A fourth-order B-spline collocation method has been applied for numerical study of Burgers–Fisher equation, which illustrates many situations occurring in various fields of science and engineering including nonlinear optics, gas dynamics, chemical physics, heat conduction, and so on. The present method is successfully applied to solve the Burgers–Fisher equation taking into consideration various parametric values. The scheme is found to be convergent. Crank–Nicolson scheme has been employed for the discretization. Quasi-linearization technique has been employed to deal with the nonlinearity of equations. The stability of the method has been discussed using Fourier series analysis (von Neumann method), and it has been observed that the method is unconditionally stable. In order to demonstrate the effectiveness of the scheme, numerical experiments have been performed on various examples. The solutions obtained are compared with results available in the literature, which shows that the proposed scheme is satisfactorily accurate and suitable for solving such problems with minimal computational efforts.

中文翻译：

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非线性Burgers–Fisher方程的四阶B样条搭配方法

一种四阶B样条搭配方法已用于Burgers–Fisher方程的数值研究，该方法说明了科学和工程学各个领域中发生的许多情况，包括非线性光学，气体动力学，化学物理学，热传导等。在考虑各种参数值的情况下，本方法已成功应用于求解Burgers-Fisher方程。发现该方案是收敛的。Crank–Nicolson方案已用于离散化。拟线性化技术已被用来处理方程的非线性。使用傅里叶级数分析（冯·诺伊曼法）讨论了该方法的稳定性，并观察到该方法是无条件稳定的。为了证明该计划的有效性，已经在各种示例上进行了数值实验。将获得的解决方案与文献中的结果进行比较，这表明所提出的方案是令人满意的准确的，并且适合以最小的计算量来解决这些问题。