Financial theory could introduce a fractional differential equation (FDE) that presents new theoretical research concepts, methods and practical implementations. Due to the memory factor of fractional derivatives, physical pathways with storage and inherited properties can be best represented by FDEs. For that purpose, reliable and effective techniques are required for solving FDEs. Our objective is to generalize the collocation method for solving time fractional Black–Scholes European option pricing model using the extended cubic B-spline. The key feature of the strategy is that it turns these type of problems into a system of algebraic equations which can be appropriate for computer programming. This is not only streamlines the problems but speed up the computations as well. The Fourier stability and convergence analysis of the scheme are examined. A proposed numerical scheme having second-order accuracy via spatial direction is also constructed. The numerical and graphical results indicate that the suggested approach for the European option prices agree well with the analytical solutions.

金融理论可以引入一个分数微分方程（FDE），它提出了新的理论研究概念、方法和实际实施。由于分数导数的记忆因素，具有存储和遗传特性的物理路径可以最好地由 FDE 表示。为此，需要可靠且有效的技术来解决 FDE。我们的目标是推广使用扩展三次 B 样条求解时间分数 Black-Scholes 欧式期权定价模型的搭配方法。该策略的关键特征是将这些类型的问题转化为适合计算机编程的代数方程组。这不仅简化了问题，还加快了计算速度。检查了该方案的傅立叶稳定性和收敛性分析。还构建了一个通过空间方向具有二阶精度的拟议数值方案。数值和图形结果表明，建议的欧式期权价格方法与解析解非常吻合。