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Numerical solution of the time‐dependent Schrödinger equation with a position‐dependent effective mass is challenging to compute due to the presence of the non‐constant effective mass. To tackle the problem we present operator splitting‐based numerical methods. The wavefunction will be propagated either by the Krylov subspace method‐based exponential integration or by an asymptotic Green's function‐based time propagator. For the former, the wavefunction is given by a matrix exponential whose associated matrix–vector product can be approximated by the Krylov subspace method; and for the latter, the wavefunction is propagated by an integral with retarded Green's function that is approximated asymptotically. The methods have complexity O(NlogN)$$ O\left(N\log N\right) $$ per step with appropriate algebraic manipulations and fast Fourier transform, where N$$ N $$ is the number of spatial points. Numerical experiments are presented to demonstrate the accuracy, efficiency, and stability of the methods.

The optimal problem exists in the individual's self‐protection activity. Generally, people consider a two‐period model within the decision equation through some constraints. In this work, we extend the two‐period model in risk decision‐making to a ‐period model with . Prevention and saving are two main factors discussed in this model. Then we apply the coordinate descent algorithm to find the “optimal” solution of the model equation. To the best knowledge we know, it is the first time to give the approximation of the optimality from a theoretical view. Some substitution results based on the first derivative are also presented. The numerical examples are tested to verify the result.

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