Abstract In this article, a fast algorithm is developed for the numerical solution of twelfth-order boundary value problems (BVPs). The Haar technique is applied to both linear and nonlinear BVPs. In Haar technique, the twelfth-order derivative in BVP is approximated using Haar functions, and the process of integration is used to obtain the expression of lower-order derivatives and approximate solution for the unknown function. Three linear and two nonlinear examples are taken from literature for checking the convergence of the proposed technique. A comparison of the results obtained by the present technique with results obtained by other techniques reveals that the present method is more effective and efficient. The maximum absolute and root mean square errors are compared with the exact solution at different collocation and Gauss points. The convergence rate using different numbers of collocation points is also calculated, which is approximately equal to 2.

中文翻译：

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一种用于处理十二阶边值问题的强大数值技术

摘要 在本文中，开发了一种用于 12 阶边值问题 (BVP) 数值求解的快速算法。Haar 技术适用于线性和非线性 BVP。在Haar技术中，BVP中的十二阶导数是用Haar函数逼近的，通过积分的过程得到未知函数的低阶导数的表达式和近似解。三个线性和两个非线性示例取自文献，用于检查所提出技术的收敛性。将本技术获得的结果与其他技术获得的结果进行比较表明，本方法更有效和高效。将最大绝对误差和均方根误差与不同搭配和高斯点的精确解进行比较。