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

B-spline differential quadrature method for the modified burgers’ equation

In this study, the Quintic B-spline Differential Quadrature method (QBDQM) is applied to find the numerical solution of the modified Burgers’ equation (MBE). The efficiency and accuracy of the method are measured by calculating the maximum error norm L∞ and the discrete root mean square error L2. The obtained numerical results are compared with published numerical results and the comparison shows that the method is an effective numerical scheme to solve the MBE. A rate of convergence analysis is also give

Two different methods for numerical solution of the modified burgers’ equation

A numerical solution of the modified Burgers’ equation (MBE) is obtained by using quartic B-spline subdomain finite element method (SFEM) over which the nonlinear term is locally linearized and using quartic B-spline differential quadrature (QBDQM) method. The accuracy and efficiency of the methods are discussed by computing \u1d43f����2 and \u1d43f����∞ error norms. Comparisons are made with those of some earlier papers. The obtained numerical results show that the methods are effective numerical schemes to solve the MBE. A linear stability analysis, based on the von Neumann scheme, shows the SFEM is unconditionally stable. A rate of convergence analysis is also given for the DQM

A numerical solution of the modified regularized long wave (MRLW) equation using quartic B-splines

In this paper, a numerical solution of the modified regularized long wave (MRLW) equation is obtained by subdomain finite element method using quartic B-spline functions. Solitary wave motion, interaction of two and three solitary waves and the development of the Maxwellian initial condition into solitary waves are studied using the proposed method. Accuracy and efficiency of the proposed method are tested by calculating the numerical conserved laws and error norms L₂ and L∞. The obtained results show that the method is an effective numerical scheme to solve the MRLW equation. In addition, a linear stability analysis of the scheme is found to be unconditionally stable.Publisher's Versio

A mixed method approach to Schrödinger equation: Finite difference method and quartic B-spline based differential quadrature method

The present manuscript includes finite difference method and quartic B-spline based differential quadrature method (FDM-DQM) for getting the numerical solutions for the nonlinear Schrödinger (NLS) equation. To solve complex NLS equation firstly we have separated NLS equation into the two real value partial differential equations. After that they are discretized in time using special type of classical finite difference method namely, Crank-Nicolson scheme. Then, for space integration differential quadrature method has been implemented. So, partial differential equation turn into simple a system of algebraic equations. To display the accuracy of the present hybrid method, the error norms L 2 and L ? and two lowest invariants I 1 and I 2 and relative changes of invariants have been calculated. As a last step, the numerical result already obtained have been compared with earlier studies by using same parameters. The comparison has clearly indicated that the presently used method, namely FDM-DQM, is an appropriate and accurate numerical scheme and allowed us to present for solving a wide class of partial differential equations

An effective application of differential quadrature method based on modi edcubic B-splines to numerical solutions of the KdV equation

In this study, numerical solutions of the third-order nonlinear Korteweg{de Vries (KdV) equation are obtainedvia differential quadrature method based on modi ed cubic B-splines. Five different problems are solved. To show theaccuracy of the proposed method,L2andL1error norms of the problem, which has an analytical solution, and threelowest invariants are calculated and reported. The obtained solutions are compared with some earlier works. Stabilityanalysis of the present method is also given.In this study, numerical solutions of the third-order nonlinear Korteweg{de Vries (KdV) equation are obtainedvia differential quadrature method based on modi ed cubic B-splines. Five different problems are solved. To show theaccuracy of the proposed method,L2andL1error norms of the problem, which has an analytical solution, and threelowest invariants are calculated and reported. The obtained solutions are compared with some earlier works. Stabilityanalysis of the present method is also given

Modifiye edilmiş burgers’ denkleminin kuintik b-spline diferensiyel quadrature metot ile nümerik çözümleri

Zamana bağlı kısmi türevli diferensiyel denklemlerde önce konum türevi içeren terim DQM ‘un kullanımıyla nümerik olarak ayrıştırılır. Böylece; kısmi türevli diferensiyel denklem, adi diferensiyel denkleme dönüştürülür. Sonraki aşamada ise elde edilen adi diferensiyel denklem; kararlılığı, doğruluğunun yüksek olması ve programlama maliyetinin düşük olması sebebiyle dördüncü mertebe Runge-Kutta metodu yardımıyla nümerik integrasyonu yapılıp çözüm elde edilir. Bu çalışmada, \u1d448������\u1d461������ + \u1d700������\u1d448������ 2\u1d448������\u1d465������ − \u1d708������\u1d448������\u1d465������\u1d465������ = 0 denklemi ile ifade edilen modifiye edilmiş Burgers’ denkleminin nümerik çözümleri kuintik B-spline baz fonksiyonlar kullanılarak DQM ile elde edildi. Metodun etkinliği ve doğruluğu \u1d43f������2 ve \u1d43f������∞ hata normlarının hesaplanması ile ölçüldü. Mevcut metot ile elde edilen nümerik sonuçlar literatürde bulunan bazı nümerik sonuçlar ile karşılaştırıldı ve yapılan karşılaştırmalardan metodun modifiye edilmiş Burgers’ denkleminin nümerik çözümleri için etkili bir yöntem olduğu görüldü. Ayrıca, yakınsama oran analizi ve kararlılık analizi incelendi