Numerical schemes are presented for a class of fourth order diffusion problems. These problems arise in lubrication theory for thin films of viscous fluids on surfaces. The equations being in general fourth order degenerate parabolic, additional singular terms of second order may occur to model effects of gravity, molecular interactions or thermocapillarity. Furthermore, we incorporate nonlinea...

This paper is a result of comparison of some available numerical methods for solving nonlinear fractional order ordinary differential equations. These methods are compared according to their computational complexity, convergence rate, and approximation error. The present study shows that when these methods are applied to nonlinear differential equations of fractional order, they have different ...

We consider the numerical evaluation of the Evans function, a Wronskian-like determinant that arises in the study of the stability of travelling waves. Constructing the Evans function involves matching the solutions of a linear ordinary differential equation depending on the spectral parameter. The problem becomes stiff as the spectral parameter grows. Consequently, the Gauss–Legendre method ha...

In this paper we present a shooting algorithm to solve fractional terminal (or boundary) value problems. We provide a convergence analysis of the numerical method, derived based upon properties of the equation being solved and without the need to impose smoothness conditions on the solution. The work is a sequel to our recent investigation where we constructed a nonpolynomial collocation method...

Groundwater is a hugely important resource in both developed and developing countries and makes up over 90% of the world’s freshwater. The idea behind our work is to use wave scattering in saturated porous media to quantify aquifer state and hydraulic properties. We initially focused on solving related inverse problem using the Specfem software to model coupled elastic/poroelastic waves. Of par...

In this paper we design and analyze a class of high order numerical methods to two dimensional Heaviside function integrals. Inspired by our high order numerical methods to two dimensional delta function integrals [19], the methods comprise approximating the mesh cell restrictions of the Heaviside function integral. In each mesh cell the two dimensional Heaviside function integral can be rewrit...

For more than fifteen years, spectral finite elements (i.e. finite element methods on hexahedral meshes with mass-lumping) showed their efficiency to model the propagation of acoustic and elastic waves in the time domain, in particular in terms of accuracy. Moreover, their mixed formulation [1] dramatically increases their efficiency in terms of storage and computational time. This approach, wh...

a new two-step implicit linear obrechkoff twelfth algebraic order method with vanished phase-lag and its first, second, third and fourth derivatives is constructed in this paper. the purpose of this paper is to develop an efficient algorithm for the approximate solution of the one-dimensional radial schrodinger equation and related problems. this algorithm belongs in the category of the multist...