Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:   introduce a new numerical method for solving time fractional partial differential equation. The time discretization is based on Diethelm’s method where the Hadamard finite-part integral is approximated by using the piecewise quadratic interpolation polynomials. The space discretization is based on the standard finite element method. The error estimates with the convergence order \(O(\tau ^{3-\alpha } + h^2), 0<\alpha <1\) are proved in detail by using the argument developed recently by Lv and Xu (SIAM J Sci Comput 38:A2699–A2724, 2016), where \(\tau \) and h denote the time and space step sizes, respectively. Numerical examples in both one- and two-dimensional cases are given.
['Zhiqiang Li', 'Zongqi Liang', 'Yubin Yan']
I am writing a research paper on a idea High-Order Numerical Methods for Solving Time Fractional Partial Differential Equations
[{"from": "human", "value": "I am writing a research paper on a idea High-Order Numerical Methods for Solving Time Fractional Partial Differential Equations"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:   introduce a new numerical method for solving time fractional partial differential equation. The time discretization is based on Diethelm\u2019s method where the Hadamard finite-part integral is approximated by using the piecewise quadratic interpolation polynomials. The space discretization is based on the standard finite element method. The error estimates with the convergence order \\(O(\\tau ^{3-\\alpha } + h^2), 0<\\alpha <1\\) are proved in detail by using the argument developed recently by Lv and Xu (SIAM J Sci Comput 38:A2699\u2013A2724, 2016), where \\(\\tau \\) and h denote the time and space step sizes, respectively. Numerical examples in both one- and two-dimensional cases are given."}]