A New Special 15-Step Block Method for Solving General Fourth Order Ordinary Differential Equations

A new higher-implicit block method for the direct numerical solution of fourth order ordinary differential equation is derived in this research paper. The formulation of the new formula which is 15-step, is achieved through interpolation and collocation techniques. The basic numerical properties of the method such as zero-stability, consistency and A-stability have been examined. Investigation showed that the new method is zero stable, consistent and A-stable, hence convergent. Test examples from recent literature have been used to confirm the accuracy of the new method.

Wigner-function-based solution schemes for electromagnetic wave beams in fluctuating media

Electromagnetic waves are described by Maxwell’s equations together with the constitutive equation of the considered medium. The latter equation in general may introduce complicated operators. As an example, for electron cyclotron (EC) waves in a hot plasma, an integral operator is present. Moreover, the wavelength and computational domain may differ by orders of magnitude making a direct numerical solution unfeasible, with the available numerical techniques. On the other hand, given the scale separation between the free-space wavelength $$\lambda _0$$ λ 0 and the scale L of the medium inhomogeneity, an asymptotic solution for a wave beam can be constructed in the limit $$\kappa = 2\pi L / \lambda _0 \rightarrow \infty$$ κ = 2 π L / λ 0 → ∞ , which is referred to as the semiclassical limit. One example is the paraxial Wentzel-Kramer-Brillouin (pWKB) approximation. However, the semiclassical limit of the wave field may be inaccurate when random short-scale fluctuations of the medium are present. A phase-space description based on the statistically averaged Wigner function may solve this problem. The Wigner function in the semiclassical limit is determined by the wave kinetic equation (WKE), derived from Maxwell’s equations. We present a paraxial expansion of the Wigner function around the central ray and derive a set of ordinary differential equations (phase-space beam-tracing equations) for the Gaussian beam width along the central ray trajectory.

Resurgence in the O(4) sigma model

We analyze the free energy of the integrable two dimensional O(4) sigma model in a magnetic field. We use Volin’s method to extract high number (2000) of perturbative coefficients with very high precision. The factorial growth of these coefficients are regulated by switching to the Borel transform, where we perform several asymptotic analysis. High precision data allowed to identify Stokes constants and alien derivatives with exact expressions. These reveal a nice resurgence structure which enables to formulate the first few terms of the ambiguity free trans-series. We check these results against the direct numerical solution of the exact integral equation and find complete agreement.

A Family of Functionally-Fitted Third Derivative Block Falkner Methods for Solving Second-Order Initial-Value Problems with Oscillating Solutions

One of the well-known schemes for the direct numerical integration of second-order initial-value problems is due to Falkner (Falkner, 1936. Phil. Mag. S. 7, 621). This paper focuses on the construction of a family of adapted block Falkner methods which are frequency dependent for the direct numerical solution of second-order initial value problems with oscillatory solutions. The techniques of collocation and interpolation are adopted here to derive the new methods. The study of the properties of the proposed adapted block Falkner methods reveals that they are consistent and zero-stable, and thus, convergent. Furthermore, the stability analysis and the algebraic order conditions of the proposed methods are established. As may be seen from the numerical results, the resulting family is efficient and competitive compared to some recent methods in the literature.

Simulation of the Scattering Process Effect on High-Temperature Jet Radiation Intensity by the Monte Carlo Method

The purpose of the paper was to study the scattering effect in the gas jet model on the angular dependence of the radiation intensity. Along with the Monte Carlo method used as the main calculation method, we applied a direct numerical solution of the equation of radiation transfer in a non-scattering medium, known as the discrete directions method, or Ray-Tracing Method. We compared the results obtained using the two methods when calculating a non-scattering medium in order to verify the solution according to the Monte Carlo scheme. Furthermore, we calculated the medium with an increasing value of the local scattering coefficient. Findings of research show the significant effect of scattering processes on the redistribution of radiation energy from the surface of the object. The computational algorithm is implemented on the CUDA C architecture. The use of analytical jet models, e.g. according to Abramovich's theory, and the results of calculations in the computational gas dynamics packages makes it possible to calculate the values of the radiation intensity for a wide class of objects

A Numerical Method for Calculating the Nonlinear Elastic Characteristic of Longitudinal-Transverse Transducers

This paper presents a numerical method for studying the stress-strain state of longitudinal-transverse transducers and obtaining their nonlinear elastic characteristic. The authors propose a mathematical model that uses a direct numerical solution of the boundary value problem based on the plain curved rod equations in MATLAB. The stress-strain state and the nonlinear elastic characteristic of the system are obtained using a method of successive loading based on linearized equations of the curved rod. The proposed model can be considered as an initial approximation to the solution of the spatial problem of the longitudinal-torsional transducer.

An Optimized 5-Point Block Formula for Direct Numerical Solution of First Order Stiff Initial Value Problems

In this research article, we focus on the formulation of a 5-point block formula for solving first order ordinary differential equations (ODEs). The method is formulated via interpolation and collocation approach using power series expansion as the approximate solution. It has been established that the derived method is of order six. Basic properties such zero and absolute stabilities, convergence, order and error constant have also been investigated. The accuracy of the method was verified on some selected stiff IVPs, compared with some existing methods (DIBBDF, SDIBBDF, BBDF(4), BBDF(5) and odes15s) and test performance showed that the new method is viable.