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We present a method of finding weighted Koppelman formulas for ($p$,$q$)-forms on$n$-dimensional complex manifolds$X$which admit a vector bundle of rank$n$over$X$×$X$, such that the diagonal of$X$×$X$has a defining section. We apply the method to ℙ^{$n$}and find weighted Koppelman formulas for ($p$,$q$)-forms with values in a line bundle over ℙ^{$n$}. As an application, we look at the cohomology groups of ($p$,$q$)-forms over ℙ^{$n$}with values in various line bundles, and find explicit solutions to the $\overline{\partial}$ -equation in some of the trivial groups. We also look at cohomology groups of (0,$q$)-forms over ℙ^{$n$}×ℙ^{$m$}with values in various line bundles. Finally, we apply our method to developing weighted Koppelman formulas on Stein manifolds.

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In this paper, we consider the time-dependent Maxwell’s equations when Cole–Cole dispersive medium is involved. The Cole–Cole model contains a fractional time derivative term, which couples with the standard Maxwell’s equations in free space and creates some challenges in developing and analyzing time-domain finite element methods for solving this model as mentioned in our earlier work [J. Li, J. Sci. Comput., 47 (2001), pp. 1–26]. By adopting some techniques developed for the fractional diffusion equations [V.J. Ervin, N. Heuer, and J.P. Roop, SIAM J. Numer. Anal., 45 (2007), pp. 572–591], [Y. Lin and C. Xu, J. Comput. Phys., 225 (2007), pp. 1533–1552], [F. Liu, P. Zhuang, V. Anh, I. Turner, and K. Burrage, Appl. Math. Comput., 191 (2007), pp. 12–20], we propose two fully discrete mixed finite element schemes for the Cole–Cole model. Numerical stability and optimal error estimates are proved for both schemes. The proposed algorithms are implemented and detailed numerical results are provided to justify our theoretical analysis.