Semi-Analytical and Numerical Solutions of the KdV-BBM Equation Using Differential Transformation Method and iRK-5 Based Method of Lines
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KdV-BBM, iRK-5, Cauchy boundary, DTM, MOL
Abstract
The Korteweg-de Vries-Benjamin-Bona-Mahony (KdV-BBM) equation is a fundamental mathematical model describing nonlinear long-wave propagation in dispersive media. This study presents a comprehensive computational investigation employing two distinct and complementary approaches: a tailored semi-analytical Differential Transformation Method (DTM) and a numerical Method of Lines (MOL) coupled with an improved fifth-order Runge-Kutta (iRK-5) scheme to solve both homogeneous and inhomogeneous KdV-BBM equations. The scientific novelty of this work lies in optimizing these two independent frameworks to achieve high accuracy without requiring restrictive linearization or excessively small time steps. Semi-analytically, DTM is optimized using prescribed Cauchy boundary conditions for the homogeneous model to supply essential derivative data that guarantees coefficient uniqueness and accelerates series convergence, while Nayar’s Theorem is incorporated to efficiently circumvent complex recurrence relations in the inhomogeneous case via variable separability. Numerically, the continuous PDE is discretized spatially via centered differences and integrated in time using the high-order iRK-5 scheme. Both approaches demonstrate exceptional performance in resolving the KdV-BBM model. Comparative evaluations demonstrate that the MOL(iRK-5) scheme superiorly preserves physical wave integrity, soliton peak height, and phase characteristics over extended simulation periods (T = 30), significantly outperforming standard explicit Fourth-Order Runge-Kutta (RK-4) and implicit Crank-Nicolson (C-N) methods. Unlike lower-order implicit schemes like C-N, which exhibit limited temporal accuracy (O(Δt2)) and high matrix inversion costs, or RK-4 which faces severe CFL constraints, iRK-5 achieves higher temporal precision (O(Δt5)) and an expanded stability domain, allowing larger time steps (Δt =0.5) with minimal error (E ≈6.58 ×10−4). Von Neumann stability analysis applied to the spatial discretization of the MOL framework confirms neutral stability, with iRK-5 providing a larger stability margin along the imaginary axis compared to RK-4. Grid refinement tests for the inhomogeneous model further confirm that spatial resolution (Δx) dominates total accuracy; reducing Δx from 0.01 to 0.0003 successfully decreases the Mean Square Error from 10−3 to 10−7. Overall, both the semi-analytical DTM and numerical MOL(iRK-5) frameworks prove to be highly accurate, robust, and effective standalone tools for investigating complex nonlinear dispersive wave dynamics.
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