# Download e-book for kindle: A unified approach to boundary value problems by Athanassios S. Fokas By Athanassios S. Fokas

ISBN-10: 0898716519

ISBN-13: 9780898716511

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Extra info for A unified approach to boundary value problems

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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, .

14) and let q satisfy the following initial and boundary conditions: q(x, 0) = q0 (x), ∂xj q(0, t) = gj (t), 0 < x < ∞, 0 < t < T, j = 0, 1, . . , N − 1. 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, .

Representing the product of δ functions in terms of Fourier integrals, δ(x − x )δ(y − y ) = 1 4π 2 R2 eik1 (x−x )+ik2 (y−y ) dk1 dk2 , (61) we see that the fundamental solution of (54) is given by G=− 1 4π 2 R2 eik1 (x−x )+ik2 (y−y ) dk1 dk2 . 4. Green’s Identities, Images, Transforms, and the Wiener–Hopf Technique fokas 2008/7/24 page 21 ✐ 21 Substituting this expression in the LHS of (56a) and comparing the resulting equation with the formula for q given by (51a) (the representation in the spectral plane), we observe that the former equation involves integrations with respect to both dk1 and dk2 , whereas the latter equation involves integration only with respect to dk.