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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)