Two-term Kummer function solutions of the 1D Schrödinger equation

Some remarks on the location of non-asymptotic zeros of Whittaker and Kummer hypergeometric functions

The paper is devoted to the study of the function E γ ρ,μ(z) defined for complex ρ, μ, γ (Re(ρ) > 0) by which is a generalization of the classical Mittag-Leffler function E ρ,μ(z) and the Kummer confluent hypergeometric function Φ(γ, μ; z). The properties of E γ ρ,μ(z) including usual differentiation and integration, and fractional ones are proved. Further the integral operator with such a function kernel is studied in the space L(a, b). Compositions of the Riemann–Liouville fractional integration and differentiation operators with E γ ρ,μ,ω;a+ are established. An analogy of the semigroup property for the composition of two such operators with different indices is proved, and the results obtained are applied to construct the left inversion operator to the operator E γ ρ,μ,ω;a+. Since, for γ = 0, E 0 ρ,μ,ω;a+ coincides with the Riemann–Liouville fractional integral of order μ, the above operator and its inversion can be considered as generalized fractional calculus operators involving the generalized Mittag-Leffler function E γ ρ,μ(z) in the kernels. Similar assertions are presented for the integral operators containing the Mittag-Leffler and Kummer functions, E ρ,μ(z) and Φ(γ, μ; z), in the kernels, and applications are given to obtain solutions in closed form of the integral equations of the first kind.

The basic parabolic cylinder (bpc) distributions of the firstBasic parabolic cylinder (bpc) distributionof the first kind and second kindsBasic parabolic cylinder (bpc) distributionof the second kind were derived by OgasawaraOgasawara (Commun Stat Theor Methods [17]) to obtain the absolute momentsMomentsabsolute of the normal variable underStripe truncation stripe (zebraicTruncationzebraic or tigerishTruncationtigerish) truncationTruncationstripe. The bpc distribution of the third kindBasic parabolic cylinder (bpc) distributionof the third kind or simply bpc is introduced as a general bpc, whose special cases are the bpc’s of the first and second kinds. TheWeighted or incomplete parabolic cylinder function weighted or incomplete parabolic cylinder functionParabolic cylinder functionincomplete is obtained using theWeighted or incomplete Kummer confluent hypergeometric function weighted or incomplete Kummer confluent hypergeometric functionKummer confluent hypergeometric function given by OgasawaraOgasawara [J Multivar Anal [18]) to have the absolute momentsMomentsabsolute. The multivariate bpc distributionMultivariate basic parabolic cylinder (bpc) distribution is also derived to obtain the absolute momentsMomentsabsolute of the normal vectorNormalvector underTruncationsectional sectional truncationSectional truncation, which is the multivariate version ofTruncationstripe stripe truncationStripe truncation.

Examples of the application of confluent hypergeometric functions in miscellaneous areas of theoretical physics are presented. Emphasis is placed on the use of Kummer function in the field of microwaves: the cases of normal and slow rotationally symmetric TE modes propagation in the azimuthally magnetized circular ferrite waveguide are considered. Some of the properties of function mentioned in the complex (real) domain, of importance in the solution of boundary-value problem stated for normal (slow) waves, are established. A theorem for the identity of purely imaginary and real zeros of the complex resp. real Kummer function for certain parameters, is proved numerically. The terms for wave transmission are obtained as four bilaterally open intervals of variation of quantities, specifying the fields. It turns out that the normal (slow) modes may exist in one (two) region(s). The theoretically predicted phase curves for the first waves of the two TE sets examined show that the structure explored is suitable for ferrite control components design.

The borderlines of the area in which the circular waveguide, entirely filled with azimuthally magnetized ferrite, works as a phase shifter for the normal TE <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">01</sub> mode, are determined. For the purpose the special features of the cut-off regime of the configuration and of its limiting one, observed at higher frequencies in which the propagation stops, provided the magnetic bias is negative, are used. For a fixed numerical equivalent of the normalized in an appropriate way guide radius, the off-diagonal ferrite permeability tensor element and the phase constant of the wave are counted by the roots of the structure's characteristic equation, derived through the Kummer confluent hypergeometric function. This is done repeatedly for a varying negative (positive) value of the imaginary part of its complex first parameter. The process continues, until the computed value of the element coincides with that of the same, corresponding to the cut-off state for the radius chosen (to the limiting one, if the latter is greater than certain number) with a prescribed accuracy. The relevant calculated numerical equivalent of the phase constant is accepted as the one, searched for. The differential phase shift equals the constant mentioned in the first case, whereas in the second one, it is found from it by a simple arithmetic. Changing the normalized radius, the boundaries of the domain studied are traced. The outcomes are presented graphically in a normalized form.

The propagation of circularly symmetrical mode in an infinitely long thin-wire helix, wound on a latching ferrite cylinder, magnetised azimuthally to remanence and surrounded by homogeneous, isotropic, lossless dielectric and perfectly conducting metallic shield is studied analytically. The characteristic equation of the system, involving the normalized phase constant, structure geometry, parameters of both media and the frequency is derived in terms of modified Bessel functions and Kummer confluent hypergeometric functions. The nonreciprocal phase characteristics of the structure, assuming Marconi latching ferrite L1A and thermal conducting dielectrics are presented in normalized form for opposite directions of remanent flux, showing application of the system as practical remanent latching phaser.

ABSTRACTGeneralized Abel type integral equations with Gauss, Kummer's and Humbert's confluent hypergeometric functions in the kernel and generalized Abel type integral equations with localized fractional integrals are considered. The left-hand sides of these equations are inversed by using generalized fractional derivatives. Explicit solutions of the equations in the class of locally summable functions are obtained. They are represented in terms of hypergeometric functions. Asymptotic power exponential type expansions of the generalized and localized fractional integrals are obtained. The base solutions of the generalized Abel type integral equation are given in the form of asymptotic series.

The climate change mitigation benefits of biochar systems arise largely from carbon storage in biochar. However, while biochar is increasingly recognized as a carbon dioxide removal technology, there are on-going scientific discussions on how to estimate the persistence of biochar carbon when biochar is used in soils. Estimates vary from decades to millennia, building on different modelling approaches and evidence. Here, we revisited the persistence estimates derived from extrapolation of biochar incubation experiments, with the aims of making incubation data available, modelling choices transparent, and results reproducible. An extensive dataset of biochar incubations, including 129 biochar decomposition time series, was compiled and is made available alongside code for its analysis. Biochar persistence correlations were sensitive to data selection procedures and to the curve fitting modelling step, while soil temperature adjustments methods had less impact. Biochar H/C ratio remained the main predictor of biochar persistence, in line with previous research, regardless of the extrapolation assumptions (multi-pool exponential functions or power function) used in curve fitting. The relation between H/C and percentage of biochar carbon remaining after 100 years (BC100) was better explained by a power model than a linear model, with R2 values between 0.5 and 0.9. Using multi-pool exponential functions, estimated BC100 varied between 90 % and 60 % for H/C from 0 to 0.7. However, using power functions, BC100 was constrained between 90 % and 80 % for the same H/C range. Additional information about the biochar, the pyrolysis conditions or the environmental incubation conditions did not significantly increase explained variance. Notably, the dataset lacks observations at H/C ratios below 0.2, of biochar made from manure and biosolids, biochar from processes other than slow pyrolysis, field studies, and incubation temperatures below 10 °C, which should guide future experimental work. The detailed analysis performed in this study does not cast doubts on the longevity of biochar carbon storage; rather, it confirms previous knowledge by critically examining the modelling, elucidating the assumptions and limitations, and making the analysis fully reproducible. There is a need for further interdisciplinary work on integration of various theories and approaches to biochar persistence, ultimately leading to the formulation of policy-relevant conclusions.

Rational Bezier surface is a widely used surface fitting tool in CAD. When all the weights of a rational Bezier surface go to infinity in the form of power function, the limit of surface is the regular control surface induced by some lifting function, which is called toric degenerations of rational Bezier surfaces. In this paper, we study on the degenerations of the rational Bezier surface with weights in the exponential function and indicate the difference of our result and the work of Garc´ia-Puente et al. Through the transformation of weights in the form of exponential function and power function, the regular control surface of rational Bezier surface with weights in the exponential function is defined, which is just the limit of the surface. Compared with the power function, the exponential function approaches infinity faster, which leads to surface with the weights in the form of exponential function degenerates faster.

Estimating WRF model computation time is necessary because of the need for domain expansion for weather prediction. However, the optimum computation time is not simply gained by enlarging the domain and adding processor numbers. Thus, an investigation was carried out to determine the correlation between computation time on domain size, number of grids, and number of processors used to run the WRF model. This study uses a collection of computation time as the data input from running the WRF model with two domain group ratios, 2: 1 and 1: 1, and various processors. Negative Exponential Function (NEF) and Power Function (PF) as exponential decay functions are evaluated to represent the curve formed from the computation time against the number of processors in one domain case. This study also evaluates the speed up and efficiency of the use of processor numbers against the tested domains. NEF represents the decrease in computation time curve in a domain case better than PF since this function has a steeper slope, better initial value, and k constant that keeps the computation time falling below 0. The computation time can be optimally saved by adding approximately eight processors at the same domain ratio with four times larger grid size and the same amount of grid number. Investigations for other domain ratios need to be carried out to determine the characteristics of computation time on the number of processors, grid size, and domain size.

Shear strength of granular materials based on fractal fragmentation of particles

Effects of erodent flow energy and local exposure time on the erosion of cement-based composites at high-speed hydro-abrasive flow

A one-dimensional integral model that predicts the auto-ignition time of wood under linearly increasing radiant heat flux is developed. The solution to the integral model for the ignition time is compared with measured data. Reasonable comparative results are shown for the ignition time of wood exposed to different heat flux increase rates (HFIR). Both the solutions and the measured data also show that the ignition time can be expressed as a power function and the power exponent is about 0.67.

Computing the Kummer function U(a, b, z) for small values of the arguments

In this paper, we solve the Duffin–Kemmer–Petiau (DKP) equation in the presence of the Woods–Saxon potential barrier and well for spin-one particles. We derive the scattering solution in terms of the Gaussian hypergeometric function [Formula: see text], the regularized Gaussian hypergeometric function [Formula: see text], and the Gamma function [Formula: see text]. Our analysis reveals the presence of transmission resonances. To observe these resonances, we calculate and plot the transmission T and reflection R coefficients for various parameters of the Woods–Saxon potential barrier. Our results are compared with those obtained for the square potential barrier and the cusp potential barrier, which represent limiting cases of the Woods–Saxon potential barrier. Furthermore, we investigate the bound state solutions, determining the critical turning point [Formula: see text] to study particle–antiparticle creation and compute the norm N. Finally, we also compare our results with those obtained for the square potential well and the cusp potential well, confirming that pair creation occurs in both potential wells.