In this article, we propose a fully-discrete scheme for the numerical solution of a nonlinear time-fractional biharmonic problem. This problem is first converted into an equivalent system by introducing a new variable. Then spatial and temporal discretizations are done by the weighted $b$-spline method and $L2$-$1_\sigma$ approximation, respectively. The weighted $b$-spline method uses weighted $b$-splines on a tensor product grid as basis functions for the finite element space and by construction, it is a mesh-free method. This method combines the computational benefits of $b$-splines and standard mesh-based elements. We derive $\alpha$-robust \emph{a priori} bound and convergence estimate in the $L^2(\Omega)$ norm for the proposed scheme. Finally, we carry out few numerical experiments to support our theoretical findings.

翻译：本文针对非线性时间分数阶双调和问题的数值求解，提出一种全离散格式。首先引入新变量将原问题转化为等价方程组，随后分别采用加权B样条方法与L2-1_σ逼近进行空间和时间离散。加权B样条方法以张量积网格上的加权B样条作为有限元空间的基函数，其构造本质属于无网格方法，兼具B样条的计算优势与标准网格单元的灵活性。我们推导了该格式在L^2(Ω)范数下的α-稳健先验界与收敛性估计。最后通过数值实验验证了理论分析结果。