7 research outputs found
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
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
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
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
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