Abstract
The first order of accuracy difference scheme for the numerical solution of the boundary value problem for the differential equation with parameterp,i(du(t)/dt)+Au(t)+iu(t)=f(t)+p,0<t<T,u(0)=φ,u(T)=ψ, in a Hilbert spaceHwith self-adjoint positive definite operatorAis constructed. The well-posedness of this difference scheme is established. The stability inequalities for the solution of difference schemes for three different types of control parameter problems for the Schrödinger equation are obtained.
Highlights
Difference SchemeThe theory and applications of well-posedness of inverse problems for partial differential equations have been studied extensively in a large cycle of papers.Our goal in this paper is to investigate Schrodinger equations with parameter
The proof of Theorem 4 is based on Theorem 3 and the symmetry property of the operator Axh is defined by formula (34) and the following theorem on the coercivity inequality for the solution of the elliptic difference problem in L2h
Dh is the approximation of the operator ∂⋅/∂n.⃗ With the help of the difference operator Axh, we arrive to the following boundary value problem: iuth (t, x) + Axhuh (t, x) + iuh (t, x) = ph (x) + fh (t, x), 0 < t < T, x ∈ Ωh, uh (0, x) = φh (x), uh (T, x) = ψh (x), x ∈ Ωh (52)
Summary
The theory and applications of well-posedness of inverse problems for partial differential equations have been studied extensively in a large cycle of papers (see, e.g., [1–24] and the references therein). In the paper [25], the boundary value problem for the differential equation with parameter p i du (t) dt. The stability inequalities for the solution of three determinations of control parameter problems for the Schrodinger equation were obtained. U0 = φ, uN = ψ for the approximate solution of the boundary value problem (1) for the differential equation with parameter p is presented. The stability inequalities for the solution of difference schemes for the approximate solution of three different types of control parameter problems are obtained.
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