Analytical inversion of general periodic tridiagonal matrices

An analysis of the linear waves in infinitely-long square and triangular lattice strips of identical particles with nearest neighbour interactions for all combinations of fixed and free boundary conditions, as well as the periodic boundary, is presented. Expressions for the dispersion relations and the associated normal modes in these waveguides are provided in the paper; some of which are expressed implicitly in terms of certain linear combinations of the Chebyshev polynomials. The effect of next-nearest-neighbour interaction is also included for the square lattice waveguides. It is found that localized propagating waves, so called surface wave modes, occur in the triangular lattice strips, as well as square lattice strips with next-nearest-neighbour interactions, when either or both boundaries are free. In this paper, the even and odd modes are also discussed separately, wherever applicable. Graphical illustrations of the dispersion curves are included for all waveguides. The discrete waveguides analysed in the paper have broad applications in physics and engineering, including their merit in classical problems in elasticity, acoustics and electromagnetism, as well as recent technological issues involving various transport phenomena in quasi-one-dimensional nano-structures.

/pdf/on-linear-waveguides-of-square-and-triangular-lattice-strips-4qirvemqsu.pdf

On linear waveguides of square and triangular lattice strips: an application of Chebyshev polynomials

An efficient computational algorithm is developed for solving linear matrix equations whose characteristic interaction matrix ${\bf H}'$ differs by a matrix of rank r from an interaction matrix ${\bf H}$ of a system whose solution is known, or easily computed. The formulation clearly manifests the effects of physical parameters characterizing the interactions in the system. The exact inverse is given for a nonsymmetric Toeplitz matrix with arbitrary corner elements.

Solution of modified matrix equations

We present a sixth-order non-polynomial spline method for the solutions of two- point nonlinear boundary value problem u(4) + f (x;u) = 0;u(a) = 1;u''(a) = 2;u(b) = 1;u '' (b) = 2; in off step points. Numerical method of sixth-order with end conditions of the order 6 is derived. The convergence analysis of the method has been discussed. Numerical examples are presented to illustrate the applications of method, and to compare the computed results with other known methods.

http://journals.iau.ir/article_521694_1c6825ac0abb0d3c70714cd7759a0928.pdf

Non-polynomial spline solutions for special nonlinear fourth-order boundary value problems

Abstract In this paper, four-band Toeplitz matrices and four-band Hankel matrices of type I and type II with perturbed rows are introduced. Explicit determinants, inverses and eigenvalues for these matrices are derived by using a nice inverse formula of block bidiagonal Toeplitz matrices.

/pdf/explicit-determinants-inverses-and-eigenvalues-of-four-band-4r7acwp4wu.pdf

Explicit determinants, inverses and eigenvalues of four band Toeplitz matrices with perturbed rows

In some problems in numerical analysis one is faced with solving a linear system of equations in which the matrix of the linear system is symmetric, tridiagonal and Toeplitz, except for elements at or near the corners. The anomalous elements usually arise from boundary conditions in the original problem. This paper is concerned with expressing the inverse of this type of matrix in such a way that some of its theoretical properties may be obtained.

The Inverses of Some Matrices Deviating Slightly from a Symmetric, Tridiagonal, Toeplitz Form

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The Inverses of Some Matrices Deviating Slightly from a Symmetric, Tridiagonal, Toeplitz Form