Limiting Cases of Subdiffusion
We study the limiting cases with respect to the order of the time derivative of a time fractional diffusion equation (TFDE), equipped with the zero initial condition and the Dirichlet boundary condition, in a subdiffusive case, i.e. the order of the time derivative is 0<α<1.
The properties of the fractional time derivative allow us to view (TFDE) as a reasonable interpolation of the usual diffusion equation and a Helmholz equation.
We use the double layer approach to reduce (TFDE) to a corresponding Volterra integral equation of the second kind. Then, using the well-known results of functional analysis, we show that the unique continuous solution of (TFDE) converges to the unique continuous solution of the diffusion equation as α tends to the upper limit, whereas the lower limit is more involved. Therefore, in this sense (TFDE) is more likely to be a parabolic type equation rather than being of elliptic type.
KeywordsDiffusion Equation Fundamental Solution Heat Equation Boundary Integral Equation Fractional Differential Equation
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