We study a spline wavelet alternative direction implicit (SW-ADI) algorithm for solving two-dimensional reaction diffusion equations. This algorithm is based on a collocation method for PDEs with a specially designed spline wavelet for the Sobolev space H2.I/on a closed interval I. By using the tensor product nature of adaptive wavelet meshes, we propose a SW-ADI method for two-dimensional problems.The proposed SW-ADI method is an efficient time-dependent adaptive method with second-order accuracy for solutions with localized phenomena, such as in flame propagations. Issues like new boundary wavelets for more accurate boundary conditions,adaptive strategy for two-dimensional meshes, data structure and storage and implementation details, and numerical results will be discussed.

We study asymptotic behavior of solutions for a class of second order nonlinear differential equations. Using Bihari's inequality, we obtain conditions under which all continuable solutions are asymptotic to at+b ast→ +∞, wherea, b are real constants.

The advance of very large scale integrated (VLSI)systems has been continuously challenging today's circuit simulators in both computational speed and stability. A novel numerical method, the fast wavelet collocation method (FWCM), was first proposed in [1] to explore a new direction of circuit simulation.The FWCM uses a totally different numerical means from the classical time-marching or frequency-domain methods and has demonstrated several superior computational properties, such as uniform error distribution and better computational stability,as compared to that provided by the conventional simulation methods.