In recent years, fractional calculus has been widely used in abnormal diffusion, signal processing and control, fluid mechanics, image processing, soft matter research, earthquake analysis, viscoelastic damper, power fractal network, fractional sinusoidal oscillator, fractal theory and fractional PID controller design. However, due to the historical dependence and global correlation of fractional calculus, the numerical calculation complexity of fractional derivative equation is increased.

The main problem in numerical algorithm is that (1) long-term problem has not found a satisfactory solution. In numerical simulation, the amount of calculation increases exponentially with the increase of time course. At the same time, the short-term memory method proposed by some scholars is only effective in a few cases and is not universal. Therefore, solving long-term problems has a long way to go. ⑵ Based on the original algorithm, the algorithm and software of spatio-temporal mixed fractional derivative equation are developed. If a mathematical tool is to be widely used in engineering, it must have mature algorithms and software, such as finite element calculation and simulation software, so that finite element can be widely used in engineering. ⑶ The definition of fractional derivative is not perfect. There are many definitions of fractional derivative now, but none of them can be accepted by most scholars.

At present, the numerical algorithms of fractional derivative equation mainly include: (1) finite difference method: explicit scheme, implicit scheme, Crank-Nicholson scheme, predictor-corrector algorithm, linear algorithm and so on. ⑵ Series approximation methods: variational iteration method, Adomian decomposition method, homotopy perturbation method, general theoretical analysis method, differential transformation method, etc. (3) finite element method; (4) Meshless method; 5. Some new algorithms: matrix transformation, extrapolation, etc.

These numerical algorithms have their own advantages and disadvantages, and different conditions and equations are suitable for different algorithms, which requires us to be familiar with various methods and apply them flexibly. Otherwise, it is easy to get wrong calculation results. Another difficulty is which definition of fractional derivative to use in numerical calculation, which involves the choice of definition. According to a large number of references, the calculation of time fractional derivative is generally defined by Caputo, and the numerical calculation of spatial fractional derivative equation is often defined by Riemann-Liouville and series.