A Note on Nonlocal Boundary Value Problems for Hyperbolic Schrödinger Equations

Abstract

The nonlocal boundary value problem d 2 u t /dt 2 Au t f t 0 ≤ t ≤ 1 , i du t /dt Au t g t −1 ≤ t ≤ 0 , u 0 u 0 − , u t 0 u t 0 − , Au −1 αu μ ϕ, 0 < μ ≤ 1, for hyperbolic Schrödinger equations in a Hilbert space H with the self-adjoint positive definite operator A is considered. The stability estimates for the solution of this problem are established. In applications, the stability estimates for solutions of the mixed-type boundary value problems for hyperbolic Schrödinger equations are obtained.

On the numerical solutions of high order stable difference schemes for the hyperbolic multipoint nonlocal boundary value problems

“…By rearranging problem (18) we arrive to the system of linear equation (15 For the solution of linear system (15) we use the method of third order of accuracy difference scheme (4). Now let us present some numerical results for the approximate solutions of problem (8) that are obtained by matlab implementation.…”

“…. ; x m Þ 2 X; 0 < t < 1; Note that many scientists have been studied on the solutions of boundary value problems such as parabolic equations, elliptic equations and equations of mixed types extensively (see, e.g., [17][18][19][20][21][22][23][24][25][26][27][28] and the references therein).…”

Second Order Equations in Functional Spaces: Qualitative and Discrete Well-Posedness

“…Section 7.3 is based on results of [70][71][72][73][74][75][76]. Section 7.4 is based on results of [77][78][79][80][81][82][83][84][85][86][87][88][89][90][91]. Section 7.5 is devoted to stochastic hyperbolic equations.…”

On Stability of a Third Order of Accuracy Difference Scheme for Hyperbolic Nonlocal BVP with Self-Adjoint Operator

“…The well posedness of nonlocal boundary value problems for parabolic equations, elliptic equations, and equations of mixed types have been studied extensively by many scientists (see, e.g., [11][12][13][14][19][20][21][22][23][24][25][26][27][28][29][30][31][32] and the references therein).…”

“…Nonlocal boundary value problems have been a major research area in the case when it is impossible to determine the boundary conditions of the unknown function. Over the last few decades, the study of nonlocal boundary value problems is of substantial contemporary interest (see, e.g., [6][7][8][9][10][11][12][13][14] and the references given therein).…”