We study both boundary and internal stabilization problems for the fourth order Schrödinger equation in a smooth bounded domain Ω of Rⁿ. We first consider the boundary stabilization problem. By introducing suitable dissipative boundary conditions, we prove that the solution decays exponentially in an appropriate energy space. In the internal stabilization problem, by assuming that the damping term is effective on the neighborhood of the boundary, we prove the exponential decay of the L²(Ω)-energy of the solution. Both results are established by using multipliers technique and compactness/uniqueness arguments.

Three main contributions are presented in this paper. First, the septic quasi-interpolants are calculated with all their coefficients. Second, we explore the results to solve a generalized and broad class of Fredholm integral equations of the second kind. Finally, we present three degenerate kernel methods; the latter is a combination of the two previously established methods in the literature. Moreover, we provide a convergence analysis and we give new error bounds. Finally, we exhibit some numerical examples and compare them with previous results in the literature.

In this work, we present piecewise constant Galerkin method for a class of Cauchy singular integral equations of the second kind with constant coefficients in L²([0,1],C), using a sequence of orthogonal finite rank projections. We prove the existence and uniqueness theorems for the Cauchy integral equation and the approximate equation, respectively. We perform the error analysis for which we give new and improved estimates for the rates of convergence. Numerical example illustrates the theoretical results.

Fuglede-Putnam Theorem have been proved for a considerably large number of class of operators. In this paper by using the spectral theory, we obtain a theoretical and general framework from which Fuglede-Putnam theorem may be promptly established for many classes of operators.