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A new modified conformal expansion method to some nonlinear conformal partial differential equations Cover

A new modified conformal expansion method to some nonlinear conformal partial differential equations

International Journal of Mathematics and Computer in Engineering
AHEAD OF PRINT
By: Lama Alhakim,  Alaaeddin Moussa,  Boubekeur Gasmi,  Yazid Mati and  Md Nurul Raihen  
Open Access
|Dec 2025

Figures & Tables

Fig. 1

(a) The 3D plot of u3.1 (ξ) when λ = 5, μ = 0.01, C1 = −5, C2 = −11, β = 6, γ = 1, k1 = 1; k2 = −1 and α=15 \alpha = {1 \over 5} , showcasing it as a multi-soliton solution, (b) The 3D plot of v3.1 (ξ)) when λ = 3, μ=16 \mu = {1 \over 6} , C1 = 2, C2 = 3, β = 11, γ = −3, k1 = 1; k2 = 2, and α = 1, showcasing it as a bright soliton solution.
(a) The 3D plot of u3.1 (ξ) when λ = 5, μ = 0.01, C1 = −5, C2 = −11, β = 6, γ = 1, k1 = 1; k2 = −1 and α=15 \alpha = {1 \over 5} , showcasing it as a multi-soliton solution, (b) The 3D plot of v3.1 (ξ)) when λ = 3, μ=16 \mu = {1 \over 6} , C1 = 2, C2 = 3, β = 11, γ = −3, k1 = 1; k2 = 2, and α = 1, showcasing it as a bright soliton solution.

Fig. 2

(a) The 3D plot of u3.2 (ξ) when λ = 0.1, μ = 1, C1 = −5, C2 = −11, β = 63, γ = 1, k1 = 1; k2 = −1 and α=12 \alpha = {1 \over 2} , showcasing it as a kink wave solution, (b) The 3D plot of v3.2 (ξ) when λ = 0.3, μ = 111, C1 = 3, C2 = 2, β = −2, γ = −1, k1 = 1, k2 = 1, and α=12 \alpha = {1 \over 2} , showcasing it as a multi-soliton solution.
(a) The 3D plot of u3.2 (ξ) when λ = 0.1, μ = 1, C1 = −5, C2 = −11, β = 63, γ = 1, k1 = 1; k2 = −1 and α=12 \alpha = {1 \over 2} , showcasing it as a kink wave solution, (b) The 3D plot of v3.2 (ξ) when λ = 0.3, μ = 111, C1 = 3, C2 = 2, β = −2, γ = −1, k1 = 1, k2 = 1, and α=12 \alpha = {1 \over 2} , showcasing it as a multi-soliton solution.

Fig. 3

(a) The 3D plot of u3.3 (ξ) when λ = 4, μ = 4, C1 = −3, C2 = −5, β = 3, γ = 3, k1 = 3, k2 = 3 and α = 1, showcasing it as a solitary wave. (b) The 3D plot of v3.3 (ξ) when λ = 2, μ = 1, C1 = 1.1, C2 = 2, β = 15, γ = −1, k1 = 2, k2 = −2 and α=13 \alpha = {1 \over 3} , showcasing it as a single wave behavior.
(a) The 3D plot of u3.3 (ξ) when λ = 4, μ = 4, C1 = −3, C2 = −5, β = 3, γ = 3, k1 = 3, k2 = 3 and α = 1, showcasing it as a solitary wave. (b) The 3D plot of v3.3 (ξ) when λ = 2, μ = 1, C1 = 1.1, C2 = 2, β = 15, γ = −1, k1 = 2, k2 = −2 and α=13 \alpha = {1 \over 3} , showcasing it as a single wave behavior.

Fig. 4

(a) The 3D plot of h2.1 (ξ) when λ = 4, μ=12 \mu = {1 \over 2} , β0 = 2, α2 = 2, C1 = −1, C2 = 1, p = 8, q = 2 and α=12 \alpha = {1 \over 2} , showcasing it as multi-soliton solution, (b) The 3D plot of g2.1 (ξ)) when λ = 2, μ = 0.1, β0 = 1, α2 = 1, C1 = 4, C2 = 2, p = 6, q = 0.05 and α=12 \alpha = {1 \over 2} , showcasing it as a as soliton solution.
(a) The 3D plot of h2.1 (ξ) when λ = 4, μ=12 \mu = {1 \over 2} , β0 = 2, α2 = 2, C1 = −1, C2 = 1, p = 8, q = 2 and α=12 \alpha = {1 \over 2} , showcasing it as multi-soliton solution, (b) The 3D plot of g2.1 (ξ)) when λ = 2, μ = 0.1, β0 = 1, α2 = 1, C1 = 4, C2 = 2, p = 6, q = 0.05 and α=12 \alpha = {1 \over 2} , showcasing it as a as soliton solution.

Fig. 5

(a) The 3D plot of h2.2 (ξ) when λ = 2, μ = 8, α=12 \alpha = {1 \over 2} , β0 = 1, α2 = 2, C1 = −5, C2 = −11, p = 6 and q = 3, showcasing it as multi-soliton solution, (b) The 3D plot of g2.2 (ξ) when λ = 1, μ = 4, β0 = 2, α2 = 1, C1 = −11, C2 = 2, p = −4, q = 3 and α=12 \alpha = {1 \over 2} , showcasing it as multi-soliton solution.
(a) The 3D plot of h2.2 (ξ) when λ = 2, μ = 8, α=12 \alpha = {1 \over 2} , β0 = 1, α2 = 2, C1 = −5, C2 = −11, p = 6 and q = 3, showcasing it as multi-soliton solution, (b) The 3D plot of g2.2 (ξ) when λ = 1, μ = 4, β0 = 2, α2 = 1, C1 = −11, C2 = 2, p = −4, q = 3 and α=12 \alpha = {1 \over 2} , showcasing it as multi-soliton solution.

Fig. 6

(a) The 3D plot of of h3.3 (ξ) when λ = 2, μ = 1, α=12 \alpha = {1 \over 2} , β0 = 1, α2 = 1, C1 = 1, C2 = 1, p = 6 and q = 5, showcasing it as Singular Kink wave solution, (b) The 3D plot of g3.3 (ξ) when λ = 2, μ = 1, β0 = 1, α2 = 1, C1 = 111, C2 = 5, p = 2, q = 5 and α=12 \alpha = {1 \over 2} , showcasing it as singular anti Kink wave solution of
(a) The 3D plot of of h3.3 (ξ) when λ = 2, μ = 1, α=12 \alpha = {1 \over 2} , β0 = 1, α2 = 1, C1 = 1, C2 = 1, p = 6 and q = 5, showcasing it as Singular Kink wave solution, (b) The 3D plot of g3.3 (ξ) when λ = 2, μ = 1, β0 = 1, α2 = 1, C1 = 111, C2 = 5, p = 2, q = 5 and α=12 \alpha = {1 \over 2} , showcasing it as singular anti Kink wave solution of

Fig. 7

(a) The 3D plot of ψ1.1 (ξ) when λ = 2, μ = 0.2, C1 = −5, C2 = 11, β = 1, k1 = 1 and α=12 \alpha = {1 \over 2} , showcasing it as Singular Kink wave solution, (b) The 3D plot of ϕ1.1 (ξ)) when λ = 2, μ = 0.2, C1 = −1, C2 = 1, β = 1, k1 = 1, δ = 2, and α=15 \alpha = {1 \over 5} , showcasing it as multi-soliton solution.
(a) The 3D plot of ψ1.1 (ξ) when λ = 2, μ = 0.2, C1 = −5, C2 = 11, β = 1, k1 = 1 and α=12 \alpha = {1 \over 2} , showcasing it as Singular Kink wave solution, (b) The 3D plot of ϕ1.1 (ξ)) when λ = 2, μ = 0.2, C1 = −1, C2 = 1, β = 1, k1 = 1, δ = 2, and α=15 \alpha = {1 \over 5} , showcasing it as multi-soliton solution.

Fig. 8

(a) The 3D plot of ψ1.2 (ξ) when λ = 2, μ = 0.2, C1 = −5, C2 = 11, β = 1, k1 = 1 and α=12 \alpha = {1 \over 2} , showcasing it as muli-soliton solution, (b) The 3D plot of ϕ1.2 (ξ) when λ = 1, μ = 4, C1 = −11, C2 = 2, β = 1, k1 = 1, δ = −4, and α=13 \alpha = {1 \over 3} , showcasing it as multi-soliton solution.
(a) The 3D plot of ψ1.2 (ξ) when λ = 2, μ = 0.2, C1 = −5, C2 = 11, β = 1, k1 = 1 and α=12 \alpha = {1 \over 2} , showcasing it as muli-soliton solution, (b) The 3D plot of ϕ1.2 (ξ) when λ = 1, μ = 4, C1 = −11, C2 = 2, β = 1, k1 = 1, δ = −4, and α=13 \alpha = {1 \over 3} , showcasing it as multi-soliton solution.

Fig. 9

(a) The 3D plot of ψ1.3 (ξ)) when λ = 2, μ = 0.2, C1 = −5, C2 = 11, β = 1, k1 = 1 and α=12 \alpha = {1 \over 2} , showcasing it as soliton solution, (b)The 3D plot of ϕ1.3 (ξ)) when λ = 2, μ = 1, C1 = 2, C2 = −2, β = −2, k1 = 2, δ = 2, and α=13 \alpha = {1 \over 3} , showcasing it as soliton solution.
(a) The 3D plot of ψ1.3 (ξ)) when λ = 2, μ = 0.2, C1 = −5, C2 = 11, β = 1, k1 = 1 and α=12 \alpha = {1 \over 2} , showcasing it as soliton solution, (b)The 3D plot of ϕ1.3 (ξ)) when λ = 2, μ = 1, C1 = 2, C2 = −2, β = −2, k1 = 2, δ = 2, and α=13 \alpha = {1 \over 3} , showcasing it as soliton solution.
DOI: https://doi.org/10.2478/ijmce-2025-0029 | Journal eISSN: 2956-7068
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Language: English
Submitted on: May 31, 2024
Accepted on: Sep 3, 2024
Published on: Dec 17, 2025
Published by: Harran University
In partnership with: Paradigm Publishing Services
Publication frequency: 2 issues per year
Keywords:
Modified conformal expansion method,
Katugampola's derivative,
Whitham-Broer-Kaup equations,
coupled Burgers equations,
coupled mKdV equations,
exact solutions,
partial differential equations,
nonlinear equations
Related subjects:
Computer sciences,
Computer sciences, other,
Engineering,
Introductions and overviews,
Engineering, other,
Mathematics,
General mathematics,
Physics,
Physics, other

© 2025 Lama Alhakim, Alaaeddin Moussa, Boubekeur Gasmi, Yazid Mati, Md Nurul Raihen, published by Harran University
This work is licensed under the Creative Commons Attribution 4.0 License.

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