This paper is concerned with numerical solution of time fractional stochastic advection-diffusion type equation where the first order derivative is substituted by a Caputo fractional derivative of order \(\alpha \) (\(0 <\alpha \le 1\)). This type of equations due to randomness can rarely be solved, exactly. In this paper, a new approach based on finite difference method and spline approximation is employed to solve time fractional stochastic advection-diffusion type equation, numerically. After implementation of proposed method, the under consideration equation is transformed to a system of second order differential equations with appropriate boundary conditions. Then, using a suitable numerical method such as the backward differentiation formula, the resulting system can be solved. In addition, the error analysis is shown in some mild conditions by ignoring the error terms \(O(\Delta t^2)\) in the system. In order to show the pertinent features of the suggested algorithm such as accuracy, efficiency and reliability, some test problems are included. Comparison achieved results via proposed scheme in the case of classical stochastic advection-diffusion equation (\(\alpha =1\)) with obtained results via wavelets Galerkin method and obtained results for other values of \(\alpha \) with the values of exact solution confirm the validity, efficiency and applicability of the proposed method.

本文涉及时间分数阶随机对流扩散型方程的数值解，其中一阶导数由阶\（\ alpha \）（\（0 <\ alpha \ le 1 \）的Caputo分数阶导数代替）。由于随机性，这类方程很难精确地求解。本文采用一种基于有限差分法和样条逼近的新方法，对时间分数形式的随机对流扩散方程进行了数值求解。在实施所提出的方法之后，考虑中的方程被变换为具有适当边界条件的二阶微分方程系统。然后，使用诸如后向微分公式之类的合适数值方法，可以求解所得的系统。此外，通过忽略误差项\（O（\ Delta t ^ 2）\）可以在某些中等条件下显示误差分析。在系统中。为了显示建议算法的相关特征，例如准确性，效率和可靠性，其中包括一些测试问题。在经典随机对流扩散方程（\（\ alpha = 1 \）的情况下，通过拟议方案获得的结果与通过小波Galerkin方法获得的结果进行比较，对于其他值（\ alpha \）的结果与通过精确的解决方案证实了所提方法的有效性，有效性和适用性。