Thanasis Fokas

The Unified Transform, Imaging and AsymptoticsDepartment of Applied Mathematics and Theoretical Physics, University of Cambridge

I. The Unified Transform

ut + ux + uxxx = 0, 0 < x <∞, t > 0,

u(x , 0) = u0(x), 0 < x <∞,

u(0, t) = g0(t), t > 0.

Example 1: The Heat Equation

ut = uxx , 0 < x <∞, 0 < t < T , T > 0,

u(x , 0) = u0(x), 0 < x <∞, u(0, t) = g0(t), 0 < t < T .

The classical sine transform yields

u(x , t) =2

π

∫ ∞0

e−k2t sin(kx)

[ ∫ ∞0

sin(kξ)u0(ξ)dξ−k∫ t

0

ek2sg0(s)ds

]dk.

The unified transform yields

u(x , t) =1

2π

∫ ∞−∞

e ikx−k2t u0(k)dk − 1

2π

∫∂D+

e ikx−k2t[u0(−k) + 2ikG0(k2)

]dk,

u0(k) =

∫ ∞0

e−ikxu0(x)dx , Imk ≤ 0, G0(k) =

∫ T

0

eksg0(s)ds, k ∈ C.

Alternatively, G0(k, t) =∫ t

0eksg0(s)ds, k ∈ C.

Example 2: The Stokes Equation

The unified transform yields

u(x , t) =1

2π

∫ ∞−∞

e ikx−(ik−ik3)t u0(k)dk − 1

2π

∫∂D+

e ikx−(ik−ik3)t g(k)dk,

g(k) =1

ν1 − ν2[(ν1 − k)u0(ν2) + (k − ν2)u0(ν1)] + (3k2 − 1)G0(ω(k)),

ω(k) = ik − ik3, k ∈ D+, D+ ={Reω(k) < 0

}∩ C+.

ν1, ν2 : ω(k) = ω(νk).

ν2j + kνj + k2 − 1 = 0, j = 1, 2.

Numerical Implementation

Consider the heat equation with

u0(x) = x exp(−a2x), g0(t) = sin bt, a, b > 0.

Then,

u(x , t) =1

2π

∫L

e ikx−k2t

[1

(ik + a)2− 1

(−ik + a)2

−k(e(k+ib)t − 1

k + ib− e(k−ib)t − 1

k − ib

)]dk .

• N. Flyer and A.S. Fokas, A hybrid analytical numerical method forsolving evolution partial differential equations. I. The half-line, Proc.R. Soc. 464, 1823-1849 (2008)

References

• B. Fornberg and N. Flyer, A numerical implementation of Fokas boundaryintegral approach: Laplace’s equation on a polygonal domain, Proc. R.Soc. A 467, 2983–3003 (2011)

• A.C.L. Ashton, On the rigorous foundations of the Fokas method forlinear elliptic partial differential equations, Proc. R. Soc. A 468,1325–1331 (2012)

• P. Hashemzadeh, A.S Fokas and S.A. Smitheman, A numerical techniquefor linear elliptic partial differential equations in polygonal domains, Proc.R. Soc. A 471, 20140747 (2015)

• M.J. Ablowitz, A.S. Fokas and Z.H. Musslimani, On a new nonlocalformulation of water waves, J. Fluid Mech. 562, 313–344 (2006)

• D. Crowdy, Geometric function theory: a modern view of a classicalsubject, Nonlinearity 21, 205–219 (2008)

• N.E. Sheils and D.A. Smith, Heat equation on a network using the Fokasmethod, J. Phys. A 48(33), 335001 (2015)

• D.M. Ambrose and D.P. Nicholls, Fokas integral equation for threedimensional layered-media scattering, J. Comput. Phys. 276, 1–25 (2014)

Summary1747, d’ Alembert and Euler: Separation of Variables1807, Fourier: Transforms1814, Cauchy: Analyticity1828, Green: Green’s Representations1845, Kelvin: Images

Physical Space

TransformsMethod of Images

Fourier (spectral) Space

Green’s Integral Representations

New Method

Invariance of Global Relation and Jordan’s LemmaIntegral Representations in the Fourier Space

http://www.personal.reading.ac.uk/ smr07das/UTM/

• A.S. Fokas, A Unified Approach to Boundary Value Problems,CBMS-NSF, SIAM (2008)

• A.S. Fokas and E.A. Spence, Synthesis as Opposed to Separation ofVariables, SIAM Review 54, 291–324 (2012)

• B. Deconinck, T. Trogdon and V. Vasan, The Method of Fokas forSolving Linear Partial Differential Equations, SIAM Review 56,159–186 (2014)

II. Medical Imaging

PET - SPECT

F-Gelfand (1994) : Novel derivation of 2D FT via ∂

(∂x1 + i∂x2 − k)µ(x1, x2, k) = u(x1, x2).

F-Novikov (1992) : Novel derivation of Radon transform[1

2

(k +

1

k

)∂x1 +

1

2i

(k − 1

k

)∂x2

]µ(x1, x2, k) = f (x1, x2).

Novikov (2002) : Derivation of attenuated Radon transform[1

2

(k +

1

k

)∂x1 +

1

2i

(k− 1

k

)∂x2

]µ(x1, x2, k) + f (x1, x2)µ(x1, x2, k) = g(x1, x2).

• G.A. Kastis, D. Kyriakopulou, A. Gaitanis, Y. Fernandez, B.F. Hutton andA.S. Fokas, Evaluation of the spline reconstruction technique for PET,Med. Phys. 41, 042501 (2014)

• G.A. Kastis, A. Gaitanis, A. Samartzis and A.S. Fokas, The SRTreconstruction algorithm for semiquantification in PET imaging, Med.Phys. 42, 5970–5982 (2015)

ATTENUATED RADON TRANSFORM

Direct Attenuated Radon transform

gf (ρ, θ) =

∫ ∞−∞

e−∫∞τ

fdsgdτ.

Inverse Attenuated Radon transform

P±g(ρ) = ±g(ρ)

2+

1

2iπ

∮ ∞−∞

g(ρ′)

ρ′ − ρdρ′.

H(θ, τ, ρ) = e∫∞τ fds

{eP− f (ρ,θ)P−e−P− f (ρ,θ) + e−P+ f (ρ,θ)P+eP

+ f (ρ,θ)}gf (ρ, θ).

g(x1, x2) =1

4π(∂x1−i∂x2 )

∫ 2π

0

e iθH(θ, x1cos θ+x2sin θ, x2cos θ−x1sin θ)dθ.

III. Asymptotics of the Riemann Zeta Function

The Laplace equation in the exterior of the Hankel contour and theRiemann hypothesis

g()

(r)g+

(r)g-

α

α

g±(r) =(re±iα)su

(re±iα

), g(θ) = − i

a

(ae iθ

)su(ae iθ

), u(z) =

1

e−z − 1.

The Riemann hypothesis is valid if and only if the above Neumannboundary value problem does not have a solution which is bounded asr →∞.

• A.S. Fokas and M.L. Glasser, The Laplace equation in the exterior ofthe Hankel contour and novel identities for the hypergeometricfunctions, Proc. R. Soc. A 469, 20130081 (2013)

On the Lindelof Hypothesis

t

π

∮ ∞−∞<{

Γ(it − iτ t)

Γ(σ + it)Γ(σ + iτ t)

}|ζ(σ + iτ t)|2 dτ + G(σ, t) = 0,

0 ≤ σ ≤ 1, t > 0, (1)

G(σ, t) =

{ζ(2σ) +

(Γ(1−s)

Γ(s) + Γ(1−s)Γ(s)

)Γ(2σ − 1)ζ(2σ − 1) + 2(σ−1)ζ(2σ−1)

(σ−1)2+t2 , σ 6= 12 ,

<{

Ψ(

12 + it

)}+ 2γ − ln 2π + 2

1+4t2 , σ = 12 ,

(2)

with Ψ(z) denoting the digamma function, i.e.,

Ψ(z) =ddzΓ(z)

Γ(z), z ∈ C,

and γ denoting the Euler constant.

We conclude that

∑∑m1,m2∈N

1

ms−itδ3

1

1

ms+itδ3

2

1

ln[m2

m1(t1−δ3 − 1)

] =

{O(tδ3 ln t

), σ = 1

2 ,

O(t

δ32

), 1

2 < σ < 1,

t →∞,

where N is defined by

N =

{m1 ∈ N+, m2 ∈ N+, 1 ≤ m1 ≤ [T ], 1 ≤ m2 < [T ],

m2

m1>

1

t1−δ3 − 1

(1 + c(t)

), t−

δ32 ≺ c(t) ≺ 1, t > 0, T =

t

2π

}.

References

• A. S. Fokas and J. Lenells, On the asymptotics to all orders of theRiemann zeta function and of a two-parameter generalization of theRiemann zeta function, https://arxiv.org/abs/1201.2633 (preprint)

• A. S. Fokas, On the proof of a variant of Lindelof’s hypothesis (preprint)