Abstract Using approximation techniques, long-wave length framework and boundary-layer, the effects of electrostatic force induced shear stress on flow behavior down an inclined solid substrate are investigated. In general case, considered model accounts in presence inertia regime streamwise viscous diffusion with influence normal electric field imposed stress. Galerkin weighted residual, two c...

The dynamics of patterns in large two-dimensional domains remains a challenge in nonequilibrium phenomena. Often it is addressed through mild extensions of one-dimensional equations. We show that full two-dimensional generalizations of the latter can lead to unexpected dynamic behavior. As an example we consider the anisotropic Kuramoto-Sivashinsky equation, which is a generic model of anisotro...

An algorithm is presented to integrate nonlinear partial differential equations, which is particularly useful when accurate estimation of spatial derivatives is required. It is based on an analytic approximation method, referred to as distributed approximating functionals (DAF's), which can be used to estimate a function and a finite number of derivatives with a specified accuracy. As an applic...

In this work we investigate a technique for accelerating convergence of adjoint–based optimization of PDE systems based on a nonlinear change of variables in the control space. This change of variables is accomplished in the differentiate–then–discretize approach by constructing the descent directions in a control space not equipped with the Hilbert structure. We show how such descent direction...

Most evolution equations are partially integrable and, in order to explicitly integrate all possible cases, there exist several methods of complex analysis, but none is optimal. The theory of Nevanlinna and Wiman-Valiron on the growth of the meromorphic solutions gives predictions and bounds, but it is not constructive and restricted to meromorphic solutions. The Painlevé approach via the a pri...

Modern dynamical systems theory has previously had little to say about finite difference and finite element approximations of partial differential equations (pdes) [1]. However, recently I have shown one way that centre manifold theory may be used to create and support the spatial discretisation of pdes such as Burgers’ equation [2] and the Kuramoto-Sivashinsky equation [3]. In this paper the g...

We process snapshots of trajectories of evolution equations with intrinsic symmetries, and demonstrate the use of recently developed eigenvector-based techniques to successfully quotient out the degrees of freedom associated with the symmetries in the presence of noise. Our illustrative examples include a one-dimensional evolutionary partial differential (the Kuramoto-Sivashinsky) equation with...

A simple model for dendritic growth is given by S2d'" + 9' — cos(9). For S ss 1 we prove that there is no bounded, monotonic solution which satisfies d(-oo) = -7t/2 and Q(oo) = n/2. We also investigate the existence of bounded, monotonic solutions of an equation derived from the Kuramoto-Sivashinsky equation, namely y" + y = 1 y1 /2. We prove that there is no monotonic solution which satisfies ...

where μ, ν are positive constants. This equation, in the case μ = 0, was derived independently by Sivashinsky [1] and Kuramoto [2] with the purpose to model amplitude and phase expansion of pattern formations in different physical situations, for example, in the theory of a flame propagation in turbulent flows of gaseous combustible mixtures, see Sivashinsky [1], and in the theory of turbulence...

We report the first observations of numerical "hopping" cellular flame patterns found in computer simulations of the Kuramoto-Sivashinsky equation. Hopping states are characterized by nonuniform rotations of a ring of cells, in which individual cells make abrupt changes in their angular positions while they rotate around the ring. Until now, these states have been observed only in experiments b...