Derivation of a fractional Boussinesq equation for modelling unconfined groundwater

In this manuscript, a fractional Boussinesq equation is obtained by assuming power-law changes of flux in a control volume and using a fractional Taylor series. Furthermore, it was assumed that the average thickness of the watery layer of an aquifer is constant, and the linear fractional Boussinesq equation was derived. Unlike classical Boussinesq equation, due to the non-locality property of fractional derivatives, the parameters of the fractional Boussinesq equation are constant and scale-invariant. In addition, the fractional Boussinesq equation has two various fractional orders of differentiation with respect to x and y that indicate the degree of heterogeneity in the x and y directions, respectively.

[1]  Jacob Bear,et al.  Physical principles of water percolation and seepage , 1968 .

[2]  Inverse techniques for estimating transmissivity and drainable pore space utilizing data from subsurface drainage experiments , 1994 .

[3]  H. S. Chauhan,et al.  An analytical solution for design of bi-level drainage systems , 1998 .

[4]  I. Podlubny.  Fractional differential equations , 1998 .

[5]  Ji-Huan He.  Approximate analytical solution for seepage flow with fractional derivatives in porous media , 1998 .

[6]  Christine Doughty.  Investigation of conceptual and numerical approaches for evaluating moisture, gas, chemical, and heat transport in fractured unsaturated rock , 1999 .

[7]  D. Benson,et al.  Simulating Scale-Dependent Solute Transport in Soils with the Fractional Advective–Dispersive Equation , 2000 .

[8]  D. Benson,et al.  The fractional‐order governing equation of Lévy Motion , 2000 .

[9]  D. Benson,et al.  Application of a fractional advection‐dispersion equation , 2000 .

[10]  Rina Schumer,et al.  Fractional Dispersion, Lévy Motion, and the MADE Tracer Tests , 2001 .

[11]  D. Benson,et al.  Eulerian derivation of the fractional advection-dispersion equation. , 2001, Journal of contaminant hydrology.

[12]  A. M. Garcia,et al.  Drainage equations for random and irregular tile drainage systems , 2001 .

[13]  Y. Pachepsky,et al.  Generalized Richards' equation to simulate water transport in unsaturated soils , 2003 .

[14]  H. M. Selim,et al.  Application of the Fractional Advection‐Dispersion Equation in Porous Media , 2003 .

[15]  M. Meerschaert,et al.  Fractal travel time estimates for dispersive contaminants , 2005, Ground water.

[16]  H. Srivastava,et al.  Theory and Applications of Fractional Differential Equations , 2006 .

[17]  A. Manglik,et al.  Water table fluctuation owing to time-varying recharge, pumping and leakage , 2006 .

[18]  J. Stillman,et al.  A semi-analytical model for transient flow to a subsurface tile drain , 2006 .

[19]  Zaid M. Odibat,et al.  Generalized Taylor's formula , 2007, Appl. Math. Comput..

[20]  Guanhua Huang,et al.  A finite element solution for the fractional advection–dispersion equation , 2008 .

[21]  M. Meerschaert,et al.  Fractional conservation of mass , 2008 .

[22]  Y. Pachepsky,et al.  Modelling solute transport in soil columns using advective-dispersive equations with fractional spatial derivatives , 2010, Adv. Eng. Softw..

[23]  José António Tenreiro Machado,et al.  Control of a heat diffusion system through a fractional order nonlinear algorithm , 2010, Comput. Math. Appl..

[24]  Shaher Momani,et al.  Solutions of a fractional oscillator by using differential transform method , 2010, Comput. Math. Appl..

[25]  Vasily E. Tarasov,et al.  Fractional Vector Calculus , 2010 .

[26]  Aiguo Xiao,et al.  Weighted finite difference methods for a class of space fractional partial differential equations with variable coefficients , 2010, J. Comput. Appl. Math..

[27]  F. Mainardi,et al.  Recent history of fractional calculus , 2011 .

[28]  Roberto Garrappa,et al.  Generalized exponential time differencing methods for fractional order problems , 2011, Comput. Math. Appl..

[29]  JinRong Wang,et al.  Nonlocal problems for fractional integrodifferential equations via fractional operators and optimal controls , 2011, Comput. Math. Appl..

[30]  V. E. Tarasov.  Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media , 2011 .

[31]  K. Diethelm,et al.  Fractional Calculus: Models and Numerical Methods , 2012 .