Enhanced-accuracy difference schemes for one nonclassical boundary-value problem

On Some Spectral Properties of a Nonlocal Boundary Value Problem A. M. A. El-Sayed, Zaki. F. A. El-Raheem and N. A. O. Buhalima

“…It is well-known that many topics in mathematical physics require the investigation of the eigenvalues and eigenfunctions of Sturm-Liouville type boundary value problems. Eigenvalue problems with nonlocal conditions are closely linked with boundary problems for differential equations with nonlocal conditions ( [10], [11]). Eigenvalue problems for differential operators with nonlocal conditions are considerably less investigated than the classical boundary condition cases.…”

The Spectrum of a Nonlocal Difference Operator with Variable Coefficients

“…Ionkin [1,2] was the first to study the well-posing properties of finite-difference schemes with nonlocal boundary conditions for the case of constant coefficients k(x) ≡ 1 . He has shown that in this case all the eigenvalues of the matrix A are real, but the eigenvector system is not complete.…”

“…He has shown that in this case all the eigenvalues of the matrix A are real, but the eigenvector system is not complete. Expanding the sought solution in a biorthogonal sum in eigen-and adjoined functions of the difference operator and applying two-sided inequalities for the biorthogonal expansion coefficients, Ionkin [1,2] has obtained prior bounds on the solution of the finite-difference problem in the grid L 2 -норме in terms of the initial conditions and the right-hand side. The stability theory of nonlocal difference schemes with constant coefficients is described in detail in [3,4].…”

“…The stability theory of nonlocal difference schemes with constant coefficients is described in detail in [3,4]. The stability of nonlocal difference schemes with variable coefficients has been analyzed by the maximum principle in [5][6][7][8]. Here we can find, in particular, examples of coefficients for which the matrix A has complex real values.…”

Stability of Nonlocal Finite-Difference Problems

“…The expansion in the Riesz basis has been used to prove the existence and uniqueness of many problems with nonlocal boundary conditions. Difference schemes for problem (1), (2) were first considered by Ionkin [5] in 1977. An explicit basis of eigen-and adjoint functions of the finite-difference operator was constructed and sufficient stability conditions were derived for weighted finite-difference schemes.…”