A unified approach to boundary value problems
A novel approach to analysing initial-boundary value problems for integrable partial differential equations (PDEs) in two dimensions, based on ideas of the inverse scattering transform that the author introduced in 1997. This method is unique in also yielding novel integral representations for linear PDEs. Several new developments are addressed in the book, including a new transform method for linear evolution equations on the half-line and on the finite interval; analytical inversion of certain integrals such as the attenuated Radon transform and the Dirichlet-to-Neumann map for a moving boundary; integral representations for linear boundary value problems; analytical and numerical methods for elliptic PDEs in a convex polygon; and integrable nonlinear PDEs. An epilogue provides a list of problems on which the author's new approach has been used, offers open problems, and gives a glimpse into how the method might be applied to problems in three dimensions.
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Evolution Equations on the HalfLine
Evolution Equations on the Finite Interval
Asymptotics and a Novel Numerical Technique
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A.S. Fokas analytic function associated asymptotic boundary conditions boundary value problems bounded and analytic Chapter complex fc-plane compute convex polygon defined in terms denotes depicted in Figure derivation differential form Dirichlet boundary Dirichlet boundary condition Dirichlet problem Dirichlet to Neumann evaluated example finite interval following equations formulated Fourier transform functions A(k Furthermore given global relation half-line heat equation Hence implies initial-boundary value problem integral equation integrand inverse Jordan's lemma KdVII Laplace equation Lax pair linear PDE linearizable modified Helmholtz equation Neumann boundary values Neumann map novel integral representations polygon Proposition qo(k qo(x quadrant Radon transform real axis relevant replaced residue conditions respectively resulting equation Robin boundary conditions Schwarz conjugate semi-infinite strip solitons solution q(x solved spectral functions spectral plane theorem transform method unknown boundary values unknown functions valid vector Volterra integral equation yields