Spline algorithms are described for solving the radial equation for continuum states. The Galerkin method leads to a generalized eigenvalue problem for which the eigenvalue is known so that inverse iteration can be used to determine the eigenvector. Three cases are considered: the equation whose solution is sin κr, the Coulomb problem, and the hydrogen scattering problem. Plots are presented for the first two cases that show the dependence of the error in the phase shift on spline parameters and execution time. The results for the scattering problem are compared with earlier values.
In this article, we have briefly examined the entropy generation in magnetohydrodynamic (MHD) Eyring-Powell fluid over an unsteady oscillating porous stretching sheet. The impact of thermal radiation and heat source/sink are taken in this investigation. The impact of embedded parameters on velocity function, temperature function, entropy generation rate, and Bejan number are deliberated through graphs, and discussed as well. By studying the entropy generation in magnetohydrodynamic Eyring-Powell fluid over an unsteady oscillating porous stretching sheet, the entropy generation rate is reduced with escalation in porosity, thermal radiation, and magnetic parameters, while increased with the escalation in Reynolds number. Also, the Bejan number is increased with the escalation in porosity and magnetic parameter, while increased with the escalation in thermal radiation parameter. The impact of skin fraction coefficient and local Nusselt number are discussed through tables. The partial differential equations are converted to ordinary differential equation with the help of similarity variables. The homotopy analysis method (HAM) is used for the solution of the problem. The results of this investigation agree, satisfactorily, with past studies.
Spline-based methods for continuum state wavefunctions are evaluated by applying these techniques to the study of the resonance structure in the photoionisation cross section of He, both from the 1s2 1S ground state and the 1s2p 1P excited state. The Galerkin method is used for obtaining approximate solutions of Schrodinger's equation for a wavefunction defined in terms of a single continuum orbital and a set of bound orbitals. Eigenvectors of interaction matrices for a range of energies are obtained using inverse iteration. This method is an extension of the MCHF method in which the SCF procedure allows bound orbitals to vary with the energy. Photoionisation cross sections, resonance positions and widths are compared with other theories and experiment. Good agreement is obtained, but the studies suggest that 'weakly closed' channels be treated in the same manner as the continuum orbitals, reducing the iterative SCF nature of the calculation and expanding the dimension of the interaction matrices.
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