Exponential-sum-approximation technique for variable-order time-fractional diffusion equations

In this paper, we study the variable-order (VO) time-fractional diffusion equations. For a VO function $$\alpha (t)\in (0,1)$$ , develop an exponential-sum-approximation (ESA) technique to approach Caputo fractional derivative. The ESA keeps both quadrature exponents and number of exponentials in summation unchanged at different time level. Approximating parameters are properly selected achieve efficient accuracy. Compared with general direct method, proposed method reduces storage requirement from $${\mathcal {O}}(n)$$ {O}}(\log ^2 n)$$ computational cost {O}}(n^2)$$ $$\mathcal {O}(n\log respectively, n being levels. When fast algorithm is exploited construct scheme for equations, complexity only {O}}(mn\log {O}}(m\log ^2n)$$ requirement, where m denotes spatial grid points. Theoretically, unconditional stability error analysis given. effectiveness verified by numerical examples.