Numerical and dynamical behaviors of nonlinear traveling wave solutions of the Kudryashov–Sinelshchikov equation

In this article, a mathematical model symbolizing numerical solutions of the nonlinear Kudryashov–Sinelshchikov (KS) equation has been investigated. For this aim, we use collocation finite element method with septic B-spline functions and appropriate initial and boundary conditions on uniform mesh points. The proficiency and effectiveness of the presented method are examined in four test examples, consisting; behavior of a single soliton, collision of two solitons, Maxwellian initial condition and growth of a wavy hole. L2 and L∞ error norms are used to validate practicality and reliability of our numerical algorithm. In addition, two invariants are calculated to determine the conservation properties of the presented algorithm and the numerical results have been shown to be unconditionally stable. Obtained numerical results have been illustrated with tables and graphics for easy visualization of properties of the problem modeled. Numerical calculations to solve such problems prove the accuracy and effectiveness of the finite element method. In collocation method, using B-spline functions gives rise to a technique that needs only the unknown parameters at certain node points to create the solution. Collocation method has two excellent advantages: establishing method does not include integrations and the resulting matrix system is banded with small band width. Consequently, B-splines associate with the collocation provide a simple solution procedure of linear and nonlinear partial differential equations (PDEs). After that, we have analyzed error estimation for the Galerkin finite element method. Furthermore, dynamical properties of traveling waves of the KS equation are investigated in absence and presence of external periodic perturbation. In absence of perturbation, periodic traveling wave solution of the KS equation are studied. On the other hand, a collection of various quasiperiodic wave features of the KS equation is presented in presence of periodic perturbation. Therefore, it can be said that the numerical technique is effective and valid in obtaining numerical solutions of such PDEs. In addition, the study on dynamical properties of the KS equation provides qualitative behavior of the corresponding wave features.