This paper presents a numerical solution of the temporal-fractional Black–Scholes equation governing European options (TFBSE-EO) in the finite domain so that the temporal derivative is the Caputo fractional derivative. For this goal, we firstly use linear interpolation with the $$(2-\alpha)$$
(
2
-
α
)
-order in time. Then, the Chebyshev collocation method based on the second kind is used for approximating the spatial derivative terms. Applying the energy method, we prove unconditional stability and convergence order. The precision and efficiency of the presented scheme are illustrated in two examples.
This paper develops a numerical method for approximating the space fractional diffusion equation in Caputo derivative sense. In this discretization process, firstly, the compact finite difference with convergence order O(δτ 2) is used to obtain the semi-discrete in time derivative. Afterward, the spatial fractional derivative is discretized by using the Chebyshev collocation method of the third-kind. This collocation scheme is based on the operational matrix. In addition, time-discrete stability and convergence are theoretically proved in detail. We solve two examples by the proposed method and the obtained results are compared with other numerical methods. The numerical results show that our method is much more accurate than existing methods.
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