We study numerical approximations to solutions of a system of two nonlinear diffusion equations in a bounded interval, coupled at the boundary in a nonlinear way. In certain cases the system develops a blow-up singularity in finite time. Fixed mesh methods are not well suited to approximate the problem near the singularity. As an alternative to reproduce the behaviour of the continuous solution...

Finite time singularity formation in a fourth order nonlinear parabolic partial differential equation (PDE) is analyzed. The PDE is a variant of a ubiquitous model found in the field of microelectromechanical systems (MEMS) and is studied on a one-dimensional (1D) strip and the unit disc. The solution itself remains continuous at the point of singularity while its higher derivatives diverge, a ...

In a paper titled "A topologically related singularity suggests a maximum preferred size for protein domains" (Zbilut et al., Proteins 2007;66:621-629), Zbilut et al. claim to have found a singularity in certain geometrical properties of protein structures, and suggest that this singularity may limit the maximum size of protein domains. They find further support for the singularity in their ana...

When all the inputs of a parallel manipulator (PM) are locked, the manipulator is usually turned into a structure. When an uncertainty singularity occurs for a PM, the latter structure is unstable or, in other words, it may undergo infinitesimal or finite motion. Hence, the investigation of the uncertainty singularities of a PM can be reduced to the instability analysis of its corresponding str...

In this work, a second-order approximation of the fractional substantial derivative is presented by considering a modified shifted substantial Grünwald formula and its asymptotic expansion. Moreover, the proposed approximation is applied to a fractional diffusion equation with fractional substantial derivative in time. With the use of the fourth-order compact scheme in space, we give a fully di...

Optimal pointwise estimates are derived for the biharmonic Green function in arbitrary C4,γ -smooth domains. Maximum principles do not exist for fourth order elliptic equations and the Green function may change sign. It prevents using a Harnack inequality as for second order problems and hence complicates the derivation of optimal estimates. The present estimate is obtained by an asymptotic ana...

Singular initial-value problems in second-order ordinary differential equations of Lane–Emden type are investigated. In this study, we shall introduce a further development in the Adomian decomposition method to overcome the difficulty at the singular point of non-homogeneous, linear and non-linear Lane–Emden-like equations; especially when the singularity appears on the right-hand side of this...

The nature of gravitational singularities, long mysterious, has now become clear through a combination of mathematical and numerical analysis. As the singularity is approached, the time derivative terms in the field equations dominate, and the singularity behaves locally like a homogeneous oscillatory spacetime. email address: [email protected]

This paper reports the discovery of a new type of singularity that is as signijlcant as other singularities like kinematic singularity, algorithmic singularity, and semi-singularity. This singularity, named semi algorithmic singularity(SAS), occurs when a redundant manipulator, faced with inequality constraints arising from kinematic limits such as joint angle limits and obstacles, proceeds to ...

The topology of a normal surface singularity does not determine the analytical invariants of its equisingularity class, but recent partial results indicated that this should be true under two restrictions, a topological one, that the link of the singularity is a rational homology sphere, and an analytical one, that the singularity is Q-Gorenstein. Neumann and Wahl conjectured that the singulari...