direction, and speed, which are essential for understanding the impact of waves on coastal areas, marine ecosystems, and human activities at sea. In coastal engineering, the unidirectional wave model plays a vital role in designing coastal structures that can withstand the forces exerted by waves. Engineers use this model to determine the wave conditions at a given site, which is crucial for designing breakwaters, seawalls, and other coastal defenses that protect shorelines from erosion and damage during storms. By employing the unidirectional wave model, engineers can make informed decisions about the size, shape, and placement of coastal structures, ensuring their effectiveness in minimizing wave impact. Meteorologists also rely on the unidirectional wave model to forecast and track ocean waves and their effects on weather patterns. By incorporating wave data into their models, meteorologists can better predict the intensity and path of storms, hurricanes, and other extreme weather events that are fueled by ocean waves. This information is essential for issuing timely warnings and alerts to coastal communities, helping to mitigate the potential damage and loss of life caused by these natural disasters. The UWM with beta-derivative operator (UWM-BDO) takes the following form [18]: Dκ t W + 1 6 a1Wxxx + 3 2 a2WWx + 15 32 a22W2Wx + a3Wx + a4Wy + a5Wz = 0, (1) where W = W(x, y, z, t) presents a wave profile; a1, a2, a3, a4 and a5 are constants. Dκ is the beta-derivative operator (BDO) of order κ. The BDO is a novel conformable fractional derivative that was recently proposed by Atangana et al [19]. Now, let us BDO of order κ ∈ (0, 1] for the function W : (0,∞) → R as follows: Dκ t W(t) = lim ϵ→0 W(t+ ϵ(t+ 1 Γ(β)) 1−β)−W(t) ϵ . For any constants c1 and c2, the BDOmeets the following characteristics [19]: (1)Dκ t [c1u(t)+ c2v(t)] = c1Dκ t u(t)+c2Dκ t v(t), (2)Dκ t [c1] = 0, (3)Dκ t u(t) = (t+ 1 Γ(κ)) 1−κ du dt , (4)D κ t u(v(t)) = (t+ 1 Γ(κ)) 1−κv′(t)u′(v(t)) (5) If θ = c1 κ (t+ 1 Γ(κ)) κ, then Dκ t u(t) = c1 du dt . W. W. Mohammed et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6422 3 of 13 The main contribution of this work is to acquire the exact solutions to the UWM-BDO (1). The F-expansion method is used to create fractional solutions in the form of elliptic, trigonometric and hyperbolic functions. Numerous important scientific phenomena may be investigated using the achieved solutions of the UWM-BDO (1) because this equation has a significant applications in various fields such as oceanography, coastal engineering, and meteorology. To analyze the impact of the BDO on the solutions of UWM-BDO (1), many graphs are created with the MATLAB program. The following is an outline of the paper: We obtain the wave equation for Eq. (1) in Section 2. The solutions of UWM-BDO (1) are obtained in Section 3. In Section 4, we discuss the influence of BDO on the solutions of (1). Finally, the conclusion of the study is stated. 2. Traveling Wave Equation To get the wave equation for UWM-BDO (1), we apply W(x, y, z, t) = V(ρ), ρ = ρ1x+ ρ2y + ρ3z + λ κ (t+ 1 Γ(κ) )κ, (2) where ρ1, ρ2, ρ3 are non-zero constants and they represent the wave number component in the x, y and z directions, respectively; λ is the wave speed. Differentiating Eq. (2) with respect to t, x, y and z, we get Dκ t W = λV ′, Wx = ρ1V ′, Wy = ρ2V ′, Wz = ρ3V ′, Wxxx = ρ31V ′′′. (3) Putting Eqs (2) and (3) into Eq. (1), we get 1 6 a1ρ 3 1V ′′′ + (λ+ a3ρ1 + a4ρ2 + a5ρ3)V ′ + 3 2 a2ρ1VV ′ + 15 32 a22ρ1V2V ′ = 0. (4) Integrating Eq. (4) once, we have V ′′ +A1V +A2V2 +A3V3 = 0, (5) where A1 = 6(λ+ a3ρ1 + a4ρ2 + a5ρ3) a1ρ31 , A2 = 9a2 2a1ρ21 , A3 = 15a22 16a1ρ21 . 3. Exact solutions of the UWM-BDO Let us utilize the F-expansion method ( for more information see [20–22]). Assuming the solution of Eq. (5) has the form V(ρ) = N∑ j=0 ℓjF j(ρ), ℓN ̸= 0, (6) W. W. Mohammed et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6422 4 of 13 where ℓ0, ℓ1, ℓ2, , ℓN−1 and ℓN are unknown constants to be calculated and F solves the following auxiliary equation: (F ′)2 = ℏ1F4 + ℏ2F2 + ℏ3, (7) where ℏ1, ℏ2, and ℏ3 are real numbers. First, let us balance V3 with V ′′ in Eq. (5) to calculate the parameters N as 3N = N + 2 ⇒ N = 1. With N = 1, Eq. (6) becomes V(ρ) = ℓ0 + ℓ1F(ρ), (8) Putting Eq. (8) into Eq. (5), we get +[2ℏ1ℓ1 +A3ℓ 3 1]F3 + [A2ℓ 2 1 + 3A3ℓ0ℓ 2 1]F2 +[ℏ2ℓ1 +A1ℓ1 + 2ℓ0ℓ1A2 + 3ℓ20ℓ1A3]F +[A1ℓ0 +A2ℓ 2 0 +A3ℓ 3 0] = 0. For j = 3, 2, 1, 0, we balance each coefficient of F j with zero to have 2ℏ1ℓ1 +A3ℓ 3 1 = 0, A2ℓ 2 1 + 3A3ℓ0ℓ 2 1 = 0, ℏ2ℓ1 +A1ℓ1 + 2ℓ0ℓ1A2 + 3ℓ20ℓ1A3 = 0, and A1ℓ0 +A2ℓ 2 0 +A3ℓ 3 0 = 0. By solving the above equations, we get ℓ0 = −A2 3A3 = −8 5a2 , ℓ1 = ± √ −2ℏ1 A3 = ± √ −32a1ρ21ℏ1 15a22 , and ℏ2 = A2 2 9A3 = 12 5a1ρ21 . (9) Substituting (9) into Eq. (8), we have the solution of the traveling wave Eq. (5) as: V (ρ) = −8 5a2 ± √ −128ℏ1 25a22ℏ2 F (ρ) . (10) Consequently, putting Eq. (10) into Eq. (2), we get the next solutions for the UWM-BDO (1): W(x, y, z, t) = { −8 5a2 ± √ −128ℏ1 25a22ℏ2 F (ρ) } , (11) where ρ = ρ1x+ ρ2y + ρ3z + λ κ(t+ 1 Γ(κ)) κ. W. W. Mohammed et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6422 5 of 13 There are several cases for the solutions of Eq. (9) that depending on ℏ1, ℏ2 and ℏ3 as follows: Case 1: If ℏ1 = ϖ2, ℏ2 = − ( 1 +ϖ2 ) and ℏ3 = 1, then F (ρ) = sn(ρ), and Eq. (11) has the form W(x, y, z, t) = { −8 5a2 ± √ 128ϖ2 25a22 (1 +ϖ2) sn(ρ1x+ ρ2y + ρ3z + λ κ (t+ 1 Γ(κ) )κ) } . (12) If ϖ → 1, then Eq. (12) changes to W(x, y, z, t) = { −8 5a2 ± 8 5a2 tanh(ρ1x+ ρ2y + ρ3z + λ κ (t+ 1 Γ(κ) )κ) } . (13) Case 2: If ℏ1 = ϖ2, ℏ2 = − ( 1 +ϖ2 ) and ℏ3 = 1, then F (ρ) = cd(ρ), and Eq. (11) takes the form W(x, y, z, t) = { −8 5a2 ± √ 128ϖ2 25a22 (1 +ϖ2) cd(ρ1x+ ρ2y + ρ3z + λ κ (t+ 1 Γ(κ) )κ) } . (14) Case 3: If ℏ1 = −ϖ2, ℏ2 = 2ϖ2 − 1 and ℏ3 = 1−ϖ2, then F (ρ) = cn(ρ), and Eq. (11) has the form W(x, y, z, t) = { −8 5a2 ± √ 128ϖ2 25a22(2ϖ 2 − 1) cn(ρ1x+ ρ2y + ρ3z + λ κ (t+ 1 Γ(κ) )κ) } , (15) for 1√ 2 < ϖ < 1. If ϖ → 1, then Eq. (15) becomes W(x, y, z, t) = { −8 5a2 ± √ 128 25a22 sech(ρ1x+ ρ2y + ρ3z + λ κ (t+ 1 Γ(κ) )κ) } . (16) Case 4: If ℏ1 = −1, ℏ2 = 2−ϖ2 and ℏ3 = ϖ2 − 1, then F (ρ) = dn(ρ), and Eq. (11) has the form W(x, y, z, t) = { −8 5a2 ± √ 128 25a22(2−ϖ2) dn(ρ1x+ ρ2y + ρ3z + λ κ (t+ 1 Γ(κ) )κ) } . (17) If ϖ → 1, then Eq. (17) becomes Eqs (16). Case 5: If ℏ1 = 1, ℏ2 = − ( 1 +ϖ2 ) and ℏ3 = ϖ2, then F (ρ) = ns(ρ), and Eq. (11) has the form W(x, y, z, t) = { −8 5a2 ± √ 128 25a22 (1 +ϖ2) ns(ρ1x+ ρ2y + ρ3z + λ κ (t+ 1 Γ(κ) )κ) } . (18) W. W. Mohammed et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6422 6 of 13 If ϖ → 1, then Eq. (18) tends to W(x, y, z, t) = { −8 5a2 ± 8 5a2 coth(ρ1x+ ρ2y + ρ3z + λ κ (t+ 1 Γ(κ) )κ) } . (19) While if ϖ → 0, then Eq. (18) becomes W(x, y, z, t) = { −8 5a2 ± √ 128 25a22 csc(ρ) } . (20) Case 6: If ℏ1 = 1, ℏ2 = − ( 1 +ϖ2 ) and ℏ3 = ϖ2, then F (ρ) = dc(ρ) = dn(ρ) cn(ρ) , and Eq. (11) has the form W(x, y, z, t) = { −8 5a2 ± √ 128 25a22 (1 +ϖ2) dc(ρ1x+ ρ2y + ρ3z + λ κ (t+ 1 Γ(κ) )κ) } . (21) If ϖ → 0, then Eq. (21) becomes W(x, y, z, t) = { −8 5a2 ± √ 128 25a22 sec(ρ1x+ ρ2y + ρ3z + λ κ (t+ 1 Γ(κ) )κ) } . (22) Case 7: If ℏ1 = 1−ϖ2, ℏ2 = 2ϖ2 − 1 and ℏ3 = −ϖ2, then F (ρ) = nc(ρ), and Eq. (11) has the form W(x, y, z, t) = { −8 5a2 ± √ 128(1−ϖ2) 25a22(2ϖ 2 − 1) nc(ρ1x+ ρ2y + ρ3z + λ κ (t+ 1 Γ(κ) )κ) } , (23) for ϖ < 1√ 2 . If ϖ → 0, then Eq. (23) becomes Eqs (22). Case 8: If ℏ1 = ϖ2 − 1, ℏ2 = 2−ϖ2 and ℏ3 = −1, then F (ρ) = nd(ρ), and Eq. (11) takes the form W(x, y, z, t) = { −8 5a2 ± √ 128(1−ϖ2) 25a22(2−ϖ2) nd(ρ1x+ ρ2y + ρ3z + λ κ (t+ 1 Γ(κ) )κ) } . (24) Case 9: If ℏ1 = −ϖ2(1 − ϖ2), ℏ2 = 2ϖ2 − 1 and ℏ3 = 1, then F (ρ) = sd(ρ), and Eq. (11) takes the form W(x, y, z, t) = { −8 5a2 ± √ 128ϖ2(1−ϖ2) 25a22(1− 2ϖ2) sd(ρ1x+ ρ2y + ρ3z + λ κ (t+ 1 Γ(κ) )κ) } , (25) for ϖ > 1√ 2 . Case 10: If ℏ1 = 1, ℏ2 = 2ϖ2 − 1 and ℏ3 = −ϖ2(1 −ϖ2), then F (ρ) = ds(ρ), and Eq. (11) takes the form W(x, y, z, t) = { −8 5a2 ± √ −128ℏ1 25a22ℏ2 ds(ρ1x+ ρ2y + ρ3z + λ κ (t+ 1 Γ(κ) )κ) } , (26) W. W. Mohammed et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6422 7 of 13 for ϖ < 1√ 2 . Case 12: If ℏ1 = 1 4 , ℏ2 = ϖ2−2 2 and ℏ3 = ϖ2 4 , then F (ρ) = ns(ρ) ± ds(ρ), and Eq. (11) takes the form W(x, y, z, t) = −8 5a2 ± √ −128ℏ1 25a22ℏ2 ( ns(ρ1x+ ρ2y + ρ3z + λ κ (t+ 1 Γ(κ) )κ) ± ds(ρ1x+ ρ2y + ρ3z + λ κ (t+ 1 Γ(κ) )κ) ) . (27) If ϖ → 1, then Eq. (27) changes to W(x, y, z, t) = −8 5a2 ± √ 1 9a23 ( coth(ρ1x+ ρ2y + ρ3z + λ κ (t+ 1 Γ(κ) )κ) ± csch(ρ1x+ ρ2y + ρ3z + λ κ (t+ 1 Γ(κ) )κ) ) . (28) While, if ϖ → 0, then Eq. (27) tends to W(x, y, z, t) = −8 5a2 ± √ −128ℏ1 25a22ℏ2 ( csc(ρ1x+ ρ2y + ρ3z + λ κ (t+ 1 Γ(κ) )κ) ± cot(ρ1x+ ρ2y + ρ3z + λ κ (t+ 1 Γ(κ) )κ) ) . (29) Case 13: If ℏ1 = ϖ2 4 , ℏ2 = ϖ2−2 2 and ℏ3 = ϖ2 4 , then F (ρ) = √ 1−ϖ2 (sd(ρ)± cd(ρ)) , and Eq. (11) takes the form W(x, y, z, t) = −8 5a2 ± √ −128ℏ1 25a22ℏ2 ( sd(ρ1x+ ρ2y + ρ3z + λ κ (t+ 1 Γ(κ) )κ) ± cd(ρ1x+ ρ2y + ρ3z + λ κ (t+ 1 Γ(κ) )κ) ) . (30) Case 14: If ℏ1 = ϖ2−1 4 , ℏ2 = ϖ2+1 2 and ℏ3 = ϖ2−1 4 , then F (ρ) = ϖsd(ρ) ± nd(ρ), and Eq. (11) has the form W(x, y, z, t) = −8 5a2 ± √ −128ℏ1 25a22ℏ2 ( ϖsd(ρ1x+ ρ2y + ρ3z + λ κ (t+ 1 Γ(κ) )κ) ± nd(ρ1x+ ρ2y + ρ3z + λ κ (t+ 1 Γ(κ) )κ) ) . (31) Case 15: If ℏ1 = ϖ2 4 , ℏ2 = ϖ2−2 2 and ℏ3 = 1 4 , then F (ρ) = sn(ρ) 1±dn , and Eq. (11) takes the form W(x, y, z, t) = −8 5a2 ± √ −128ℏ1 25a22ℏ2 ( sn(ρ1x+ ρ2y + ρ3z + λ κ(t+ 1 Γ(κ)) κ) 1± dn(ρ1x+ ρ2y + ρ3z + λ κ(t+ 1 Γ(κ)) κ) ) . (32) W. W. Mohammed et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6422 8 of 13 If ϖ → 1, then Eq. (32) becomes W(x, y, z, t) = −8 5a2 ± √ 1 9a23 ( tanh(ρ1x+ ρ2y + ρ3z + λ κ(t+ 1 Γ(κ)) κ) 1± sech(ρ1x+ ρ2y + ρ3z + λ κ(t+ 1 Γ(κ)) κ) ) . (33) Case 16: If ℏ1 = −1 4 , ℏ2 = ϖ2+1 2 and ℏ3 = (1−ϖ2) 2 4 , then F (ρ) = ϖcn(ρ) ± dn(ρ), and Eq. (11) takes the form W(x, y, z, t) = −8 5a2 ± √ 64ℏ1 25a22(ϖ 2 + 1) ( ϖcn(ρ1x+ ρ2y + ρ3z + λ κ (t+ 1 Γ(κ) )κ) ± dn(ρ1x+ ρ2y + ρ3z + λ κ (t+ 1 Γ(κ) )κ) ) . (34) Case 17: If ℏ1 = 1 4 , ℏ2 = 1−2ϖ2 2 and ℏ3 = 1 4 , then F (ρ) = sn(ρ) 1±cn(ρ) , and Eq. (11) has the form W(x, y, z, t) = −8 5a2 ± √ 64ℏ1 25a22(2ϖ 2 − 1) sn(ρ1x+ ρ2y + ρ3z + λ κ(t+ 1 Γ(κ)) κ) 1± cn(ρ1x+ ρ2y + ρ3z + λ κ(t+ 1 Γ(κ)) κ) , (35) for 1√ 2 < ϖ < 1. Where cn(ρ) = cn(ρ,ϖ), dn(ρ) = dn(ρ,ϖ), sn(ρ) = sn(ρ,ϖ), sc(ρ) = sc(ρ,ϖ), ds(ρ) = ds(ρ,ϖ), are the Jacobi elliptic functions (JEFs) for 0 < ϖ < 1 and ϖ is the elliptic modulus. We note that JEFs ϖ → 0 ϖ → 1 sn(ρ) sin(ρ) tanh(ρ) cs(ρ) cot(ρ) csch(ρ) cn(ρ) cos(ρ) sech(ρ) ds(ρ) csc(ρ) csch(ρ) dn(ρ) 1 sech(ρ) sc(ρ) tan(ρ) sinh(ρ) ns(ρ) csc(ρ) coth(ρ) 4. Discussion and Effect of BDO The UWM-BDO provides a powerful tool for modeling wave propagation in a wide range of physical systems. By capturing non-local effects and memory effects, this mathe- matical model offers a more comprehensive understanding of complex wave phenomena in heterogeneous media. The incorporation of fractional derivatives into the wave equation opens up new avenues for research and applications in areas such as acoustics, electromag- netics, and geophysics, where accurate and efficient wave modeling is essential. We discuss the impact of the beta-derivative operator on the obtained solutions of UWM-BDO (1). To show how these solutions behave, various graphical representations W. W. Mohammed et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6422 9 of 13 are offered. For some obtained solutions, such as Eqs. (12), (13) and (16), we simulate the graphical representations for ρ1 = ρ2 = ρ3 = 1, a2 = 1 and for varying values of κ. We deduce from these figures that when the fractional order decreases, the surface moves into the right as follows: (a) κ = 1 (b) κ = 0.8 (c) κ = 0.6 (d) κ = 0.5 (e) κ = 0.4 (f) κ = 1, 0.8, 0.6, 0.5, 0.4 Figure 1. (a-e) describe 3D-profile of solution W(x, y, z, t) in Eq (12) with ϖ = 0.5, a2 = 1, y = z = 0, ρ1 = ρ2 = ρ3 = 1, x ∈ [−4, 4], and t ∈ [0, 3] (f) display 2D-profile of Eq. (12) with various κ W. W. Mohammed et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6422 10 of 13 (a) κ = 1 (b) κ = 0.8 (c) κ = 0.6 (d) κ = 0.5 (e) κ = 0.4 (f) κ = 1, 0.8, 0.6, 0.5, 0.4 Figure 2. (a-e) describe 3D-profile of solution W(x, y, z, t) in Eq (13) with a2 = 1, y = z = 0, ρ1 = ρ2 = ρ3 = 1, x ∈ [−4, 4], and t ∈ [0, 3] (f) display 2D-profile of Eq. (13) with various κ W. W. Mohammed et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6422 11 of 13 (a) κ = 1 (b) κ = 0.8 (c) κ = 0.6 (d) κ = 0.5 (e) κ = 0.4 (f) κ = 1, 0.8, 0.6, 0.5, 0.4 Figure 3. (a-e) describe 3D-profile of solution W(x, y, z, t) in Eq (33) with a2 = 1, y = z = 0, ρ1 = ρ2 = ρ3 = 1, x ∈ [−4, 4], and t ∈ [0, 3] (f) display 2D-profile of Eq. (33) with various κ W. W. Mohammed et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6422 12 of 13 5. Conclusions In this paper, the unidirectional wave model (UWM) (1) with beta-derivative operator (BDO) was considered. We obtained many various kind of solutions including periodic soliton, bright soliton, kink soliton, anti-kink soliton, singular soliton, and dark-bright soli- ton by using the F-expansion method. 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