Ghafoor Abdul, Haq Sirajul, Hussain Manzoor, Abdeljawad Thabet, Alqudah Manar A
Institute of Numerical Sciences, Kohat University of Science and Technology, Kohat 26000, KP, Pakistan.
Faculty of Engineering Sciences, GIK Institute, Topi 23640, KP, Pakistan.
Entropy (Basel). 2022 Apr 18;24(4):567. doi: 10.3390/e24040567.
In this work, an efficient and robust numerical scheme is proposed to solve the variable coefficients' fourth-order partial differential equations (FOPDEs) that arise in Euler-Bernoulli beam models. When partial differential equations (PDEs) are of higher order and invoke variable coefficients, then the numerical solution is quite a tedious and challenging problem, which is our main concern in this paper. The current scheme is hybrid in nature in which the second-order finite difference is used for temporal discretization, while spatial derivatives and solutions are approximated via the Haar wavelet. Next, the integration and Haar matrices are used to convert partial differential equations (PDEs) to the system of linear equations, which can be handled easily. Besides this, we derive the theoretical result for stability via the Lax-Richtmyer criterion and verify it computationally. Moreover, we address the computational convergence rate, which is near order two. Several test problems are given to measure the accuracy of the suggested scheme. Computations validate that the present scheme works well for such problems. The calculated results are also compared with the earlier work and the exact solutions. The comparison shows that the outcomes are in good agreement with both the exact solutions and the available results in the literature.
在这项工作中，我们提出了一种高效且稳健的数值格式，用于求解欧拉 - 伯努利梁模型中出现的变系数四阶偏微分方程（FOPDEs）。当偏微分方程（PDEs）为高阶且包含变系数时，数值解是一个相当繁琐且具有挑战性的问题，这也是本文主要关注的内容。当前的格式本质上是混合的，其中二阶有限差分用于时间离散化，而空间导数和求解通过哈尔小波进行近似。接下来，利用积分矩阵和哈尔矩阵将偏微分方程（PDEs）转化为线性方程组，该方程组易于处理。除此之外，我们通过拉克斯 - 里希特迈尔准则推导了稳定性的理论结果，并通过计算进行了验证。此外，我们探讨了计算收敛速度，其接近二阶。给出了几个测试问题来衡量所提格式的精度。计算结果验证了本格式对于此类问题效果良好。还将计算结果与早期工作及精确解进行了比较。比较结果表明，所得结果与精确解以及文献中的现有结果都非常吻合。
