Wavelet collocation methods for solving neutral delay differential equations

In this paper, we proposed wavelet based collocation methods for solving neutral delay differential equations. We use Legendre wavelet, Hermite wavelet, Chebyshev wavelet and Laguerre wavelet to solve the neutral delay differential equations numerically. We solved five linear and one nonlinear problem to demonstrate the accuracy of wavelet series solution. Wavelet series solution converges fast and gives more accurate results in comparison to other methods present in literature. We compare our results with Runge–Kutta-type methods by Wang et al. (Stability of continuous Runge–Kutta-type methods for nonlinear neutral delay-differential equations,” Appl. Math. Model, vol. 33, no. 8, pp. 3319–3329, 2009) and one-leg θ methods by Wang et al. (Stability of one-leg θ method for nonlinear neutral differential equations with proportional delay,” Appl. Math. Comput., vol. 213, no. 1, pp. 177–183, 2009) and observe that our results are more accurate.

On the best $L_2$-approximations of multivariable functions by means of splines

We obtain sharp inequalities of Jackson type for the best approximations of functions in $L_2(\mathbb{R}^m)$ by means of partial sums of wavelet series in case of multidimensional analogues of Shannon-Kotelnikov wavelets.