The traveling wave behavior of the nonlinear third and fourth-order advection-diffusion equation has been elaborated. In this study, the effect of dispersion and dissipation processes was mainly analyzed thoroughly. In the thorough analysis, strictly permanent short waves to breaking waves, having comparative higher amplitudes, have been observed. The governed problem was employed with the space-splitting method for a coupled system of equations to conduct the computational process. For the time derivative, the Crank-Nicolson difference approximation was studied. An orthogonal collocation method using Hermite splines has been implemented to approximate the solution of the semi-discretized coupled problem. The proposed method reduces the equation to an iterative scheme of an algebraic system of collocation equations, which reduced the computational complexity. The proposed scheme is found to be unconditionally stable, and the numerical demonstrations and comparisons represented the computational efficiency.

The present study proposes a hybrid numerical technique to discuss the solution of non-linear reaction-diffusion equations with variable coefficients. The perturbation parameter was assumed to be time-dependent. The spatial domain was discretized using the cubic Hermite splines collocation method. These splines are smooth enough to interpolate the function as well as its tangent at the node points. The temporal domain was discretized using the Crank-Nicolson scheme, commonly known as the CN scheme. The cubic Hermite splines are convergent of order $h^4$, and the CN scheme is convergent of order $\mathrm{\Delta }{t}^2$. The technique is found to be convergent of order $O(h^2({\gamma }_2{\epsilon }_j\mathrm{\Delta }t+{\gamma }_0(1+\overline{\alpha })h^2)+\mathrm{\Delta }{t}^2)$. The step size in the space direction is taken to be $h$, and the step size in the time direction is $\mathrm{\Delta }t$. Stability of the proposed scheme was studied using the $L_2$ and $L_{\mathrm{\infty }}$ norms. The proposed scheme has been applied to different sets of problems and is found to be more efficient than existing schemes.