A Numerical Method in the Hilbert Space for Solving the Time-Fractional Reaction-Diffusion Equation with Time Delay

This paper aims to obtain an approximate solution through a simplified reproducing kernel method (SRKM) for the time-fractional delay reaction-diffusion equation. By a simple transformation, we first homogenize the considered reaction-diffusion equation with delay. Then, we recall some reproducing kernel Hilbert spaces and their properties and build up the reproducing kernel Hilbert space that we need throughout the solution scheme. This new reproducing kernel space satisfies the delay condition, the property that reduces the computational complexity. Next, the nth-term approximation Un of the exact solution U is obtained without the Gram–Schmidt orthogonalization process. The properties of completeness and orthogonal projection of the considered basis are stated and proved. Eventually, given to the various examples represented, the efficiency and accuracy of the method are scrutinized. It is shown that the proposed method works well for various values of fractional order derivatives and even for large mode N.