A new eigenvalue problem for the difference operator with nonlocal conditions

In the paper, the spectrum structure of one-dimensional differential operator with nonlocal conditions and of the difference operator, corresponding to it, has been exhaustively investigated. It has been proved that the eigenvalue problem of difference operator is not equivalent to that of matrix eigenvalue problem Au = λu, but it is equivalent to the generalized eigenvalue problem Au = λBu with a degenerate matrix B. Also, it has been proved that there are such critical values of nonlocal condition parameters under which the spectrum of both the differential and difference operator are continuous. It has been established that the number of eigenvalues of difference problem depends on the values of these parameters. The condition has been found under which the spectrum of a difference problem is an empty set. An elementary example, illustrating theoretical expression, is presented.

APPLICATION OF DM METHODS FOR PROBLEMS WITH PARTIAL DIFFERENTIAL EQUATIONS

Two variants of applications of the Degenerate Matrix Method for solving problems with PDB are considered. Solutions of the simple testing problem and of one more complicated nonlinear problem with PDB of the fifth order are presented as examples.

ANALYSIS OF GENERALIZED MULTISTEP ADAM'S METHODS BY DEGENERATE MATRIX METHOD FOR ORDINARY DIFFERENTIAL EQUATIONS

Adam's methods in the multistep mode are considered by means of general schemes of the degenerate matrix method. The stability function for these methods is computed by the residue theory on the complex plane. Performance of uniformly and non‐uniformly distributed nodes in the standardized interval is compared.

MULTISTEP DEGENERATE MATRIX METHOD FOR ORDINARY DIFFERENTIAL EQUATIONS

Two of the simplest general schemes of the degenerate matrix method in the multistep mode are considered. The stability function for these methods is computed by the residue theory in the complex plane. Performances of uniformly and nonuniformly distributed nodes in the standardized interval are compared.

DEGENERATE MATRIX METHOD WITH CHEBYSHEV NODES FOR SOLVING NONLINEAR SYSTEMS OF DIFFERENTIAL EQUATIONS

One of the simplest schemes of the degenerate matrix method with nodes as zeroes of Chebyshev polynomials of the second kind is considered. Performance of simple iterations and some modifications of Newton method for the discrete problem is compared.

DEGENERATE MATRIX METHOD FOR SOLVING NONLINEAR SYSTEMS OF DIFFERENTIAL EQUATIONS

Degenerate matrix method for numerical solving nonlinear systems of ordinary differential equations is considered. The method is based on an application of special degenerate matrix and usual iteration procedure. The method, which is connected with an implicit Runge‐Kutta method, can be simply realized on computers. An estimation for the error of the method is given.