From discrete time random walks to numerical methods for fractional - PowerPoint PPT Presentation

From discrete time random walks to numerical methods for fractional order differential equations Click to edit Present’s Name Dr. Christopher Angstmann SANUM 2016 Never Stand Still Faculty of Science School of Mathematics and Statistics

Collaborators Everything Bruce Henry, UNSW Fractional Fokker-Planck equation and reaction sub-diffusion Isaac Donnelly, and James Nichols, UNSW Byron Jacobs, University of the Witwatersrand Fractional SIR models Anna McGann, UNSW

References C. N. Angstmann et al. A discrete time random walk model for anomalous diffusion, J. Comput. Phys. (2015), doi:10.1016/j.jcp.2014.08.003 C. N. Angstmann et al. From stochastic processes to numerical methods: A new scheme for solving reaction subdiffusion fractional partial differential equations J. Comput. Phys. (2016) doi:10.1016/j.jcp.2015.11.053 C. N. Angstmann et al. A fractional order recovery SIR model from a stochastic process. Bulletin of Mathematical Biology , (2016) doi:10.1007/s11538-016-0151-7

The Idea Consider a particle that is undergoing a random walk. At every point in time we have a probability distribution for the location of the particle. This probability distribution evolves in time according to a governing equation.

The Idea For a discrete time system the governing equation is a difference equation. For a continuous time system the governing equation is a differential equation. If we construct a sequence of discrete time random walks that tend towards a continuous time random walk, then we will also have a sequence of difference equations that tend to the differential equation.

Advantages As the approximation is always a governing equation for the random walk then we have some guarantees about its behavior. • In the absence of reactions we know that the particle is not created nor destroyed, so the scheme must conserve mass. • The approximation is always a finite distance from the solution of the differential equation. We can use the tools of stochastic processes on the numerical method.

Example: A Biased Random Walk The governing equation for finding the particle undergoing a biased random walk at position x , at time t is: X ( x, t ) = p r ( x − ∆ x, t − ∆ t ) X ( x − ∆ x, t − ∆ t ) + p l ( x + ∆ x, t − ∆ t ) X ( x + ∆ x, t − ∆ t ) Taking the “diffusion” limit of this process gives a Fokker-Planck equation as the governing equation: = D ∂ 2 X ( x, t ) ∂ X ( x, t ) − 2 D β ∂ ∂ x ( f ( x, t ) X ( x, t )) ∂ x 2 ∂ t Where p r ( x, t ) − p l ( x, t ) ∆ x 2 f ( x, t ) = lim D = lim β ∆ x 2 ∆ t ∆ x → 0 ∆ x, ∆ t → 0

Boltzmann weights Boltzmann weights are taken for the jump probabilities. exp( − β V ( x + ∆ x, t )) f ( x, t ) = − ∂ V ( x, t ) p r ( x, t ) = exp( − β V ( x + ∆ x, t )) + exp( − β V ( x − ∆ x, t )) ∂ x This guarantees that for all . 0 < p r ( x, t ) < 1 ∆ x

Example: Burgers Equation Burgers equation, = ν ∂ 2 u ( x, t ) ∂ u ( x, t ) − u ( x, t ) ∂ u ( x, t ) ∂ t ∂ x 2 ∂ x can be approximated by u ( x − ∆ x, t − ∆ t ) u ( x, t ) = 1 + exp( ∆ x 8 ν ( u ( x − 2 ∆ x, t − ∆ t ) + 2 u ( x − ∆ x, t − ∆ t ) + u ( x, t − ∆ t ))) u ( x − ∆ x, t − ∆ t ) + 1 + exp( ∆ x 8 ν ( u ( x − 2 ∆ x, t − ∆ t ) + 2 u ( x − ∆ x, t − ∆ t ) + u ( x, t − ∆ t )))

Example: Burgers Equation

Fractional Fokker-Planck Equation The fractional Fokker-Planck equation is = D ∂ 2 ∂ u ( x, t ) − 2 D β ∂ � 0 D 1 − α � � f ( x, t ) 0 D 1 − α � ( u ( x, t )) ( u ( x, t )) t t ∂ x 2 ∂ t ∂ x Where is the Riemann-Liouville fractional derivative 0 D 1 − α ( u ( x, t )) t defined by Z t 1 ∂ 0 D 1 − α ( t − v ) α − 1 u ( x, v ) dv ( u ( x, t )) = 0 < α < 1 t Γ ( α ) ∂ t 0 This equation is typically derived from the limit of a Continuous Time Random Walk.

Fractional Fokker-Planck Equation To find approximations for a fractional Fokker-Planck equation we need to use a different discrete time random walk. We modify the random walk by letting the particle wait for a random number of time steps before jumping. This introduces a waiting time probability distribution, . ψ ( n ) By choosing an appropriate heavy tailed distribution the governing equations will limit to the fractional Fokker-Planck equation.

Discrete Time Random Walk For such a random walk the governing equation is, in general, n − 1 X u ( x, n ∆ t ) = u ( x, ( n − 1) ∆ t ) + p r ( x − ∆ x, ( n − 1) ∆ t ) K ( n − m ) u ( x − ∆ x, m ∆ t ) m =0 n − 1 n − 1 X X + p l ( x + ∆ x, ( n − 1) ∆ t ) K ( n − m ) u ( x + ∆ x, m ∆ t ) − K ( n − m ) u ( x, m ∆ t ) m =0 m =0 Where K(n) is the memory kernel associated with the waiting time probability distribution. It is defined through it’s Z transform with Z{ K ( n ) } = Z{ ψ ( n ) } Z{ φ ( n ) } Where is the probability that the particle has not jumped by the n φ ( n ) time step since arriving at the site.

Sibuya Distribution To limit to the fractional Fokker-Planck equation we take the Sibuya waiting time distribution, Γ ( α + 1) n 1 − α ⇣ ⌘ ψ ( n ) = ( − 1) n +1 Y φ ( n ) = Γ ( n + 1) Γ ( α − n + 1) m m =1 0 < α < 1 n > 0 for and . This gives a tractable memory kernel

Example: Fractional Diffusion Equation The simplest fractional example is the fractional diffusion equation, α = 4 D = 1 which just f ( x,t )=0. Taking , and . 5 ∂ ρ ( x , t ) 5 ∂ 2 ρ ( x , t ) 1 − 4 = D t ∂ t ∂ x 2 With zero flux boundaries ∂ ρ ( x , t ) ∂ ρ ( x , t ) = 0 = 0 ∂ x ∂ x x = 0 x = 1 and the initial condition ρ ( x ,0) = δ ( x − 1 2 )

Example: Fractional Diffusion Equation 1.015 0.004 1.010 X H x,2 L 0.002 1.005 1.000 0.000 0.995 - 0.002 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 x x 1.000 1.000 0.500 0.500 0.100 0.100 0.050 0.050 0.010 0.010 0.005 0.005 0.10 0.15 0.20 0.30 0.50 0.70 1.00 1.50 0.10 0.15 0.20 0.30 0.50 0.70 1.00 1.50 t t

Reaction sub-diffusion equations Schemes for more complicated equations can also be found with this approach. For example the reaction sub-diffusion equation. ( ) D t ( ) u ( x , t ) ∂ u ( x , t ) = D ∂ 2 ⎡ ⎤ ⎡ ⎤ t 1 − α exp t ∫ d ( x , t ') dt ' ∫ d ( x , t ') dt ' ∂ x 2 exp − ⎥ − d ( x , t ) u ( x , t ) + b ( x , t ) ⎢ ⎢ ⎥ ∂ t ⎣ ⎦ ⎣ ⎦ 0 0 The stochastic process used is slightly different in this case. We take an ensemble of randomly walking and reacting particles.

Non-linear morphogen death rates on semi-infinite domain

References C. N. Angstmann et al. A discrete time random walk model for anomalous diffusion, J. Comput. Phys. (2015) doi:10.1016/j.jcp.2014.08.003 C. N. Angstmann et al. From stochastic processes to numerical methods: A new scheme for solving reaction subdiffusion fractional partial differential equations J. Comput. Phys. (2016) doi:10.1016/j.jcp.2015.11.053 C. N. Angstmann et al. A fractional order recovery SIR model from a stochastic process. Bulletin of Mathematical Biology , (2016) doi:10.1007/s11538-016-0151-7