A study of the general truncated m fractional derivatives and their q analogues

Abstract

The entire subject matter of the thesis is divided into seven chapters. newlineChapter 1 introduces briefly, Special functions, Fractional calculus, Image processing and q-calculus. newline newlineChapter 2 deals with a new type of conformable fractional derivative which generalizes the standard properties and results of the classical integer order calculus. Also, well known physical problems like Newton s law of cooling and Kirchoff s voltage law are generalized and solved in terms of the conformable fractional sense. Additionally, the image processing has been done with the use of this new fractional derivative. newlineThese results are published in the journal: Applied Mathematics E-Notes (2022) (Article in Press) (Mathematical Reviews, SCOPUS, ESCI). newline newlineIn Chapter 3, q-analogues of the results derived in Chapter 2 are obtained. A q-analogue of a physical problem-the population growth is also generalized and solved in this chapter. newline newlineChapter 4 contains the detailed study of a new type of fractional derivative, which is called the Right Local General Truncated M -fractional derivative that generalizes the properties and results of the integer order calculus. In context with this fractional derivative, Kirchoff s voltage law is generalized and the comparison of the solution for various parametric values is described. newlineThese results are published in the journal: Communications in Korean Mathe- matical Society (2022), vol. 37, no. 2, pp. 503 520 (Mathematical Reviews: MR4415618, SCOPUS, ESCI). newline newlineChapter 5 incorporate a q-analogue of the Right Local General Truncated M - fractional derivative whose results and properties, which have been derived in Chapter 4. Also, a q-analogue of a physical problem-the falling body problem, is solved and the corresponding q-vertical velocity and q-distance are obtained. newlineThe results of this q-analogue are published in the journal: Jordan Journal of Mathematics and Statistics (2023), vol. 16, no. 1, pp. 1 22 (Mathematical Reviews, SCOPUS, ESCI). newline newlineChapter 6 covers a novel type of fractional derivative with the non-