Narrow Results

This paper proposes a new approach to solving three-dimensional linear and nonlinear Volterra integral equations (3D-VIEs) of the second kind using the Taylor collocation method (TCM). We develop an algorithm utilizing Taylor polynomials to approximate solutions of 3D-VIEs within a finite-dimensional space. Moreover, we perform convergence analysis to validate the efficacy of our method. Additionally, we show comparison examples to demonstrate the method’s effectiveness. The comparison highlights the efficiency and reliability of our approach.

For positive $p$ and real $\alpha$ let $A_{\alpha }^p$ denote the weighted Bergman spaces of the unit ball ${𝔹}_n$ introduced in [R. Zhao and K. Zhu, Theory of Bergman Spaces on the Unit Ball in${ℂ}_n$, Mémoires de la Société Mathématique de France, Vol. 115 (2008)]. It is well known that, at least in the case $n=1$, all functions in $A_{\alpha }^p$ can be approximated in norm by their Taylor polynomials if and only if $p>1$. In this paper we show that, for $f\in A_{\alpha }^p$ with $0<p\le 1$, we always have $\parallel S_{N}f-f{\parallel }_{p,\beta }\to 0$ as $N\to \infty$, where $\beta >\alpha +n(1-p)$ and $S_{N}f$ is the $N$th Taylor polynomial of $f$. We also show that for every $f$ in the Hardy space $H^p$, $0<p\le 1$, we always have $\parallel S_{N}f-f{\parallel }_{p,\beta }\to 0$ as $N\to \infty$, where $\beta >n(1-p)-1$. This generalizes and improves a result in [J. McNeal and J. Xiong, Norm convergence of partial sums of $H^1$ functions, Internat. J. Math.29 (2018) 1850065, 10 pp.].

The main purpose of this work is to provide a numerical approach for linear second-order differential and integro-differential equations with constant delay. An algorithm based on the use of Taylor polynomials is developed to construct a collocation solution $u\in S_m^{(1)}({\mathrm{\Pi }}_N)$ for approximating the solution of second-order linear DDEs and DIDEs. It is shown that this algorithm is convergent. Some numerical examples are included to demonstrate the validity of this algorithm.