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• A Tail of Two Distributions: (Inverse Problems for Fractional Diffusion Equations)

William Rundell Texas A&M University,

• Fick’s, Darcy’s . . . law + Continuity equation: J = ρ∇u ∂u ∂t

= ∇ · J Here u = u(x, t) can be thought of as temperature, J as heat flux, and ρ is a diffusion coefficient that depends on the medium. Combining these gives:

• Fick’s, Darcy’s . . . law + Continuity equation: J = ρ∇u ∂u ∂t

= ∇ · J Here u = u(x, t) can be thought of as temperature, J as heat flux, and ρ is a diffusion coefficient that depends on the medium. Combining these gives:

• In the time-independent/steady-state case: Laplace’s equation: △u = 0 • The heat equation ut = ∇·ρ∇u and with ρ = 1 • Fundamental solution K(x, t) = 1√

4πt e−

x2

4t

• Decay rate is a Gaussian distribution: 〈x2〉 ∝ t

• Fick’s, Darcy’s . . . law + Continuity equation: J = ρ∇u ∂u ∂t

= ∇ · J Here u = u(x, t) can be thought of as temperature, J as heat flux, and ρ is a diffusion coefficient that depends on the medium. Combining these gives:

• In the time-independent/steady-state case: Laplace’s equation: △u = 0 • The heat equation ut = ∇·ρ∇u and with ρ = 1 • Fundamental solution K(x, t) = 1√

4πt e−

x2

4t

• Decay rate is a Gaussian distribution: 〈x2〉 ∝ t Many ways to argue this.

• Foundation was Brownian motion. Made explicit by Einstein in 1905: verified by Perrin to compute Avogadro’s number (Nobel Prize 1926).

• A statistician might argue this as follows: The particle jumps should be independent random variables. In the limit or aggregate, by the Central

Limit Theorem, these should approach a Gaussian distribution.

• Einstein demonstrated 〈x2〉 ∝ t2 gave straight line motion - wave equation.

• In the last 40 years accumulated evidence indicates not

all diffusion processes seem to obey this law.

So what if we allow the more general version 〈x2〉 ∝ φ(t) – and (somehow) find the φ that “best fits the data?”

• In the last 40 years accumulated evidence indicates not

all diffusion processes seem to obey this law.

So what if we allow the more general version 〈x2〉 ∝ φ(t) – and (somehow) find the φ that “best fits the data?”

Will we still get a nice partial differential equation?

• In the last 40 years accumulated evidence indicates not

all diffusion processes seem to obey this law.

So what if we allow the more general version 〈x2〉 ∝ φ(t) – and (somehow) find the φ that “best fits the data?”

Will we still get a nice partial differential equation?

• Drop the “nice” for a moment. Just will it be a pde? Unless f(t) = tn and n an integer, not under almost any setting.

• In the last 40 years accumulated evidence indicates not

all diffusion processes seem to obey this law.

So what if we allow the more general version 〈x2〉 ∝ φ(t) – and (somehow) find the φ that “best fits the data?”

Will we still get a nice partial differential equation?

• Drop the “nice” for a moment. Just will it be a pde? Unless f(t) = tn and n an integer, not under almost any setting.

Nice (in the sense of beauty) is always in the eyes of the beholder.

If in the sense of mathematical analysis, then, frankly, no.

• Where do the fractions enter?

Anomalous (fractional) Diffusion: 〈x2〉 ∝ tα , α 6= 1 leads to a continuous time random walk and a fractional derivative in the time variable Dα0,tu ,

If α < 1 we have sub-diffusion; α > 1 gives super diffusion.

• Where do the fractions enter?

Anomalous (fractional) Diffusion: 〈x2〉 ∝ tα , α 6= 1 leads to a continuous time random walk and a fractional derivative in the time variable Dα0,tu ,

If α < 1 we have sub-diffusion; α > 1 gives super diffusion.

Anything more general than simple powers?

Of course! How close to a pde do you want? One aim would be to preserve as

much structure as possible in the resulting “differential” equations and prevent

the analysis from being overly challenging or too abstract.

The fractional power law is already hard enough.

Remember Einstein’s quote

“Always keep things simple, but no simpler”

• Where do the fractions enter?

Anomalous (fractional) Diffusion: 〈x2〉 ∝ tα , α 6= 1 leads to a continuous time random walk and a fractional derivative in the time variable Dα0,tu ,

If α < 1 we have sub-diffusion; α > 1 gives super diffusion.

Anything more general than simple powers?

Of course! How close to a pde do you want? One aim would be to preserve as

much structure as possible in the resulting “differential” equations and prevent

the analysis from being overly challenging or too abstract.

The fractional power law is already hard enough.

Remember Einstein’s quote

“Always keep things simple, but no simpler”

Will lead to very different physics.

In turn will lead to interesting inverse problems.

Replace either the time or the space derivative in the heat equation by a

derivative of fractional order J = ρDβxu D α t = ∇·J

Replace either the time or the space derivative in the heat equation by a

derivative of fractional order J = ρDβxu D α t = ∇·J

Fractional differential equation

Dα0,tu−Dβ0,xu+ aux + bu = f with α ∈ (0, 1) , β ∈ (1, 2) being the order of “differentiation” at a micro- scopical level: (Can also have 1 < α < 2 for “fractional wave equation”).

Replace either the time or the space derivative in the heat equation by a

derivative of fractional order J = ρDβxu D α t = ∇·J

Fractional differential equation

Dα0,tu−Dβ0,xu+ aux + bu = f with α ∈ (0, 1) , β ∈ (1, 2) being the order of “differentiation” at a micro- scopical level: (Can also have 1 < α < 2 for “fractional wave equation”).

In the case of space fractional derivatives we might need n∑

i=1

aiD αi xi u+

n∑

i=1

biD βi xiu+ cu = f 1 < αi ≤ 2, 0 < βi ≤ 1

• • Leibniz’s letter to L’Hospital (1695): “Thus it follows that d

1 2x will be equal to x

√ [2]dx : x , an apparent

paradox, from which one day useful consequences will be drawn.”

• • Leibniz’s letter to L’Hospital (1695): “Thus it follows that d

1 2x will be equal to x

√ [2]dx : x , an apparent

paradox, from which one day useful consequences will be drawn.”

• Euler’s observation (1738): n ∈ N+ , µ, α ∈ R+ dn

dxn x

µ = Γ(µ + 1)

Γ(µ − n + 1)x µ−n

,

dxα x

µ = Γ(µ + 1)

Γ(µ − α + 1)x µ−α ⇐ αth order derivative

• • Leibniz’s letter to L’Hospital (1695): “Thus it follows that d

1 2x will be equal to x

√ [2]dx : x , an apparent

paradox, from which one day useful consequences will be drawn.”

• Euler’s observation (1738): n ∈ N+ , µ, α ∈ R+ dn

dxn x

µ = Γ(µ + 1)

Γ(µ − n + 1)x µ−n

,

dxα x

µ = Γ(µ + 1)

Γ(µ − α + 1)x µ−α ⇐ αth order derivative

• Abel’s integral equation for tautochrone problem (1823): µ ∈ (0, 1) ,

Iµ[ϕ] =

∫ x

0

(x−t)−µϕ(t)dt = f(x)

Two limiting cases:

limµ→1− 1

Γ(1−µ) Iµ[ϕ] = ϕ(x) limµ→0+

1 Γ(1−µ)

Iµ[ϕ] = ∫ x 0 ϕ(t) dt

⇒ L.H.S. is a fractional integral of order 1 − µ

• Some members of the fractional derivative zoo of order α ∈ (n− 1, n) [a] Riemann-Liouville fractional derivative

∂αt u(t) = RDαt u(t) =

1

Γ(n−α) dn

dtn

∫ t 0 (t−s)n−1−αu(s) ds

[b] Caputo fractional derivative (Dzherbashyan, 1960, Caputo, 1967)

CDαt u(t) = 1

Γ(n−α) ∫ t 0 (t−s)n−1−αu(n)(s) ds

[But this really should be called the Dzherbashyan-Caputo derivative.]

• Some members of the fractional derivative zoo of order α ∈ (n− 1, n) [a] Riemann-Liouville fractional derivative

∂αt u(t) = RDαt u(t) =

1

Γ(n−α) dn

dtn

∫ t 0 (t−s)n−1−αu(s) ds

[b] Caputo fractional derivative (Dzherbashyan, 1960, Caputo, 1967)

CDαt u(t) = 1

Γ(n−α) ∫ t 0 (t−s)n−1−αu(n)(s) ds

[But this really should be called the Dzherbashyan-Caputo derivative.]

• Since these are non-local operators they really should be written as R 0D

α t u(t) ,

C 0D

α t u(t) to denote the starting position.

• Some members of the fractional derivative zoo of order α ∈ (n− 1, n) [a] Riemann-Liouville fractional derivative

∂αt u(t) = RDαt u(t) =

1

Γ(n−α) dn

dtn

∫ t 0 (t−s)n−1−αu(s) ds

[b] Caputo fractional derivative (Dzherbashyan,