An innovative scheme of collocation having quintic Hermite splines as base functions has been followed to solve Burgers’ equation. The scheme relies on approximation of Burgers’ equation directly in non-linear form without using Hopf–Cole transformation (Hopf in Commun Pure Appl Math 3:201–216, 1950; Cole in Q Appl Math 9:225–236, 1951). The significance of the numerical technique is demonstrated by comparing the numerical results to the exact solution and published results (Asaithambi in Appl Math Comput 216:2700–2708, 2010; Mittal and Jain in Appl Math Comput 218:7839–7855, 2012). Five problems with different initial conditions have been examined to validate the efficiency and accuracy of the scheme. Euclidean and supremum norms have been reckoned to scrutinize the stability of the numerical scheme. Results have been demonstrated in plane and surface plots to indicate the effectiveness of the scheme.

中文翻译：

---

五阶Hermite样条曲线求解Burgers方程的探索

一种创新的搭配方案以五阶Hermite样条作为基本函数，用于解决Burgers方程。该方案直接依赖于非线性形式的Burgers方程的近似，而无需使用Hopf-Cole变换（Hopf在Commun Pure Appl Math 3：201-216，1950； Cole在Q Appl Math 9：225-236，1951）。通过将数值结果与精确解和已发布的结果进行比较，证明了数值技术的重要性（Asaithambi在Appl Math Comput 216：2700-2708，2010； Mittal和Jain在Appl Math Comput 218：7839-7855，2012）。研究了五个具有不同初始条件的问题，以验证该方案的效率和准确性。欧几里得准则和最高准则已被认为是对数值方案稳定性的仔细研究。