TY - JOUR
T1 - The oscillation of solutions of volterra integral and integro-differential equations with highly oscillatory kernels
AU - Brunner, Hermann
AU - Ma, Yunyun
AU - Xu, Yuesheng
N1 - This research is supported in part by the Hong Kong Research Grants Council (HKBU No. 200113), the Natural Sciences Engineering Research Council of Canada (Discovery grant No. A9406), the U.S. National Science Foundation under grant No. DMS-152233, the Guangdong Provincial Government of China through the Computational Science Innovative Research Team program and the Natural Science Foundation of China under grant No. 11471013.
PY - 2015/12
Y1 - 2015/12
N2 - We study the oscillatory structures of so-lutions of Volterra integral and integro-differential equations (VIEs, VIDEs) with highly oscillatory kernels. Based on the structured oscillatory spaces introduced in Wang and Xu [28], we first analyze the degree of oscillation of the solution of VIEs associated with the oscillatory kernels belonging to a certain structured oscillatory space by using the resolvent representation of the solution. According to a decomposition of the oscillatory integrals in the complex plane, we prove that the Volterra integral operator reduces the oscillatory order of the functions in the structured oscillatory spaces corresponding to the oscillatory structure of the kernel. The analogous oscillatory structure of solutions of VIDEs is then analyzed by representing the solution of the VIDEs by the differential resolvent kernel and by exploiting the relationship between the VIDEs and the equivalent VIE. We conclude that the solutions of the VIEs and VIDEs preserve the oscil-latory components of the kernel.
AB - We study the oscillatory structures of so-lutions of Volterra integral and integro-differential equations (VIEs, VIDEs) with highly oscillatory kernels. Based on the structured oscillatory spaces introduced in Wang and Xu [28], we first analyze the degree of oscillation of the solution of VIEs associated with the oscillatory kernels belonging to a certain structured oscillatory space by using the resolvent representation of the solution. According to a decomposition of the oscillatory integrals in the complex plane, we prove that the Volterra integral operator reduces the oscillatory order of the functions in the structured oscillatory spaces corresponding to the oscillatory structure of the kernel. The analogous oscillatory structure of solutions of VIDEs is then analyzed by representing the solution of the VIDEs by the differential resolvent kernel and by exploiting the relationship between the VIDEs and the equivalent VIE. We conclude that the solutions of the VIEs and VIDEs preserve the oscil-latory components of the kernel.
KW - Decomposition of the oscillatory integral
KW - Highly oscillatory kernel
KW - Oscillation preserving solution
KW - Oscillatory structured space
KW - Volterra integral equation
KW - volterra integro-differential equation
UR - http://www.scopus.com/inward/record.url?scp=84964692006&partnerID=8YFLogxK
U2 - 10.1216/JIE-2015-27-4-455
DO - 10.1216/JIE-2015-27-4-455
M3 - Journal article
AN - SCOPUS:84964692006
SN - 0897-3962
VL - 27
SP - 455
EP - 487
JO - Journal of Integral Equations and Applications
JF - Journal of Integral Equations and Applications
IS - 4
ER -