A new approach for solving nonlinear Thomas-Fermi equation based on fractional order of rational Bessel functions

In this paper, the fractional order of rational Bessel functions collocation method (FRBC) to solve Thomas-Fermi equation which is defined in the semi-infinite domain and has singularity at $x = 0$ and its boundary condition occurs at infinity, have been introduced. We solve the problem on semi-infinite domain without any domain truncation or transformation of the domain of the problem to a finite domain. This approach at first, obtains a sequence of linear differential equations by using the quasilinearization method (QLM), then at each iteration solves it by FRBC method. To illustrate the reliability of this work, we compare the numerical results of the present method with some well-known results in other to show that the new method is accurate, efficient and applicable

Approximative solutions of difference equations

Asymptotic properties of solutions of difference equations of the form $$\Delta^m x_n=a_nf(n,x_{\sigma(n)})+b_n$$ are studied. Using the iterated remainder operator and fixed point theorems we obtain sufficient conditions under which for any solution $y$ of the equation $\Delta^my=b$ and for any real $s\leq 0$ there exists a solution $x$ of the above equation such that $\Delta^kx=\Delta^ky+\mathrm{o}(n^{s-k})$ for any nonnegative integer $k\leq m$. Using a discrete variant of the Bihari lemma and a certain new technique we give also sufficient conditions under which for a given real $s\leq m-1$ all solutions $x$ of the equation satisfy the condition $x=y+\mathrm{o}(n^s)$ where $y$ is a solution of the equation $\Delta^my=b$. Moreover, we give sufficient conditions under which for a given natural $k<m$ all solutions $x$ of the equation satisfy the condition $x=y+u$ for a certain solution $y$ of the equation $\Delta^my=b$ and a certain sequence $u$ such that $\Delta^ku=\mathrm{o}(1)$

Asymptotically polynomial solutions of difference equations of neutral type

Asymptotic properties of solutions of difference equation of the form $$\Delta^m(x_n+u_nx_{n+k})=a_nf(n,x_{\sigma(n)})+b_n$$ are studied. We give sufficient conditions under which all solutions, or all solutions with polynomial growth, or all nonoscillatory solutions are asymptotically polynomial. We use a new technique which allows us to control the degree of approximation

Two-dimensional hydrodynamic core-collapse supernova simulations with spectral neutrino transport. I. Numerical method and results for a 15 M_sun star

Supernova models with a full spectral treatment of the neutrino transport are presented, employing the Prometheus/Vertex neutrino-hydrodynamics code with a ``ray-by-ray plus'' approximation for treating two- (or three-) dimensional problems. The method is described in detail and critically assessed with respect to its capabilities, limitations, and inaccuracies in the context of supernova simulations. In this first paper of a series, 1D and 2D core-collapse calculations for a (nonrotating) 15 M_sun star are discussed, uncertainties in the treatment of the equation of state -- numerical and physical -- are tested, Newtonian results are compared with simulations using a general relativistic potential, bremsstrahlung and interactions of neutrinos of different flavors are investigated, and the standard approximation in neutrino-nucleon interactions with zero energy transfer is replaced by rates that include corrections due to nucleon recoil, thermal motions, weak magnetism, and nucleon correlations. Models with the full implementation of the ``ray-by-ray plus'' spectral transport were found not to explode, neither in spherical symmetry nor in 2D with a 90 degree lateral wedge. The success of previous 2D simulations with grey, flux-limited neutrino diffusion can therefore not be confirmed. Omitting the radial velocity terms in the neutrino momentum equation leads to ``artificial'' explosions by increasing the neutrino energy density in the convective gain layer by about 20--30% and thus the integral neutrino energy deposition in this region by about a factor of two. (abbreviated)Comment: 46 pages plus 13 pages online material; 49 figures; referee's comments included, version accepted by Astronomy & Astrophysic

Impact of double-logarithmic electroweak radiative corrections on the non-singlet structure functions at small x

In the QCD context, the non-singlet structure functions of u and d -quarks are identical, save the initial quark densities. Electroweak radiative corrections, being flavor-dependent, bring further difference between the non-singlets. This difference is calculated in the double-logarithmic approximation and the impact of the electroweak corrections on the non-singlet intercepts is estimated numerically.Comment: 17 pages, no figure

Convergence model of interest rates of CKLS type

summary:This paper deals with convergence model of interest rates, which explains the evolution of interest rate in connection with the adoption of Euro currency. Its dynamics is described by two stochastic differential equations – the domestic and the European short rate. Bond prices are then solutions to partial differential equations. For the special case with constant volatilities closed form solutions for bond prices are known. Substituting its constant volatilities by instantaneous volatilities we obtain an approximation of the solution for a more general model. We compute the order of accuracy for this approximation, propose an algorithm for calibration of the model and we test it on the simulated and real market data