An Efficient Approximation to Numerical Solutions for the Kawahara Equation Via Modified Cubic B-Spline Differential Quadrature Method

Math Methods in App Sciences

Yağmurlu

Uçar

et al. 2021

Self Cite

Finding the approximate solutions to natural systems in the branch of mathematical modelling has become increasingly important and for this end various methods have been proposed. The purpose of the present paper is to obtain and analyze the numerical solutions of Modified Equal Width equation (MEW). This equation is one of those equations used to model nonlinear phenomena which has a significant role in several branches of science such as plasma physics, fluid mechanics, optics and kinetics. Firstly, for the discretization of spatial derivatives, a fifth‐order quintic B‐spline based scheme is directly implemented. Secondly, a forward finite difference formula is applied for the temporal discretization of derivatives with respect to time. Simulation results establish the validity and applicability of the suggested technique for a wide range of nonlinear equations. Then, the newly obtained theoretical consequences are numerically justified by the simulations and test problems. These illustrative test problems are presented verifying the superiority of the newly presented scheme compared to other existing schemes and techniques. The suggested method with symbolic computational software such as, Matlab, is proven as an effective way to obtain the soliton solutions of several nonlinear partial differential equations (PDEs). Finally, the newly obtained results are presented graphically to justify the approximate findings.

Section: Introductionmentioning

confidence: 99%

A new perspective for the numerical solution of the Modified Equal Width wave equation

Math Methods in App Sciences

Yağmurlu

Uçar

et al. 2021

Self Cite

“…Bellman et al [22] firstly supposed the DQM for the numerical solutions of the differential equations. Then, many problems are solved commonly used in the scientific area [22][23][24][25][26][27][28][29].…”

Section: Introductionmentioning

confidence: 99%

Finite difference method combined with differential quadrature method for numerical computation of the modified equal width wave equation

Numerical Methods Partial

Yağmurlu

Uçar

et al. 2020

Self Cite

The aim of this study is to improve the numerical solution of the modified equal width wave equation. For this purpose, finite difference method combined with differential quadrature method with Rubin and Graves linearizing technique has been used. Modified cubic B-spline base functions are used as base function. By the combination of two numerical methods and effective linearizing technique high accurate numerical algorithm is obtained. Three main test problems are solved for various values of the coefficients. To observe the performance of the present method, the error norms of the single soliton problem which has analytical solution are calculated. Besides these error norms, three invariants are reported. Comparison of the results displays that our algorithm produces superior results than those given in the literature.

“…Differential quadrature method (DQM) was first introduced by Bellman et al (1972) to obtain numerical solution of partial differential equations. Many researchers have developed different types of DQMs utilizing various base functions such as Legendre polynomials and spline functions (Bellman et al 1972), Hermite polynomials (Cheng et al 2005), radial basis functions (Shu and Wu 2007), harmonic functions (Striz et al 1995), Sinc functions (Korkmaz and Dag 2011), B-spline functions (Başhan et al 2018a;Karakoç et al 2014), and modified B-spline functions (Mittal and Jain 2012;Başhan et al 2017Başhan et al , 2018bBaşhan 2019).…”

Section: Introductionmentioning

confidence: 99%

An effective approximation to the dispersive soliton solutions of the coupled KdV equation via combination of two efficient methods

2020

Comp. Appl. Math.

Self Cite

The numerical solutions of the coupled Korteweg-de Vries equation is investigated via combination of two efficient methods that includes Crank-Nicolson scheme for time integration and quintic B-spline based differential quadrature method for space integration. The differential quadrature method has the advantage of using less number of grid points. And the advantage of the Crank-Nicolson scheme is the prevention of long and tedious algebraic computations for time integration. Those advantages come together and produce better results. To display the accuracy and efficiency of the present hybrid method three well-known test problems, namely single soliton, interaction of two soliton and birth of solitons are solved and the error norms L 2 and L ∞ are computed and compared with earlier works. Present hybrid method obtained superior results than earlier works by using the same parameters and less number of grid points. This situation is shown by comparison of the earlier works. At the same time, two lowest invariants and numerical and analytical values of the amplitudes of the solitons during the simulations are computed and tabulated. Besides those, relative changes of invariants are computed. Properties of solitons observed clearly at the all of the test problems and figures of the all of the simulations are given.