Improved double-integration technique to approximate integral-balance
solutions of non-linear fractional subdiffusion equations has been conceived. The
time-fraction subdiffusion equation with Dirichlet boundary condition and a powerlaw
fractional diffusivity has been chosen as a test example. Problems pertinent to
approximation of time-fractional Riemann-Liouville derivative when the distribution
is expressed as a parabolic profile with unspecified exponent and accuracy of the
solutions have been analyzed. The final solution is a closed-form can be presented
with either a similarity variable of a fractional order as independent variable or by
an effective similarity variable incorporating the effects of both the fractional order
and the nonlinearity of the diffusion coefficient. Optimization problem pertinent to
determination of the optimal exponent of the parabolic profile, dependent on both
the fractional order and the nonlinearity parameter of the diffusion coefficient, has
been developed by a modified technique transforming the time-varying domain of
integration into one with fixed boundaries. It was clearly defined that the
approximate profile can exhibit a concave behaviour, typical for subdiffusion
relaxation processes when the non-linearity of the diffusion coefficient is low and
the fractional order is high. Otherwise, the increase in the nonlinearity of the
diffusion coefficient results in convex profiles typical for the degenerate diffusion
behaviour.
Keywords: Subdiffusion, degenerate diffusion, integral method, approximate
solution.
