PPubMed15 Jul 2015

Study of coupled nonlinear partial differential equations for finding exact analytical solutions Nonlinear Partial Differential Equation

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Abstract

Title: Study of coupled nonlinear partial differential equations for finding exact analytical solutions Nonlinear Partial Differential Equation Authors: ['Kamruzzaman Khan' 'M. Ali Akbar' 'H. Koppelaar'] Exact solutions of nonlinear partial differential equations (NPDEs) are obtained via the enhanced (*G*′/*G*)-expansion method. The method is subsequently applied to find exact solutions of the Drinfel'd-Sokolov-Wilson (DSW) equation and the (2+1)-dimensional Painlevé integrable Burgers (PIB) equation. The efficiency of this method for finding these exact solutions is demonstrated. The method is effective and applicable for many other NPDEs in mathematical physics.

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The enhanced (G'/G)-expansion method provides numerous exact soliton and periodic solutions for the DSW and PIB nonlinear PDEs.

What’s new

i.Applied the enhanced (G'/G)-expansion method to coupled nonlinear PDEs
ii.Derived exact analytical solutions for the Drinfel'd‑Sokolov‑Wilson (DSW) equation
iii.Derived exact analytical solutions for the (2+1)-dimensional Painlevé integrable Burgers (PIB) equation
iv.Generated multiple families of solitary, periodic, and singular wave solutions
v.Validated all obtained solutions using Maple 13

How they did it

The authors combine the spatial and temporal variables into a single travelling‑wave variable ξ=x±ωt and assume the solution can be expressed as a rational function of G′/G, where G satisfies the linear ODE G″+μG=0. By balancing the highest‑order derivative with the nonlinear term they determine the polynomial degree n, substitute the ansatz into the transformed ODE, collect powers of G′/G and set the coefficients to zero, yielding an over‑determined algebraic system that is solved (with Maple 13) for the parameters.

What they found

Nine families of hyperbolic soliton solutions for the DSW equation when μ<0 (e.g., ω=−8qμ, ω=4qμ, etc.)
Five families of hyperbolic solutions for the PIB equation when μ<0 (e.g., ω=−2βμλ−2βαμλ−a₀−αR−αa₀)
Graphical illustrations for selected solutions using parameter values p=1, q=2, r=s=1, μ=−0.25 (Figures 1‑4) and a₀=α=β=γ=1, μ=−1 (Figures 5‑8)

Caveats

✕The derived solutions require r≠−2s (as noted in the DSW analysis)
✕For certain PIB families μ must not equal −1/λ²
✕Periodic solution families are mentioned but omitted from detailed presentation
✕Parameter choices leading to division by zero are excluded

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