Report on the thesis "Integral Transform and Its Application to the Solution of the Flow Problems"

Prof. Ian N. Sneddon, Department of Mathematics, University of Glasgow

The thesis submitted by Dr. Ram Kumar is made up of 19 published and 1 manuscript of a paper accepted for publication of which he is the sole author, and of 1 published paper and 1 manuscript of which he is a part author.  The thesis is divided into three parts of which the first two are divided to topics in pure mathematics and the remaining one to topics in the theory of fluid flow.

Part I of the thesis consists of 6 published papers (all by Dr. Ram Kumar) on transform theory.  The first two are concerned respectively with the calculation of Hankel transform and of Laplace transform. These papers are short but the results they contain are significant and useful, in particular the generalisation of Tricomi's theorem.  The third paper is a valuable contribution to the operational calculus based on the Laplace transform.  The remaining three papers are of determining functions which are self-reciprocal under the Hankel transform.  This problem had been discussed by many mathematicians of great distinction working several years before the author and it is to his credit that he was able to make significant contribution to the theory at this late date.

Part II consists of 9 published papers of which Dr. Ram Kumar is the sole author.  Together they form a substantial and significant contribution to the theory of "Agrawal's Generalised Hankel Transform".  These papers published in the years 1954-60 are well known to research workers throughout the world interested in transform theory.  Dr. Ram Kumar developed recurrence relations for this transform (7, 8), proved two theorems useful for the evaluation of integrals involving generalised Bessel functions (9), and established integral representation involving these transforms (10).  In the next three papers (11-13), Dr. Ram Kumar developed properties of the Agrawal transform which showed its connections with the transforms of Laplace, Hankel and Stieltjes and made possible the evaluation of some integrals involving generalised Bessel functions (and other functions) which would be difficult to compute otherwise.  The remaining two papers (14-15) are a discussion of certain infinite series expansions and covergence theorem connected with this transform.

These papers are already familiar to the scientific community, have been favorably reviewed and generally recognised as a major contribution to the theory of this particular generalisation of the Hankel transform.

The third and last part of the thesis is devoted to applications of the theory of Laplace, Hankel and finite Hankel transforms.  It s made up of three papers (two already published by Dr. Ram Kumar and one accepted by Dr. Ram Kumar and Mr. Warsi) on the theory of the unsteady flow of viscous liquid through circular and co-axial cylinders.  The remaining five papers are concerned with an extended discussion of small disturbances in inviscid and viscous, electrically conducting liquids in the presence of magnetic fields.  Of these five papers, two (one published and one accepted for publication) are written by Dr. Ram Kumar in collaboration with Prof. P.L. Bhatnagar and three (two published and one accepted for publication) are by Dr. Ram Kumar alone.

These eight papers are concerned with hydro-dynamical problems of real physical interest.  Although they show the author to be a master of transform theory, their interest is not confined to workers in transform theory but they have also been received by workers in fluid mechanism.

The papers comprising this thesis give ample evidence of substantial contribution of great originality made by Dr. Ram Kumar to both pure and applied mathematics and I have no hesitation in recommending that the degree of Doctor of Science of the University of Lucknow be conferred upon Dr. Ram Kumar without further examination.