Volume 15, Issue 7-8

Using integration of deterministic, matrix-valued functions with respect to vector-valued, volatility modulated Lévy bases, we construct random vector fields on R n . In the statistically homogeneous case, the vector field is given as a convolution. With applications to turbulence in mind, the convolution kernel is expressed in terms of the energy spectrum. The theory is applied to atmospheric boundary layer turbulence where, in particular, the Shkarofsky correlation family (a generalisation of the Matérn correlation family) is shown to fit the data well. A modification of the Shkarofsky correlation family, which reproduces a more rigorously derived result on the small-scale behaviour of fully developed turbulence, is introduced. Since turbulence possesses structure across a wide range of length scales, simulation is non-trivial. Using a smooth partition of unity, a simple algorithm is derived to decompose the simulation problem into computationally tractable subproblems. Applications within the wind energy industry are suggested.

In this paper, numerical simulations of coupled one-dimensional Gray-Scott model for pulse splitting process, self-replicating patterns and unsteady oscillatory fronts associated with autocatalytic reaction-diffusion equations as well as homoclinic stripe patterns, self-replicating pulse and other chaotic dynamics in Gierer-Meinhardt equations [12] are investigated. Our major approach is the use of higher order exponential time differencing Runge-Kutta (ETDRK) scheme that was earlier proposed by Cox and Matthews [5], which was later presented as a result of instability in a modified form by Krogstad [24] to solve stiff semi-linear problems. The semi-linear problems under consideration in this context are split into linear, which harbors the stiffest part of the dynamical system and nonlinear part that varies slowly than the linear part. For the spatial discretization, we employ higher-order symmetric finite difference scheme and solve the resulting system of ODEs with higher-order ETDRK method. Numerical examples are given to illustrate the accuracy and implementation of the methods, results and error comparisons with other standard schemes are well presented.

In this paper, the fractional derivatives in the sense of modified Riemann–Liouville derivative and the functional variable method, exp-function method and -expansion method are used to construct exact solutions for some nonlinear partial fractional differential equations via the nonlinear space-time fractional (2 + 1)-dimensional breaking soliton equations. These methods are also applied to derive a variety of travelling wave solutions with distinct physical structures for this nonlinear fractional equation. As a result, some new exact solutions for them are obtained. The three methods demonstrate power, reliability and efficiency.