Our objective in this article is to develop weighted extended B-spline (WEB-spline) approximation of the first eigenpair of $p\left(x\right)$-Laplacian problem. One of the reason for choosing WEB-spline based mesh-free method for solving $p\left(x\right)$-Laplacian problem is that, it has additional advantages compared to usual finite element method. For example, it does not require any mesh generation procedure and hence eliminates the difficult, time consuming preprocessing step in finite element method. Also, it combines the computational advantages of B-splines and geometric flexibility of standard mesh-based elements. To compute the first eigenpair of $p\left(x\right)$-Laplacian problem, we examine the problem numerically by applying an inverse power method with three different minimization techniques and we compare the results obtained. Finally, we study the asymptotic behaviour of the eigenvalue problem of $p\left(x\right)$-Laplacian and validate the results comparing with those given in the literature.

我们在本文中的目标是开发第一个特征对的加权扩展 B 样条（WEB 样条）近似 $磷\left(X\right)$- 拉普拉斯问题。选择基于WEB-spline的无网格法求解的原因之一$磷\left(X\right)$-Laplacian 问题在于，与通常的有限元方法相比，它具有额外的优势。例如，它不需要任何网格生成程序，因此消除了有限元方法中困难、耗时的预处理步骤。此外，它还结合了 B 样条的计算优势和基于网格的标准元素的几何灵活性。计算第一个特征对$磷\left(X\right)$-Laplacian 问题，我们通过应用具有三种不同最小化技术的逆幂方法对问题进行数值分析，并比较获得的结果。最后，我们研究了特征值问题的渐近行为$磷\left(X\right)$- Laplacian 并验证结果与文献中给出的结果相比。