ABSTRACT

In this paper, an exact method named Riccati–Bernoulli sub-ODE method and a numerical method named Subdomain finite element method are proposed for solving the nonlinear generalized regularized long wave (GRLW) equation. For this purpose, Bäcklund transformation of the Riccati–Bernoulli equation and sextic B-spline functions are used for the exact and numerical solutions, respectively. The single soliton wave motion is used to confirm the methods which are found to be correct and effective. The three invariants ($I_1$, $I_2$ and $I_3$) of motion have been assessed to indicate the conservation properties of the numerical algorithm. For the motion of single solitary $L_2$ and $L_{\mathrm{\infty }}$ error norms are handled to evaluate differences between the analytical and numerical solutions. Unconditional stability is demonstrated using von-Neumann method. The procured outcomes show that our new schemes guarantee an evident and a functional mathematical equipment for solving nonlinear evolution equations.

摘要

本文提出了一种精确的方法称为Riccati–Bernoulli sub-ODE方法，并将一种数值方法称为Subdomain有限元方法来求解非线性广义正则长波（GRLW）方程。为此，将Riccati–Bernoulli方程的Bäcklund变换和正弦B样条函数分别用于精确解和数值解。使用单个孤子波运动来确认所发现的方法是正确和有效的。三个不变量（${\mathrm{一世}}_{\mathrm{1个}}$， ${\mathrm{一世}}_2$ 和 ${\mathrm{一世}}_3$运动已经被评估以指示数值算法的守恒性质。对于单次运动${\mathrm{大号}}_2$ 和 ${\mathrm{大号}}_{\mathrm{\infty }}$处理误差准则以评估解析解和数值解之间的差异。使用von-Neumann方法证明了无条件稳定性。获得的结果表明，我们的新方案为求解非线性发展方程提供了可靠的功能性数学设备。