A coordinate-splitting economic diierence scheme is proposed for generalized parabolic equations (GPE) containing fourth-order diiusion operators and the algorithm for its implementation is developed. The performance of the scheme is demonstrated for diierent cases, e.g. for treating bifurcation phenomena. The technique is applied to the numerical solution of Swift-Hohenberg equation describing...

Fourth-order Schrödinger equations have been introduced by Karpman and Shagalov to take into account the role of small fourth-order dispersion terms in the propagation of intense laser beams in a bulk medium with Kerr nonlinearity. In this paper we investigate the cubic defocusing fourth order Schrödinger equation i∂tu +∆ 2 u + |u|u = 0 in arbitrary space dimension R for arbitrary initial data....

We use the bivariate spline method to solve the steady state Navier-Stokes equations numerically. The bivariate spline we use in this paper is the space of splines of smoothness r and degree 3r over triangulated quadrangu-lations. The stream function formulation for the steady state Navier-Stokes equations is employed. Galerkin's method is applied to the resulting nonlinear fourth order equatio...

We present a fourth-order accurate algorithm for solving Poisson’s equation, the heat equation, and the advection-diffusion equation on a hierarchy of block-structured, adaptively refined grids. For spatial discretization, finite-volume stencils are derived for the divergence operator and Laplacian operator in the context of structured adaptive mesh refinement and a variety of boundary conditio...

We use the bivariate spline nite elements to numerically solve the steady state NavierStokes equations. The bivariate spline nite element space we use in this paper is the space of splines of smoothness r and degree 3r over triangulated quadrangulations. The stream function formulation for the steady state Navier-Stokes equations is employed. Galerkin's method is applied to the resulting nonlin...

We construct the minimal effective chiral pion–nucleon SU(2) Lagrangian at fourth order in the chiral expansion. The Lagrangian contains 114 in principle measurable and 4 contact terms needed for the renormalization. We develop both the relativistic as well as the heavy baryon formulation of the effective field theory. For the latter, we also work out explicitly all 1/m corrections at fourth or...

We construct the minimal effective chiral pion–nucleon SU(2) Lagrangian at fourth order in the chiral expansion. The Lagrangian contains 118 in principle measurable terms. We develop both the relativistic as well as the heavy baryon formulation of the effective field theory. For the latter, we also work out explicitly all 1/m corrections at fourth order. We display all relevant relations needed...

We present benchmark simulations for the 8:1 di erentially heated cavity problem, the focus of a special session at the rst MIT conference on Computational Fluid and Solid Mechanics in June 2001. The numerical scheme is a fourth-order nite di erence method based on the vorticity-stream function formulation of the Boussinesq equations. The momentum equation is discretized by a compact scheme wit...

This paper is devoted to find the numerical solutions of the fourth order linear and nonlinear differential equations using piecewise continuous and differentiable polynomials, such as Bernstein, Bernoulli and Legendre polynomials with specified boundary conditions. We derive rigorous matrix formulations for solving linear and non-linear fourth order BVP and special care is taken about how the ...

and Applied Analysis 3 Motivated by 14, 15 , here we study the existence of AL-solutions for 1.1 . The approach is completely different from the one used in 15 , in which an iteration process, jointly with a comparison with the linear equation y 4 q t y 2 0, is employed. Our tools are based on a topological method, certain integral inequalities, and some auxiliary functions. In particular, for ...