# Time and Space Fractional Diffusion in Finite Systems

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## Abstract

Spatiotemporal nonlocal diffusion in a bounded system is addressed by considering fractional diffusion in a linear, composite system. By considering limiting conditions, solutions for combinations of Neumann and Dirichlet boundary conditions (either zero or nonzero) at the ends of a finite system are derived in terms of Mittag–Leffler functions by the Laplace transformation. Computational viability is demonstrated by inverting the solutions numerically and comparing resulting calculations with asymptotic solutions. Time and space fractional derivatives, defined by variables \(\alpha \) and \(\beta \), respectively, are employed in the Caputo sense; a single-sided, asymmetric space derivative is used. Inspection of the asymptotic solutions leads to insights on the structure of the solutions that may not be available otherwise; the resulting deductions are verified through the numerical inversions. For pure superdiffusion, characteristics of some of the solutions presented here are very similar to those of classical diffusion but combined effects for the corresponding situation result in power-law behaviors. Incidentally, to our knowledge, the pressure distribution for space fractional diffusion at long enough times in a finite system is derived based on first principles for the first time.

## Keywords

Fine-scale complexity Pressure behavior Fractured rocks Anomalous diffusion Fractional diffusion Finite systems## List of symbols

*c*Compressibility (

*L**T*\(^2\)/*M*)- \(E_{\alpha , \beta }( z)\)
- \(f(\beta )\)
See Eq. 45

*h*Thickness (

*L*)*k*Permeability (

*L*\(^2\))- \({k_{\alpha , \beta }}\)
See Eq. 2

*L*Width of linear system (

*L*)*p*Pressure (

*M*/*L*/*T*\(^2\))*q*Flux (

*L*\(^3\)/*T*)*T*Absolute temperature

*t*Time (

*T*)*u*See Eq. 33

- \(x_L\)
Distance to interface (

*L*)- \(\alpha \)
Exponent, time

- \(\beta \)
Exponent, space

- \(\varGamma (x)\)
Gamma function

- \(\eta \)
Diffusivity; various

- \({\tilde{\eta }} \)
‘Diffusivity’; see Eq. 18 (

*L*\(^{\beta +1}\)/*T*\(^\alpha \))- \(\kappa \)
Second term of Eq. 41

- \(\lambda \)
Mobility (

*L**T*/*M*)- \(\mu \)
Viscosity (M/

*L*/*T*)- \(\phi \)
Porosity (

*L*\(^3\)/*L*\(^3\))

## Subscripts

- c
Constant rate

- D
Dimensionless

*i*Coordinate, initial

- p
Constant pressure

- 1, 2
Index

## Superscript

- −
Laplace transform

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