Partial Integro-Differential Equations

A programme on numerical methods for both parabolic and hyperbolic integro-differential equations was carried out in CMA during Sept'90-Feb'91 in collaboration with Lars B. Wahlbin (Cornell Univ., USA) and Vidar Thomée (Chalmers Univ., Sweden). The procedure involved is to reduce the storage requirements for the fully discrete schemes without loosing accuracy.

Although such a programme was initiated by Thomée and Sloan [SIAM J. Numer. Anal. 1986], Thomée and Zhang [Math. Comp. 1989] using the spectral arguments in a crucial way , but the present study would rely on the energy arguments widening the scope to more general problems (14] in the list of publications).

The effect of numerical quadratures on finite element methods for parabolic integro-differential equations with nonsmooth initial data was carried out with Todd Peterson from George Mason University, USA.

The major objective is to see to what extent the known results on parabolic equations can be carried over to integro-differential equations (ref. [21] in the list of publications 1 and ref. [8] in the Research Reports).

Subsequently, some improved results were obtained for the nonsmooth case requiring less regularity on the exact solution [39] in the list of publications. For parabolic problems, ref. [34]. (with my Ph.D. student Dr. Rajen K. Sinha). The effect of spatial quadrature on the finite element approximations to hyperbolic integro-differential equation with minimal regularity assumptions was carried out with Dr. Sinha (ref. [37] in the list of publications).

Even for wave equation, the present analysis improves upon the earlier      results of Baker et al. (SIAM J. Numer. Anal. 13 (1976)).

It is difficulty to extend the semigroup theoretic arguments of Thomée and Zhang to time dependent parabolic integro-differential equations with nonsmooth initial data. Therefore, in [38] and [35] ( list of publications) and Thesis of R. K. Sinha [3], Pani and Sinha have discussed the finite element error analysis for the time dependent problem with nonsmooth data using energy arguments. One of the crucial step in this analysis is to prove he limited smoothing property and then to use dual problem so as to obtain the desired results. We note that this settles the open problem cited in the book ‘Finite Element Methods for Integro-differential Equations', (page 106) written by C. Chen and T. Shih.

Further, a related question on the negative norm estimates and superconvergence results is examined in [20] in the list of publications and [10] under Research Reports.