A modified equation of geodesic deviation

The equation of geodesic deviation is derived under the assumption that the geodesics are neighbouring, but their rate of separation is arbitrary, corresponding, physically, to the relative velocity of two test particles approaching that of light. Some of the consequences of the new equation are discussed.     

A modified Lippmann-Schwinger equation for Coulomb-like interactions

It is known that amplitudes which differ from the Coulomb one by an overall phase factor and by a distribution with a support at zero scattering angle, describe the same scattering process. We utilize this fact to derive new partial-wave expansions, which have finite expansion coefficients, for amplitudes of Coulomb-like interactions. A modified form of the Lippmann-Schwinger equation is derived. For the case of the Coulomb interaction this equation leads to a different amplitude from the Coulomb one, but equivalent to it as both describe the same scattering process. The method can be extended to derive (free of infinities) partial-wave expansions of some field theoretical amplitudes.     

A modified diffusion equation for room-acoustic predication

This letter presents a modified diffusion model using an Eyring absorption coefficient to predict the reverberation time and sound pressure distributions in enclosures. While the original diffusion model [Ollendorff, Acustica 21, 236-245 (1969); J. Picaut et al., Acustica 83, 614-621 (1997); Valeau et al., J. Acoust. Soc. Am. 119, 1504-1513 (2006)] usually has good performance for low absorption, the modified diffusion model yields more satisfactory results for both low and high absorption. Comparisons among the modified model, the original model, a geometrical-acoustics model, and several well-established theories in terms of reverberation times and sound pressure level distributions, indicate significantly improved prediction accuracy by the modification.     

A modified equation for single-mode superradiance kinetics

A new method is proposed for decoupling the chain of quantum equations, which incorporates the fluctuations in the population difference between levels in the atoms near the maximum in the superradiance intensity. A new system of equations is derived that describes a broader and more unsymmetrical superradiance pulse than does the results of [1–4]. The new solutions improve the agreement between theory and experiment [5].Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 27, No. 1, pp. 28–33, January, 1984.     

A hodograph transformation which applies to the Boyer–Finley equation

A hodograph transformation for a wide family of multidimensional nonlinear partial differential equations is presented. It is used to derive solutions of the Boyer–Finley equation (dispersionless Toda equation), which are not group invariant, and the corresponding family of explicit ultra-hyperbolic selfdual vacuum spaces.     

A nonstandard numerical method for the modified KdV equation

A linearly implicit nonstandard finite difference method is presented for the numerical solution of modified Korteweg–de Vries equation. Local truncation error of the scheme is discussed. Numerical examples are presented to test the efficiency and accuracy of the scheme.     

Some extensions of the Einstein–Dirac equation

We considered an extension of the standard functional for the Einstein–Dirac equation where the Dirac operator is replaced by the square of the Dirac operator and a real parameter controlling the length of spinors is introduced. For one distinguished value of the parameter, the resulting Euler–Lagrange equations provide a new type of Einstein–Dirac coupling. We establish a special method for constructing global smooth solutions of a newly derived Einstein–Dirac system called the CL-Einstein–Dirac equation of type II (see Definition 3.1).     

A new integrable equation combining the modified KdV equation with the negative-order modified KdV equation: multiple soliton solutions and a variety of solitonic solutions

A new third-order integrable equation is constructed via combining the recursion operator of the modified KdV equation (MKdV) and its inverse recursion operator. The developed equation will be termed the modified KdV-negative order modified KdV equation (MKdV–nMKdV). The complete integrability of this equation is confirmed by showing that it nicely possesses the Painlevé property. We obtain multiple soliton solutions for the newly developed integrable equation. Moreover, this equation enjoys a variety of solutions which include solitons, peakons, cuspons, negaton, positon, complexiton and other solutions.     

Fast integral equation methods for the modified Helmholtz equation

We present integral equation methods for the solution to the two-dimensional, modified Helmholtz equation, u(x) ? α²Δu(x) = 0, in bounded or unbounded multiply-connected domains. We consider both Dirichlet and Neumann problems. We derive well-conditioned Fredholm integral equations of the second kind, which are discretized using high-order, hybrid Gauss-trapezoid rules. Our fast multipole-based iterative solution procedure requires only O(N) operations, where N is the number of nodes in the discretization of the boundary. We demonstrate the performance of our methods on several numerical examples, and we show that they have both the ability to handle highly complex geometry and the potential to solve large-scale problems.     

A semi-empirical modified hypernetted chain equation for the one-component plasma

The hypernetted chain (HNC) equation for the one-component plasma is corrected by including a single-parameter expression for the bridge graphs whose functional form ensures that the correct long-wavelength limit of the HNC results is retained. The parameter is chosen to improve the short-range behaviour.     

Breather solutions of modified Benjamin-Bona-Mahony equation

New two-component vector breather solution of the modified Benjamin-Bona-Mahony(MBBM)equation is considered.Using the generalized perturbation reduction method,the MBBM equation is reduced to the coupled nonlinear Schr¨odinger equations for auxiliary functions.Explicit analytical expressions for the profile and parameters of the vector breather oscillating with the sum and difference of the frequencies and wavenumbers are presented.The two-component vector breather and single-component scalar breather of the MBBM equation is compared.     

Solitons of the modified KdV equation

We determine all reflectionless potentials for the one-dimensional charge symmetric Dirac operator, identify them as solitons of the modified KdV equation, and give the connection to the KdV solitons. An associated dynamical system is shown to be completely integrable.     

A numerical solution of the vibration equation using modified decomposition method

In the present paper the well-known vibration equation for very large membrane with the help of powerful modification of Adomian decomposition method proposed by Wazwaz [A reliable modification of Adomian decomposition method, Applied Mathematics and Computation 102 (1999) 77-86] has been solved. By using initial value, the explicit solutions of the equation for different cases have been derived, which accelerate the rapid convergence of the series solution. The present method performs extremely well in terms of efficiency and simplicity. Numerical results for different particular cases of the problem are presented graphically.     

Numerical algorithm for the time fractional Fokker–Planck equation

Anomalous diffusion is one of the most ubiquitous phenomena in nature, and it is present in a wide variety of physical situations, for instance, transport of fluid in porous media, diffusion of plasma, diffusion at liquid surfaces, etc. The fractional approach proved to be highly effective in a rich variety of scenarios such as continuous time random walk models, generalized Langevin equations, or the generalized master equation. To investigate the subdiffusion of anomalous diffusion, it would be useful to study a time fractional Fokker–Planck equation. In this paper, firstly the time fractional, the sense of Riemann–Liouville derivative, Fokker–Planck equation is transformed into a time fractional ordinary differential equation (FODE) in the sense of Caputo derivative by discretizing the spatial derivatives and using the properties of Riemann–Liouville derivative and Caputo derivative. Then combining the predictor–corrector approach with the method of lines, the algorithm is designed for numerically solving FODE with the numerical error O(k^min{1+2α,2})+O(h²), and the corresponding stability condition is got. The effectiveness of this numerical algorithm is evaluated by comparing its numerical results for α=1.0 with the ones of directly discretizing classical Fokker–Planck equation, some numerical results for time fractional Fokker–Planck equation with several different fractional orders are demonstrated and compared with each other, moreover for α=0.8 the convergent order in space is confirmed and the numerical results with different time step sizes are shown.     

A modified characteristic equation for stability analysis of periodic reciprocal systems

A linear reciprocal system with periodic coefficients is stable if the system monodromy matrix has simple structure and eigenvalues all of modulus unity. Under the assumption that the former condition is true, it is proved in this paper that this criterion is equivalent to the condition that the roots, α, of an algebraic equation all lie in the interval −1 ⩽ α ⩽ 1, and an explicit scheme is presented for the derivation of this algebraic equation in terms of the coefficients of the characteristic equation of the system monodromy matrix. The modified criterion provides considerable computational advantage over the usual form of the criterion.     

Evaporation of droplets in the two-dimensional Ginzburg–Landau equation

We consider the problem of coarsening in two dimensions for the real (scalar) Ginzburg–Landau equation. This equation has exactly two stable stationary solutions, the constant functions +1 and −1. We assume most of the initial condition is in the “−1” phase with islands of “+1” phase. We use invariant manifold techniques to prove that the boundary of a circular island moves according to Allen–Cahn curvature motion law. We give a criterion for non-interaction of two arbitrary interfaces and a criterion for merging of two nearby interfaces.