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In this paper, we study a loaded modified diffusion equation (the Hallaire equation with the fractional derivative with respect to time). The compact finite difference schemes of Crank-Nicolson type of higher order is developed for approximating the stated problem on uniform grids with the orders of accuracyO   (   h  4   +   τ   2  −  α    )    $\mathcal{O}(h^4+\tau^{2-\alpha})$    andO   (   h  4   +   τ   2    )    $\mathcal{O}(h^4+\tau^{2})$   . A priori estimates are obtained for solutions of differential and difference equations. Stability of the suggested schemes and also their convergence with the rate equal to the order of the approximation error are proved. Proposed theoretical calculations are illustrated by numerical experiments on test problems.

The Crank-Nicolson type compact difference schemes for a loaded time-fractional Hallaire equation