Existence and approximations of solutions to some fractional order functional integral equations

In this paper we shall study a fractional order functional integral equation. In the first part of the paper, we proved the existence and uniqueness of mile and global solutions in a Banach space. In the second part of the paper, we used the analytic semigroups theory oflinear operators and the fixed point method to establish the existence, uniqueness and convergence of approximate solutions of the given problem in a separable Hilbert space. We also proved the existence and convergence of Faedo-Galerkin approximate solution to the given problem. Finally, we give an example.

Approximation of solutions to fractional integral equation

In this paper we shall study a fractional integral equation in an arbitrary Banach space X. We used the analytic semigroups theory of linear operators and the fixed point method to establish the existence and uniqueness of solutions of the given problem. We also prove the existence of global solution. The existence and convergence of the Faedo–Galerkin solution to the given problem is also proved in a separable Hilbert space with some additional assumptions on the operator A. Finally we give an example to illustrate the applications of the abstract results.

Crucible Steel in South India-Preliminary Investigations on Crucibles from Some Newly Identified Sites

European accounts from the 17th century onwards have referred to the repute and manufacture of “wootz’, a traditional crucible steel made especially in parts of southern India in the former provinces of Golconda, Mysore and Salem. Pliny's Natural History mentions the import of iron and steel from the Seres which have been thought to refer to the ancient southern Indian kingdom of the Cheras. As yet the scale of excavations and surface surveys is too limited to link the literary accounts to archaeometallurgical evidence, although pioneering exploratory investigations have been made by scholars, especially on the pre-industrial production sites of Konasamudram and Gatihosahalli discussed in 18th-19th century European accounts. In 1991–2 during preliminary surveys of ancient base metal mining sites, Srinivasan came across unreported dumps with crucible fragments at Mel-Siruvalur in Tamil Nadu, and Tintini and Machnur in Karnataka and she collected surface specimens from these sites as well as from the known site of Gatihosahalli. She was also given crucible fragments by the Tamil University, Tanjavur, from an excavated megalithic site at Kodumanal, dated to ca 2nd c. Bc, mentioned in Tamil Sangam literature (ca 3rd c. BC-3rd c. AD), and very near Karur, the ancient capital of the Sangam Cheras. Analyses of crucible fragments from the surface collection at Mel-Siruvalur showed several iron prills with a uniform pearlitic structure of high-carbon hypereutectoid steel (∼1–1.5% C) suggesting that the end product was uniformly a high-carbon steel of a structure consistent with those of high-carbon steels used successfully to experimentally replicate the watered steel patterns on ‘Damascus’ swords. Investigations indicate that the process was of carburisation of molten low carbon iron (m.p. 1400° C) in crucibles packed with carbonaceous matter. The fabric of crucibles from all the above mentioned sites appears similar. Preliminary investigations on these crucibles are thus reported to establish their relationship to crucible production of carbon steel and to thereby extend the known horizons of this technology further.