Mathematics / Matematik
Permanent URI for this collectionhttps://hdl.handle.net/11147/8
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Article Citation - WoS: 22Citation - Scopus: 20Fokas Method for Linear Boundary Value Problems Involving Mixed Spatial Derivatives(Royal Society of Chemistry, 2020) Fokas, A. S.; Batal, Ahmet; Özsarı, TürkerWe obtain solution representation formulae for some linear initial boundary value problems posed on the half space that involve mixed spatial derivative terms via the unified transform method (UTM), also known as the Fokas method. We first implement the method on the second-order parabolic PDEs; in this case one can alternatively eliminate the mixed derivatives by a linear change of variables. Then, we employ the method to biharmonic problems, where it is not possible to eliminate the cross term via a linear change of variables. A basic ingredient of the UTM is the use of certain invariant maps. It is shown here that these maps are well defined provided that certain analyticity issues are appropriately addressed.Article Citation - WoS: 13Citation - Scopus: 14Nonlinear Schrödinger Equations on the Half-Line With Nonlinear Boundary Conditions(Texas State University - San Marcos, 2016) Batal, Ahmet; Özsarı, TürkerIn this article, we study the initial boundary value problem for nonlinear Schrödinger equations on the half-line with nonlinear boundary conditions ux(0, t) + λ|u(0, t)|ru(0, t) = 0, λ ∈ ℝ − {0}, r > 0. We discuss the local well-posedness when the initial data u0 = u(x, 0) belongs to an L2-based inhomogeneous Sobolev space (formula presented) with (formula presented). We deal with the nonlinear boundary condition by first studying the linear Schrödinger equation with a time-dependent inhomogeneous Neumann boundary condition ux(0, t) = h(t) where (formula presented) (0, T). © 2016 Texas State University.