In this study, for the first time, the approximate solution of Black–Scholes option pricing distributed order time-fractional partial differential equation by means of Legendre and Chebyshev wavelets is considered. The operational matrices of Legendre and Chebyshev wavelets for integer order derivative and distributed order fractional derivative are derived. Furthermore, the combination of Gauss–Legendre quadrature formula and standard Tau method along with the obtained operational matrices reduces the distributed order time-fractional Black–Scholes model (DOTFBSM) into the system of linear algebraic equations. Convergence analysis, error bounds and numerical stability of the proposed approach are discussed in detail. The presented scheme is applied on three test examples and numerical experiments confirm the theoretical results and illustrate robustness of the presented method. The results produced by current approach are found to be more accurate than some available results.

本研究首次考虑了利用Legendre和Chebyshev小波对Black-Scholes期权定价分布阶次时间分数阶偏微分方程的近似解。导出了整数阶导数和分布阶分数阶导数的勒让德小波和切比雪夫小波的运算矩阵。此外，Gauss-Legendre 求积公式和标准 Tau 方法以及获得的运算矩阵的组合将分布式阶次时间分数 Black-Scholes 模型 (DOTFBSM) 简化为线性代数方程组。详细讨论了所提出方法的收敛性分析、误差界限和数值稳定性。所提出的方案应用于三个测试实例，数值实验证实了理论结果并说明了所提出方法的鲁棒性。发现当前方法产生的结果比某些可用结果更准确。