Boundary Conditions for Two-Sided (and Tempered) Fractional - PowerPoint PPT Presentation

. . . . . . . . . . . . . . Boundary Conditions for Two-Sided (and Tempered) Fractional Difgusion James F. Kelly , Harish Sankaranarayanan, and Mark M. Meerschaert Department of Statistics and Probability Michigan State University June 22, 2018 Kelly et al. (MSU) Two-Sided BCs June 22, 2018 . . . . . . . . . . . . . . . . . . . . . . . . . . 1 / 48

. 1 . . . . . . . . . . Boundary Conditions (BCs) for One-Sided Fractional Difgusion . 2 Two-Sided Fractional Difgusion: BCs and Numerical Methods 3 Two-Sided Fractional Difgusion: Analytical Steady-State Solutions 4 One-Sided Tempered Fractional Difgusion 5 Summary and Open Problems Kelly et al. (MSU) Two-Sided BCs June 22, 2018 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 / 48

. Two-sided fractional difgusion equations are important in many . . . . . . . . . . Motivation applications: transport in heterogeneous porous media (Benson et al., . 2000), turbulence modeling (Chen, 2006), (del-Castillo Negrete et al., 2004), (Gunzburger et al., 2018), and biomedical acoustics (Treeby and Cox, 2010). Most numerical methods assume Dirichlet boundary conditions (BCs): (Meerschaert and Tadjeran, 2006) , (Mao and Karniadakis, 2018), (Samiee et al., 2018). For anomalous difgusion, a homogeneous Dirichlet BC models an absorbing boundary. Kelly et al. (MSU) Two-Sided BCs June 22, 2018 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 / 48

. . . . . . . . . . . . . . Motivation However, many of these applications involve a conserved quantity in a bounded domain. boundary and the total mass does not change. Recently, efgort has been spent on developing mass-preserving, refmecting (Neumann) BCs for space fractional difgusion equations (Ma, 2017), (Baeumer et al., 2018a,b), (Deng et al., 2018). Kelly et al. (MSU) Two-Sided BCs June 22, 2018 . . . . . . . . . . . . . . . . . . . . . . . . . . 4 / 48 From a stochastic point of view, particles are refmected at the

. . . . . . . . . . . . . . One-Sided Fractional Difgusion Equation: Riemann-Liouville The positive (left) Riemann-Liouville derivative on the bounded 1 L To derive refmecting boundary conditions , write in conservation form Kelly et al. (MSU) Two-Sided BCs June 22, 2018 . . . . . . . . . . . . . . 5 / 48 . . . . . . . . . . . . Consider one-sided space-fractional (1 < α ≤ 2) difgusion equation on [ L , R ] : ∂ ∂ tu ( x , t ) = C D α L + u ( x , t ) interval [ L , R ] is: L + u ( x , t ) = ∂ n ∂ n ∫ x u ( y , t ) D α ∂ x n I n − α L + u ( x , t ) = ( x − y ) α − n + 1 dy Γ( n − α ) ∂ x n ∂ tu ( x , t ) + ∂ ∂ ∂ xF RL ( x , t ) = 0 with fmux F RL ( x , t ) = − C D α − 1 L + u ( x , t ) .

. . . . . . . . . . . . . . Refmecting Boundary Conditions: Riemann-Liouville time t . Integrate the mass conservation equation L L mass conservation, yielding a refmecting BC (Baeumer et al., 2018a): Kelly et al. (MSU) Two-Sided BCs June 22, 2018 . . . . . . . . . . . . . . 6 / 48 . . . . . . . . . . . . ∫ R Assume some initial mass M 0 = L u ( x , t ) dx that is conserved for all ∂ M 0 ∫ R ∂ = ∂ tu ( x , t ) dx ∂ t ∫ R ∂ = − ∂ xF RL ( x , t ) dx = F RL ( L , t ) − F RL ( R , t ) . Imposing zero fmux at the boundary F RL ( L , t ) = F RL ( R , t ) = 0 ensures D α − 1 L + u ( x , t ) = 0 for x = L and x = R for all t ≥ 0

. . . . . . . . . . . . One-Sided Fractional Difgusion Equation: Patie-Simon . Also consider an alternative space-fractional difgusion equation The Patie-Simon (Patie and Simon, 2012) or mixed Caputo (Baeumer 1 L is a Caputo derivative. Applying zero fmux yields a refmecting BC: Kelly et al. (MSU) Two-Sided BCs June 22, 2018 . . . . . . . . . . . . . . . 7 / 48 . . . . . . . . . . . . ∂ ∂ tu ( x , t ) = C D α L + u ( x , t ) et al., 2018a) fractional derivative for 1 < α ≤ 2 is u ′ ( y , t ) L + u ( x , t ) = ∂ ∂ ∫ x D α ∂ x ∂ α − 1 L + u ( x , t ) = ( x − y ) α − 1 dy Γ( 2 − α ) ∂ x The corresponding fmux is F C ( x , t ) = − C ∂ α − 1 L + u ( x , t ) , where ∂ α − 1 L + ∂ α − 1 L + u ( x , t ) = 0 for x = L and x = R for all t ≥ 0 .

. . . . . . . . . . . . . . . Numerical Solutions (a) Riemann-Liouville fmux (b) Caputo fmux Figure: Numerical solution using a) Riemann-Liouville fractional derivative and b) Patie-Simon Kelly et al. (MSU) Two-Sided BCs June 22, 2018 . . . . . . . . . . . . . . 8 / 48 . . . . . . . . . . . 5 5 4 4 3 3 u(x,t) u(x,t) 2 2 1 1 0 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x x fractional derivative with refmecting BCs. α = 1 . 5, C = 1 on 0 ≤ x ≤ 1 at time t = 0 (solid line), t = 0 . 05 (dashed), t = 0 . 1 (dash dot), t = 0 . 5 (dotted).

. 1 . . . . . . . . . . Plan Boundary Conditions (BCs) for One-Sided Fractional Difgusion . 2 Two-Sided Fractional Difgusion: BCs and Numerical Methods 3 Two-Sided Fractional Difgusion: Analytical Steady-State Solutions 4 One-Sided Tempered Fractional Difgusion 5 Summary and Open Problems Kelly et al. (MSU) Two-Sided BCs June 22, 2018 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 / 48

. . . . . . . . . . . . . . Two-Sided Fractional Difgusion Equation: Riemann-Liouville source term. The positive (left) and negative (right) Riemann-Liouville fractional derivatives are given by 1 L x Kelly et al. (MSU) Two-Sided BCs June 22, 2018 . . . . . . . . . . . . . . 10 / 48 . . . . . . . . . . . . The two-sided space-fractional difgusion equation on [ L , R ] : ∂ ∂ tu ( x , t ) = pC D α L + u ( x , t ) + qC D α R − u ( x , t ) + s ( x , t ) where 1 < α ≤ 2, where C > 0, p , q ≥ 0, and p + q = 1, while s ( x , t ) is a L + u ( x , t ) = ∂ n ∂ n ∫ x u ( y , t ) D α ∂ x n I n − α L + u ( x , t ) = ( x − y ) α − n + 1 dy Γ( n − α ) ∂ x n R − u ( x , t ) = ( − 1 ) n ∂ n ( − 1 ) n ∂ n u ( y , t ) ∫ R D α ∂ x n I n − α R − u ( x , t ) = ( y − x ) α − n + 1 dy Γ( n − α ) ∂ x n

. . . . . . . . . . . Two-Sided Fractional Difgusion Equation: Patie-Simon We also consider an alternative space-fractional difgusion equation 1 . L 1 x using the Patie-Simon (Patie and Simon, 2012) or mixed Caputo 1 L x are the positive (left) and negative (right) Caputo derivatives. Kelly et al. (MSU) Two-Sided BCs June 22, 2018 . 11 / 48 . . . . . . . . . . . . . . . . . . . . . . . . . . . ∂ ∂ tu ( x , t ) = pC D α L + u ( x , t ) + qC D α R − u ( x , t ) + s ( x , t ) u ′ ( y , t ) L + u ( x , t ) = ∂ ∂ ∫ x D α ∂ x ∂ α − 1 L + u ( x , t ) = Γ( 2 − α ) ∂ x ( x − y ) α − 1 dy u ′ ( y , t ) R − u ( x , t ) = − ∂ ∂ ∫ R D α ∂ x ∂ α − 1 R − u ( x , t ) = ( y − x ) α − 1 dy Γ( 2 − α ) ∂ x (Baeumer et al., 2018a) fractional derivatives for 1 < α ≤ 2. u ( n ) ( y , t ) ∫ x ∂ α L + u ( x , t ) = ( x − y ) α − n + 1 dy Γ( n − α ) u ( n ) ( y , t ) ( − 1 ) n ∫ R ∂ α R − u ( x , t ) = ( y − x ) α − n + 1 dy Γ( n − α )

. . . . . . . . . . . . . . . Conservation Form conservation (continuity) equation nonlocal difgusion. The fmux function is given by fmux. Kelly et al. (MSU) Two-Sided BCs June 22, 2018 . . . . . . . . . . . . . . 12 / 48 . . . . . . . . . . . Physically, u ( x , t ) represents concentration governed by a local mass ∂ tu ( x , t ) + ∂ ∂ ∂ xF ( x , t ) = 0 F ( x , t ) is a fmux function (generalized Fick’s law) that accounts for F RL ( x , t ) = qC D α − 1 R − u ( x , t ) − pC D α − 1 L + u ( x , t ) F C ( x , t ) = qC ∂ α − 1 R − u ( x , t ) − pC ∂ α − 1 L + u ( x , t ) where F RL ( x , t ) is a Riemann-Liouville fmux and F C ( x , t ) is a Caputo

. . . . . . . . . . . . . . . . Refmecting (no-fmux) Boundary Conditions 1). (Baeumer et al., 2018a). Kelly et al. (MSU) Two-Sided BCs June 22, 2018 . . . . . . . . . . . . . 13 / 48 . . . . . . . . . . . Identify a no-fmux BC by setting F ( x , t ) = 0 at the boundary. Setting F ( x , t ) = 0 at x = L and x = R yields refmecting BCs: p D α − 1 L + u ( x , t ) = q D α − 1 R − u ( x , t ) for x = L and x = R for all t ≥ 0 . p ∂ α − 1 L + u ( x , t ) = q ∂ α − 1 R − u ( x , t ) for x = L and x = R for all t ≥ 0 . These boundary conditions are nonlocal since the BC at x = L or x = R depends on all values of u ( x , t ) in the interval [ L , R ] (if p ̸ = 0 or If p = 1, these BCs reduce to the refmecting BCs proposed in