Langevin simulation provides an effective way to study collisional effects in beams by reducing the six-dimensional Fokker-Planck equation to a group of stochastic ordinary differential equations. These resulting equations usually have multiplicative noise since the diffusion coefficients in these equations are functions of position and time. Conventional algorithms, e.g. Euler and Heun, give o...

Many rationally parametrized elliptic modular equations are derived. Each comes from a family of elliptic curves attached to a genus-zero congruence subgroup Γ0(N), as an algebraic transformation of elliptic curve periods, parametrized by a Hauptmodul (function field generator). The periods satisfy a Picard–Fuchs equation, of hypergeometric, Heun, or more general type; so the new modular equati...

Thermodynamic properties of a spin ice model on a Kagomé lattice are obtained from dynamic simulations and compared with properties in square lattice spin ice. The model assumes three-component Heisenberg-like dipoles of an array of planar magnetic islands situated on a Kagomé lattice. Ising variables are avoided. The island dipoles interact via long-range dipolar interactions and are restricte...

The Kardar–Parisi-Zhang (KPZ) equation is examined using the recently published leapfrog–hopscotch (LH) method as well most standard forward time centered space (FTCS) scheme and Heun method. methods are verified by reproducing an analytical solution. performance of each then compared calculating average maximum differences among results displaying runtimes. Numerical tests show that due to spe...

Of special interest are those Lamé equations with finite monodromy group, having therefore algebraic solutions, studied by Baldassarri, Beukers and van der Waall, Chudnovsky and Chudnovsky, Dwork and many others (cf. [2], [4], [8] just to mention some papers). Lamé equations also occur in the context of Grothendieck’s p-curvature conjecture (cf. [8, p. 15]). This conjecture says that if the p-c...

We seek numerical methods for second-order stochastic differential equations that accurately reproduce the stationary distribution for all values of damping. A complete analysis is possible for linear second-order equations (damped harmonic oscillators with noise), where the statistics are Gaussian and can be calculated exactly in the continuous-time and discrete-time cases. A matrix equation i...

It is known experimentally that stable few-body clusters containing negatively-charged electrons (e) and positively-charged holes (h) can exist in low-dimensional semiconductor nanostructures. In addition to the familiar exciton (e+h), three-body ‘charged excitons’ (2e+h and 2h+e) have also been observed. Much less is known about the properties of such charged excitons since three-body problems...

We derive analytical solutions of the Schrödinger wave equation for some q-deformed potentials. Most of these solutions are obtained in terms of Heun functions. However, for some particular cases the solutions are obtained in terms of geometric and hypergeometric functions. Different observations are realized for each potential. However, for all the cases which we study the oscillations are the...

We obtain isomonodromic transformations for Heun’s equation by generalizing Darboux transformation, and we find pairs and triplets of Heun’s equation which have the same monodromy structure. By composing generalized Darboux transformations, we establish a new construction of the commuting operator which ensures finite-gap property. As an application, we prove conjectures in part III.

We utilize the travelling-wave Ansatz to obtain novel analytical solutions linear diffusion–reaction equation. The reaction term is a function of time and space simultaneously, firstly in Lorentzian form secondly cosine form. new contain Heun functions first case Mathieu for second case, therefore are highly nontrivial. use these test some non-conventional explicit stable numerical methods agai...