Construction And Theoretical Analysis For Efficient Numerical Schemes Of The Time Fractional Partial Differential Equation

Because the fractional calculus was not widely accepted in the field of practical engineering in the early days,it was only in the direction of mathematical theory when it was first proposed by mathematicians.Until many physicists found that the fractional calculus models could better describe the memory and hereditary characteristics of matter,more and more scholars was interest to study the fractional calculus.In the first part,a new higher-order numerical scheme is constructed based on the idea of Block-By-Block method.It is used the quadratic Lagrangian interpolation function to approximate to the unknown function in each sub-interval and direct discrete fractional derivative.The error analysis of the scheme is carried out,and the effectiveness of the theoretical analysis of the high-order numerical scheme is verified by a specific numerical examples.In the second part,it is the extension of the first research work in time fractional partial differential equation.Firstly,it is used the first research work's scheme to time fractional derivative.Secondly,the center differential scheme is used for second derivative of space.Thirdly,the full discrete scheme of time fractional partial differential equation is obtained by coupling numerical scheme of time fractional derivative and space second derivative.Finally,the error analysis is carried out for the full discrete scheme,and the validity of the theory is verified by concrete numerical studies.