[en] In this manuscript, an approximate method for the numerical solutions of fractional order Cauchy reaction diffusion equations is considered. The concerned method is known as optimal homotopy asymptotic method (OHAM). With the help of the mentioned method, we handle approximate solutions to the aforesaid equation. Some test problems are provided at which the adapted technique has been applied. The comparison between absolute and exact solution are also provided which reveals that the adapted method is highly accurate. For tabulation and plotting, we use matlab software.

[en] Recently developed analytic techniques are introduced for the treatment of decay problems. The one-channel decay problem is treated in detail; the pole and nonexponential contributions are calculated both by numerical techniques and analytically. Conditions for the dominance of the pole contribution are specified. The asymptotic procedures used are developed systematically in a mathematical appendix

[en] Despite their potential importance for understanding astrophysical jets, physically realistic exact solutions for magnetospheres around Kerr black holes have not been found, even in the force-free approximation. Instead approximate analytical solutions such as the Blandford–Znajek (split-)monopole, as well as numerical solutions, have been constructed. In this paper we consider a new approach to the analysis and construction of such magnetospheres. We consider force-free electrodynamics close to the rotation axis of a magnetosphere surrounding a Kerr black hole assuming axisymmetry. This is the region where the force-free approximation should work the best, and where the jets are located. We perform a systematic study of the asymptotic region with (split-)monopole, paraboloidal and vertical asymptotic behaviors. Imposing asymptotics similar to a (split-)monopole, we find under certain assumptions that demanding regularity at the rotation axis and the event horizon restricts solutions of the stream equation so much that it is not possible for a solution to be continuously connected to the static (split-)monopole around the Schwarzschild black hole in the limit where the rotation goes to zero. On the one hand, this result provides independent evidence to the issues discovered with the asymptotics of the Blandford–Znajek (split-)monopole in reference (Grignani G, Harmark T and Orselli M 2018 Phys. Rev. D 98 084056). On the other hand, we also point out possible caveats in our arguments that one could conceivably exploit to amend the perturbative construction of the Blandford–Znajek (split-)monopole. (paper)

[en] Existence, uniqueness and qualitative behavior of the solution to a spatially homogeneous Boltzmann equation for particles undergoing elastic, inelastic and coalescing collisions are studied. Under general assumptions on the collision rates, we prove existence and uniqueness of an L¹ solution. This shows in particular that the cooling effect (due to inelastic collisions) does not occur in finite time. In the long time asymptotic, we prove that the solution converges to a mass-dependent Maxwellian function (when only elastic collisions are considered), to a velocity Dirac mass (when elastic and inelastic collisions are considered) and to 0 (when elastic, inelastic and coalescing collisions are taken into account). We thus show in the latter case that the effect of coalescence is dominating in large time. Our proofs gather deterministic and stochastic arguments. (authors)

[en] Analytic and numerical methods for determining the asymptotics of high-order terms of the 1/n expansion in quantum-mechanical problems are developed. It is shown that this asymptotics is always of the factorial type. The dependence of parameters of the asymptotics on the form of the potential and on the coupling constant is especially analyzed in the vicinity of the point of collision of classical solutions. 23 refs., 8 figs., 3 tabs

[en] Essentially, this article is written to present and analyse the analytical and numerical solutions of the Kadomtsev–Petviashvili (KP) equation arising in plasma physics. We derive the basic set of fluid equations governing the KP equation. The analytical solution, presented on forms of rational functions, hyperbolic functions and trigonometric functions, was analytically investigated while the numerical solution is examined here by utilizing the adaptive moving mesh method on finite differences. The stability of the obtained exact solutions is also presented and analysed. All solutions are found stable on specific intervals. The exact and numerical solutions are compared with each other to show the accuracy of the numerical solution. Under an appropriate choice of parameters, some 2D and 3D figures for the obtained analytical and numerical results are illustrated in order to compare between their accuracy. (paper)

[en] A new approach based on the Adomian decomposition and the Fourier transform is introduced. The method suggests a solution for the well-known magneto-hydrodynamic (MHD) Jeffery-Hamel equation. Results of Adomian decomposition method combined with Fourier transform are compared with exact and numerical methods. The FTADM as an exclusive and new method satisfies all boundary and initial conditions over the entire spatial and temporal domains. Moreover, using the FTADM leads to rapid approach of approximate results toward the exact solutions is demonstrated. The second derivative of Jeffery-Hamel solution related to the similar number of items of recursive terms under a vast spatial domain shows the maximum error in the order of ${10}^{-<!--\; ???\; -->5}$ comparing to exact and numerical solutions. The results also imply that the FTADM can be considered as a precise approximation for solving the third-order nonlinear Jeffery-Hamel equations. (paper)

[en] The cumulative beam breakup problem excited by the resistive-wall wake is formulated. An approximate analytic method of finding the asymptotic behavior for the transverse bunch displacement is developed and solved. Comparison between the asymptotic analytical expression and the direct numerical solution is presented. Good agreement is found. The criterion of using the asymptotic analytical expression is discussed

[en] One of the problems in creating of computers based on residue number system (RNS) is a problem of numbers translation from positional number system into the RNS and back. Accordingly, one approach to solve this problem is to choose the values of RNS bases. It is possible that this approach will help to compare the current value of numbers and determine the sign, without converting them to the positional number system

[en] Authors present periodic and antiperiodic solutions, composite solutions resulting from a nonlinear superposition of generalized separable and traveling wave solutions, and others. Some results are extended to nonlinear delay reaction-diffusion equations with time-varying delay