TY - JOUR
T1 - Spectral Collocation Methods for Second-Order Volterra Integro-Differential Equations with Weakly Singular Kernels
AU - Shi , Xiulian
AU - Chen , Yanping
AU - Huang , Yunqing
AU - Huang , Fenglin
JO - Advances in Applied Mathematics and Mechanics
VL - 2
SP - 480
EP - 502
PY - 2020
DA - 2020/01
SN - 12
DO - http://doi.org/10.4208/aamm.OA-2019-0056
UR - https://global-sci.org/intro/article_detail/aamm/13630.html
KW - Volterra integro-differential equations, weakly singular kernel, spectral collocation methods.
AB - <p style="text-align: justify;">In this paper, a Jacobi spectral collocation approximation is proposed for the solution of second-order Volterra integro-differential equations with weakly singular kernels. The solution of such equations usually exhibits a singular behaviour at the origin. By using some suitable variable transformations, we obtain a new equation which is still weakly singular, but whose solution is as smooth as we like. Then the resulting equation is solved by standard spectral methods. We establish a rigorous error analysis for the proposed method, which shows that the numerical errors decay exponentially in $L^\infty$-norm and weighted $L^2$-norm. Finally, to perform the numerical simulation, a test example is considered with non-smooth solutions.</p>