Amin Rohul, Shah Kamal, Al-Mdallal Qasem M, Khan Imran, Asif Muhammad
Department of Mathematics, University of Peshawar, Peshawar, 25120 Khyber Pakhtunkhwa Pakistan.
Department of Mathematics, University of Malakand, Dir(L), 18000 Khyber Pakhtunkhwa Pakistan.
Int J Appl Comput Math. 2021;7(2):34. doi: 10.1007/s40819-021-00975-x. Epub 2021 Feb 22.
In this paper, the Haar technique is applied to both nonlinear and linear eight-order boundary value problems. The eight-order derivative in the boundary value problem is approximated using Haar functions in this technique and the integration process is used to obtain the expression of the lower order derivative and the approximate solution of the unknown function. For the verification of validation and convergence of the proposed technique, three linear and two nonlinear examples are taken from the literature. The results are also compared with other methods available in the literature. Maximum absolute and root mean square errors at various collocation and Gauss points are contrasted with the exact solution. The convergence rate is also measured, which is almost equivalent to 2, using different numbers of collocation points.
在本文中，哈尔技术被应用于非线性和线性八阶边值问题。在该技术中，使用哈尔函数对边值问题中的八阶导数进行近似，并利用积分过程得到低阶导数的表达式以及未知函数的近似解。为了验证所提技术的有效性和收敛性，从文献中选取了三个线性和两个非线性的例子。结果还与文献中可用的其他方法进行了比较。将不同配置点和高斯点处的最大绝对误差和均方根误差与精确解进行了对比。还测量了收敛速率，使用不同数量的配置点时，收敛速率几乎等于2。
