RAO TRANSFORMS : A New Approach to Integral and Differential Equations

This book has material that is not available in the public domain. It was self-published first in 2007 but only one copy was sold, and it did not become known to researchers due to a lack of sufficient evidence of practical applications. Such evidence has now been produced by this author (using AI chatbots) and it will be published in another book soon.  The material in this book is useful for accurate and computationally efficient shift-variant image deblurring, especially in medical imaging like MRI, X-ray CT, Ultrasound, PET, etc., as well as in microscopy, astronomy, digital cameras, etc. It is also useful for solving integral and differential equations in physics, applied mathematics, engineering, etc. Parts of this book can be attached as a document for retrieval-augmentation generation (RAG) to AI Chatbots like ChatGPT, Grok, Gemini, etc. for obtaining answers to questions, research reports, programs that implement and run them on test data, etc.   For this reason, the price of this book has been set high to reflect the value of the book. In medical imaging, the results are clinically actionable easily and immediately for shift-variant-deblurring of medical images. The methods here are very useful for engineers in signal, image, and video processing, and in engineering design. The elegant closed-form explicit analytic solutions provided here are very useful for physicists and mathematicians. Researchers and practitioners in all fields of STEM could apply the methods in this book.

This book is essential for University Libraries, Research Hospital Libraries, Principal Investigators of Research Grants, doctoral and post-doctoral researchers, Research Funding Agencies like NIH, NSF, DoD, DoE, etc, Medical Image Device Manufacturers, Research labs in Aerospace industry, etc.

Research Monograph by

      Dr. Muralidhara SubbaRao  (Rao)

RAO TRANSFORMS

A New Approach to Integral and Differential Equations

“ [this research] has applications. It is guaranteed to produce doctoral dissertations.”

--An expert reviewer of this research for the U.S. Government, Feb. 2007.

 A fundamentally new approach is presented for solving integral equations and consequently differential equations. The approach is based on restating a conventional linear integral equation which is in a global form by an exactly equivalent equation   which is in a completely localized form.  This simple idea seems to have eluded all the researchers in the past. Using this simple idea, the problem of solving integral equations is completely localized and a novel closed-form analytic solution is derived that is computationally efficient. The solution provides a unified approach to a large class of diverse problems such as Fredholm’s and Volterra’s First and Second Kind integral equations as well as non-linear integral equations. Early results on basic theory and examples of applications (e.g. shift-variant image deblurring) are presented. Numerous future research topics are outlined. As differential equations and associated boundary conditions can be reformulated as integral equations, the new approach is also useful in solving differential equations. Integral and differential equations arise in many fields of science, engineering, and applied mathematics. This book is useful in all such fields to students, professionals, and researchers.

 Dr. Muralidhara SubbaRao (Rao) is a Professor of Electrical and Computer Engineering with research and teaching interests in Computer Vision, Digital Image Processing, and Computational Engineering.