A novel implementation of Petrov-Galerkin method to shallow water solitary wave pattern and superperiodic traveling wave and its multistability: generalized Korteweg-de Vries equation

This work deals with the constitute of numerical solutions of the generalized Korteweg-de Vries (GKdV) equation with Petrov-Galerkin finite element approach utilising a cubic B-spline function as the trial function and a quadratic function as the test function. Accurateness and effectiveness of the submitted methods are shown by employing propagation of single solitary wave. The L2, L∞error norms and I1, I2and I3invariants are used to validate the applicability and durability of our numerical algorithm. Implementing the Von-Neumann theory, it is manifested that the suggested method is marginally stable. Furthermore, supernonlinear traveling wave solution of the GKdV equation is presented using phase plots. It is seen that the GKdV equation supports superperiodic traveling wave solution only and it is significantly affected by velocity and nonlinear parameters. Also, considering a superficial periodic forcing multistability of traveling waves of perturbed GKdV equation is presented. It is found that the perturbed GKdV equation supports coexisting chaotic and various quasiperiodic features with same parametric values at different initial condition

A quartic subdomain finite element method for the modified kdv equation

In this article, we have obtained numerical solutions of the modified Korteweg-de Vries (MKdV) equation by a numerical technique attributed on subdomain finite element method using quartic B-splines. The proposed numerical algorithm is controlled by applying three test problems including single solitary wave, interaction of two and three solitary waves. To inspect the performance of the newly applied method, the error norms, L2 and L∞, as well as the four lowest invariants, I1, I2, I3 and I4 have been computed. Linear stability analysis of the algorithm is also examined

New exact solutionsand numerical approximations of the generalized kdv equation

This paper is devoted to create new exact and numerical solutions of the generalized Korteweg-de Vries (GKdV) equation with ansatz method and Galerkin finite element method based on cubic B-splines over finite elements. Propagation of single solitary wave is investigated to show the efficiency and applicability of the proposed methods. The performance of the numerical algorithm is proved by computing L2 and L∞ error norms. Also, three invariants I1, I2, and I3 have been calculated to determine the conservation properties of the presented algorithm. The obtained numerical solutions are compared with some earlier studies for similar parameters. This comparison clearly shows that the obtained results are better than some earlier results and they are found to be in good agreement with exact solutions. Additionally, a linear stability analysis based on Von Neumann’s theory is surveyed and indicated that our method is unconditionally stable

Numerical solutions of the modified KdV Equation with collocation method

In this article, numerical solutions of the modified Korteweg-de Vries (MKdV) equation have been obtained by a numerical technique attributed on collocation method using quintic B-spline finite elements. The suggested numerical scheme is controlled by applying three test problems involving single solitary wave, interaction of two and three solitary waves. To check the performance of the newly applied method, the error norms, L2 and L∞, as well as the three lowest invariants, I1, I2 and I3, have been calculated. The acquired numerical results are compared with some of those available in the literature. Linear stability analysis of the algorithm is also examined

A new approach for numerical solution of modified korteweg-de vries equation

In this paper, a lumped Galerkin method is applied with cubic B-spline interpolation functions to find the numerical solution of the modified Korteweg-de Vries (mKdV) equation. Test problems including motion of single solitary wave, interaction of two solitons, interaction of three solitons, and evolution of solitons are solved to verify the proposed method by calculating the error norms L2 and L1 and the conserved quantities mass, momentum and energy. Applying the von-Neumann stability analysis, the proposed method is shown to be unconditionally stable. Consequently, the obtained results are found to be harmony with the some recent result

Düzenlenmiş korteweg-de vrıes (mkdv) denkleminin sonlu elemanlar yöntemleri ile sayısal çözümleri

Bu tez çalışmasında, MKdV (Modifiye edilmiş Korteweg-deVries) denkleminin yaklaşık çözümleri dördüncü (kuartik) mertebeden B-spline fonksiyonlar kullanılarak Subdomain yöntemi ve beşinci (kuintik) mertebeden B-spline fonksiyonlar kullanılarak Kollokasyon yöntemi ile elde edilmiştir

Kollokasyon sonlu eleman yöntemi ile mkdv denkleminin sayısal çözümleri

Bu çalışmada, modifiye edilmiş Korteweg-de Vries (mKdV) denkleminin sayısal çözümleri septik B-spline kollokasyon sonlu eleman yöntemi kullanılarak elde edilmiştir. Önerilen sayısal algoritmanın doğruluğu, tek soliton dalga, iki ve üç soliton dalganın girişimi gibi üç test probleminin uygulanması ile kontrol edilmiştir. Zamana bağlı Crank Nicolson yaklaşımına dayanan sayısal algoritmamız şartsız olarak kararlıdır. Yeni uygulanan yöntemin performansını kontrol etmek için, \u1d43f��2 , \u1d43f��∞ hata normları ile \u1d43c��1, \u1d43c��2, \u1d43c��3 ve \u1d43c��4 değişmezlerinin değerleri hesaplanmıştır. Elde edilen sayısal sonuçlar literatürde bulunan diğer sonuçlarla karşılaştırılmıştır

Approximation of the KdVB equation by the quintic B-spline differential quadrature method

In this paper, the Korteweg-de Vries-Burgers’ (KdVB) equation is solved numerically by a new differential quadrature method based on quintic B-spline functions. The weighting coefficients are obtained by semi-explicit algorithm including an algebraic system with fiveband coefficient matrix. The L2 and L∞ error norms and lowest three invariants 1 2 I ,I and 3 I have computed to compare with some earlier studies. Stability analysis of the method is also given. The obtained numerical results show that the present method performs better than the most of the methods available in the literatur