Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:   consider the variable order time fractional diffusion equation.  adopt the Coimbra variable order (VO) time fractional operator, which defines a consistent method for VO differentiation of physical variables. The Coimbra variable order fractional operator also can be vied as a Caputo-type definition. Although this definition is the most appropriate definition having fundamental characteristics that are desirable for physical modeling, numerical methods for fractional partial differential equations using this definition have not yet appeared in the literature. Here an approximate scheme is first proposed. The stability, convergence and solvability of this numerical scheme are discussed via the technique of Fourier analysis. Numerical examples are provided to show that the numerical method is computationally efficient. Crown Copyright © 2012 Published by Elsevier Inc. All rights reserved.
['S. Shen', 'F. Liu', 'Ji Chen', 'Ian W. Turner', 'Vo Anh']
I am writing a research paper on a idea Numerical techniques for the variable order time fractional diffusion equation
[{"from": "human", "value": "I am writing a research paper on a idea Numerical techniques for the variable order time fractional diffusion equation"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:   consider the variable order time fractional diffusion equation.  adopt the Coimbra variable order (VO) time fractional operator, which defines a consistent method for VO differentiation of physical variables. The Coimbra variable order fractional operator also can be vied as a Caputo-type definition. Although this definition is the most appropriate definition having fundamental characteristics that are desirable for physical modeling, numerical methods for fractional partial differential equations using this definition have not yet appeared in the literature. Here an approximate scheme is first proposed. The stability, convergence and solvability of this numerical scheme are discussed via the technique of Fourier analysis. Numerical examples are provided to show that the numerical method is computationally efficient. Crown Copyright \u00a9 2012 Published by Elsevier Inc. All rights reserved."}]