Third order difference schemes (without using points outside of the domain) for one sided space tempered fractional partial differential equations

• For biological particles, the tempered power-law diffusions, instead of pure power-law diffusion, are the more general observed experimental phenomena.
• The schemes of the tempered models and their proof of the stability and convergence are much different from the ones of the corresponding non-tempered ones.
• Third-order quasi-compact schemes are derived with strict convergence and stability proof for the tempered model.
• The generation function of the matrix and Weyl's theorem play important role in the proof of convergence and stability.

Power-law probability density function (PDF) plays a key role in both subdiffusion and Lévy flights. However, sometimes because of the finiteness of the lifespan of the particles or the boundedness of the physical space, tempered power-law PDF seems to be a more physical choice and then the tempered fractional operators appear; in fact, the tempered fractional operators can also characterize the transitions among subdiffusion, normal diffusion, and Lévy flights. This paper focuses on the finite difference schemes for space tempered fractional diffusion equations, being much different from the ones for pure fractional derivatives. By using the generation function of the matrix and Weyl's theorem, the stability and convergence of the derived schemes are strictly proved. Some numerical simulations are performed to testify the effectiveness and numerical accuracy of the obtained schemes.