Применение метода Канторовича для компьютерного анализа моделей квантоворазмерных наноструктур

Within the effective mass approximation the description of dynamics of quantum-dimensional semiconductor nanostructures, e.g., quantum wells, quantum wires, and quantum dots, reduced to a multidimensional boundary-value problem for Schrödinger-type equations. For solving such problems we elaborate the symbolic-numerical algorithms realizing a computational scheme based on the generalization of Kantorovich method, reducing the problem to a set of boundary problems for second-order ordinary differential equations. The efficiency of the algorithms was demonstrated by analysis of spectral and optical characteristics of axially-symmetric spheroidal quantum dots with different confining potentials.В приближении эффективной массы динамика квантоворазмерных полупроводниковых наноструктур, таких как квантовые ямы, квантовые проволоки и квантовые точки описываются многомерной краевой задачи для уравнений шрёдингеровского типа. Для решения таких задач разработаны символьно-численные алгоритмы, реализующие вычислительную схему, основанную на обобщении метода Канторовича – редукции к набору краевых задач для обыкновенных дифференциальных уравнений второго порядка. Эффективность алгоритмов продемонстрирована анализом спектральных и оптических характеристик моделей аксиально-симметричных сфероидальных квантовых точек с различными ограничивающими потенциалами

Super-Exponentially Convergent Parallel Algorithm for a Fractional Eigenvalue Problem of Jacobi-Type

A new algorithm for eigenvalue problems for the fractional Jacobi type ODE is proposed. The algorithm is based on piecewise approximation of the coefficients of the differential equation with subsequent recursive procedure adapted from some homotopy considerations. As a result, the eigenvalue problem (which is in fact nonlinear) is replaced by a sequence of linear boundary value problems (besides the first one) with a singular linear operator called the exact functional discrete scheme (EFDS). A finite subsequence of $m$ terms, called truncated functional discrete scheme (TFDS), is the basis for our algorithm. The approach provides an super-exponential convergence rate as $m \to \infty$. The eigenpairs can be computed in parallel for all given indexes. The algorithm is based on some recurrence procedures including the basic arithmetical operations with the coefficients of some expansions only. This is an exact symbolic algorithm (ESA) for $m=\infty$ and a truncated symbolic algorithm (TSA) for a finite $m$. Numerical examples are presented to support the theory.Comment: 15 page