[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] 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] 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] The time-fractional diffusion-wave equation is considered. The time-fractional diffusion equation is obtained from the standard diffusion equation by replacing the first-order time derivative with a fractional derivative of order α element of (0,2]. The fractional derivative is described in the Caputo sense. This paper presents the analytical solutions of the fractional diffusion equations by an Adomian decomposition method. By using initial conditions, the explicit solutions of the equations have been presented in the closed form and then their numerical solutions have been represented graphically. Four examples are presented to show the application of the present technique. The present method performs extremely well in terms of efficiency and simplicity

[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

[en] A formal asymptotic expansion of a solution of the initial problem for a singularly perturbed differential-operational nonlinear equation in a small parameter has been constructed in the critical case. Splash functions of and boundary functions have been estimated of found and assessment of the residual member of the expansion has been obtained

[en] An asymptotic expansion for the solution to a nonhomogeneous difference equation of general type is obtained. The influence of the roots of the characteristic equation is taken into account. The asymptotic behavior of the remainder is established, depending on the asymptotics of the nonhomogeneous term of the equation.