# New PDF release: A unified approach to boundary value problems

By Athanassios S. Fokas

ISBN-10: 0898716519

ISBN-13: 9780898716511

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Extra resources for A unified approach to boundary value problems

Example text

The first novel integral transform constructed by this new method is the attenuated Radon transform derived by Novikov in [18] (generalizing the analogous derivation of the classical Radon transform presented in [19]). The attenuated Radon transform of a Schwartz function g(x1 , x2 ) with attenuation f (x1 , x2 ), where f is a given Schwartz function, is defined by gˆ f (ρ, θ ) = ∞ e− ∞ τ f (s cos θ −ρ sin θ,s sin θ +ρ cos θ )ds −∞ × g(τ cos θ − ρ sin θ, τ sin θ + ρ cos θ)dτ, ρ ∈ R, θ ∈ (0, 2π ).

17), where g˜ j (k) = Gj (k), j = 0, 1, . . 30) + and the functions {g˜ j }n−1 N are determined as follows: The domain DR consists of N sectors, + + N {DR,m }1 , and in each of these sectors, say DR,m , m fixed, {g˜ j (w(k))}n−1 satisfy n − N N linear algebraic equations. , for fixed m, + w(k) = w(νl,m (k)), k ∈ DR,m , νl,m (k) ∈ DR− , l = 1, . . , n − N. 31) Proof. 1 that D consists of the union of disjoint unbounded simply connected open sets. For k large, w(k) ∼ αn k n and D approaches DR which consists of the following n sectors: arg αn + n arg k ∈ π 3π , 2 2 + 2mπ, m = 0, .

L−k Then (74) yields (83). ˜ are directly related to the x-part and t-part, The spectral functions qˆ0 (k) and g(k) respectively, of the Lax pair (73): Let ψ(x, k) μ(x, 0, k), ϕ(t, k) μ(0, t, k). (85) Then ⎧ ⎨ ψ (x, k) − ikψ(x, k) = q (x), 0 < x < ∞, Im k ≤ 0, x 0 qˆ0 (k) = −ψ(0, k); −ikx ⎩ lim e ψ(x, k) = 0; (86) x→∞ ✐ ✐ ✐ ✐ ✐ ✐ ✐ 30 fokas 2008/7/24 page 30 ✐ Introduction ⎧ ⎨ϕ (t, k) + ik 2 ϕ(t, k) = iq (0, t) − kq(0, t), 0 < t < T , 2 t x g(k) ˜ = eik T ϕ(T , k) ⎩ϕ(0, k) = 0. k ∈ C, (87) We emphasize that the conceptual steps needed for the solution of the NLS equation (81) in the domain are identical to those used above for the solution of the linearized version of the NLS.