e-Testing with Interactive Images - Opportunities and Challenges

KANTBP 4M Program for Solving the Scattering Problem for a System of Ordinary Second-Order Differential Equations

“…For the general multichannel scattering problem, the nonsymmetry in the coefficient functions leads to the different numbers of open channels in the left-hand and right-hand asymptotic regions, i.e., a scattering matrix composed of square matrices of reflection amplitudes and rectangular matrices of transmission amplitudes [1]. To solve this problem, as well as the problems of bound or metastable states [2], we elaborated a new version of the program KANTBP 4M [3] implementing a finite element method (FEM) with Lagrange or Hermite interpolation polynomials [4] in the computer algebra system (CAS) MAPLE.…”

Construction of Multivariate Interpolation Hermite Polynomials for Finite Element Method

“…In recent decade we presented a new algorithm for the calculation of high-order Lagrange and Hermite interpolation polynomials (LIP and HIP) [1] of the simplex in analytical form, their classification and a typical example of the triangle element for high-accuracy finite element method (FEM) schemes [2][3][4]. We have illustrated the efficiency of the schemes using high-order accuracy LIP and HIP on benchmark calculations of exactly solvable boundary value problems(BVPs)for a triangle membrane, a hypercube and a helium atom [5,6].…”

“…Such a choice of values allows us to construct a piecewise polynomial basis continuous at the boundaries of the finite elements together with the derivatives up to a given order. In the case of a d-dimensional cube, it is shown that the basis functions are determined by products of d HIPs depending on each of the d variables given by means of the recurrence relations in analytical form [2]. Using this fact we propose a new symbolic algorithm implemented in Maple for analytical calculation of the basis functions, i.e.…”

“…, κ max r − 1, where κ max r is referred to as the multiplicity [1] of the node z r , are determined by Eqs. (2). These 1D HIPs are calculated analytically from the recurrence relations derived in [2]…”