Research On Adaptive Time Step Method For Time Fractional Partial Differential Equations

Fractional differential equation is a field of mathematical analysis,is developed from integer differential equation,is the extension of integer differential equation theory.In the past decades,the application of fractional differential equations in many fields has attracted wide attention.Many mathematical models with fractional derivatives have been successfully applied in the fields of control theory,biology and finance.For Caputo time fractional partial differential equation,fractional derivative is a definition of convolution form and has non-locality.Due to the convolution form,the calculation has the problem of large computation and large storage capacity,so it is necessary to require the numerical solution of fractional partial differential equations.As an important partial differential equation of time fractional order,it plays an important role in numerical solution.This paper takes this problem as a model and carries out the following research work:For the fractional partial differential equation of time,we first convert it into the form of Volterra integral equation,and use the fractal trapezoid formula to approximate the kernel function of Volterra integral equation to get the semi-discrete format,and then use the finite element method to discretization in space to get the fully discrete format of the fractional partial differential equation of time.The stability of the fully discrete scheme is proved.Then,the spatial and temporal posterior error estimators are derived,and the adaptive time step algorithm is constructed based on the posterior error estimators.Finally,the accuracy of the error and posterior error estimators and the effectiveness of the adaptive time step algorithm are verified by numerical examples.For Caputo time fractional partial differential equation,fractional derivative is a definition of convolution form with non-locality.Since the convolution form causes the problem of excessive computation and storage of historical terms,we adopt a fast algorithm,that is,its numerical approximation can be regarded as the sum of historical terms and current terms.The fractional derivative operator is approximated by L1 formula and SOE method.In space,we use finite element method to discretization and get the full discretization scheme of Caputo time fractional partial differential equation.Then,a posteriori error estimator on time and space is derived,and an adaptive time step algorithm coupled with a fast algorithm is constructed.Finally,two numerical examples are used to verify the effectiveness of the adaptive time step algorithm,and the error is relatively small under the condition of less computation.