INTRODUCING THE UPADHYAYA INTEGRAL TRANSFORM

Lalit Mohan Upadhyaya

doi:10.48165/

Authors

Lalit Mohan Upadhyaya Department of Mathematics, Municipal Post Graduate College, Mussoorie, Dehradun, Uttarakhand, India - 248179.

DOI:

https://doi.org/10.48165/

Keywords:

Upadhyaya transform, Laplace transform, Laplace-Carson transform, Sumudu transform, Elzaki transform, Kashuri and Fundo transform, Mahgoub transform, ZZ- transform, Sadik transform, Kamal transform, Natural transform, Mohand transform, Aboodh transform, Ramadan Group transform, Shehu transform, Sawi transform, Tarig transform, Yang transform

Abstract

Through this introductory paper we announce to the worldwide mathematics community a new type of integral transform, which we call the Upadhyaya Integral Transform or, the Upadhyaya transform (UT), in short. The new transform which we propose to proclaim through this paper, is, in fact a generalized form of the celebrated Laplace transform. The power of this generalization is that this most general form of the Laplace transform generalizes and unifies, besides the classical Laplace transform and the Laplace-Carson transform, most of the very recently introduced integral transforms of this category like, the Sumudu transform, the Elzaki transform, the Kashuri and Fundo transform, the Mahgoub transform, the ZZtransform, the Sadik transform, the Kamal transform, the Natural transform, the Mohand transform, the Aboodh transform, the Ramadan Group transform, the Shehu transform, the Sawi transform, the Tarig transform, the Yang transform, etc. We develop the general theory of the Upadhyaya transform in a way which exactly parallels the existing theory of the classical Laplace transform and also provide a number of possible generalizations of this transform and thus we prepare a firm ground for future researches in this field by employing this most generalized, versatile and robust form of the classical Laplace transform – the Upadhyaya transform – in almost all the areas wherever, the classical Laplace transform and its various aforementioned variants are currently being employed for solving the vast multitude of problems arising in the areas of applied mathematics, mathematical physics and engineering sciences and other possible fields of study inside and outside the realm of mathematics.

References

. Erdélyi, A., Magnus, W., Oberhettinger, F. and Tricomi, F. G. (1954). Tables of Integral Transforms, Vols.I, II, McGraw Hill Book Company, New York, Toronto and London.

. Oberhettinger, F. and Badii, L.( 1973). Tables of Laplace Transforms, Springer-Verlag, New York Heidelberg.

. Spiegel, M.R. (1968). Schaum’s Handbook of Mathematical Formulas, McGraw Hill , New York . [4]. Gradshteyn, I.S. and Ryzhik, I.M. (2007). Tables of Integrals, Series and Products, Seventh Edition, (Edited by Alan Jeffrey and Daniel Zwillinger), Academic Press, California, U.S.A.

. Prudnikov, A.P. and Brychkov, Yu.A. and Marichev O.I. (1992). Integrals and Series, Vol. 4, Direct Laplace Transforms, Gordon and Breach Science Publishers, New York, Paris, Melbourne, Tokyo. [6]. Spiegel, M.R. (1965). Laplace Transforms (Schaum’s Outlines), McGraw Hill , New York. [7]. Debnath, Lokenath and Bhatta, Dambaru (2015). Integral Transforms and Their Applications, Third Edition, CRC Press, Taylor and Francis Group, Boca Raton, London, New York.

. Patra, Baidyanath (2018). An Introduction to Integral Transforms (ISBN 9781138588035), CRC Press, Boca Raton, FL 334872742. Zbl 06840501.

. Kim, T. and Kim, D.S. (2017). Degenerate Laplace Transform and Degenerate Gamma Function, Russ. J. Math. Phys., 24(2), 241-248. MR3658414.

. Upadhyaya, Lalit Mohan (2018). On The Degenerate Laplace Transform – I, Bulletin of Pure and Applied Sciences –Section E- Mathematics and Statistics, Vol. 37(E), No. 1, 1-8.

. Upadhyaya, Lalit Mohan (2017). On The Degenerate Laplace Transform – II, International Journal of Engineering & Scientific Research, Vol. 5, Issue 12, December 2017, 63-71. (http://esrjournal.com/uploads/91/4403_pdf.pdf)

. Upadhyaya, Lalit Mohan (2018). On The Degenerate Laplace Transform – III, International Journal of Engineering, Science and Mathematics, Vol. 7, Issue 1, January 2018, 400-410. (http://ijesm.co.in/abstract.php?article_id=4613&title=ON%20THE%20DEGENERATE%20L APLACE%20TRANSFORM%20%20III )

. Upadhyaya, Lalit Mohan (2018). On The Degenerate Laplace Transform – IV, International Journal of Engineering & Scientific Research, Vol. 6, Issue 2, February 2018, 198-209. (http://esrjournal.com/uploads/91/4863_pdf.pdf)

. Mathai, A.M. (1997). Jacobians of Matrix Transformations and Functions of Matrix Argument. World Scientific Publishing Co. Pte. Ltd., Singapore.

. Upadhyaya, Lalit Mohan (Nov. 2003): Matrix Generalizations of Multiple Hypergeometric Functions By Using Mathai's Matrix Transform Techniques (Ph.D. Thesis, Kumaun

University, Nainital, Uttarakhand, India) #1943, IMA Preprint Series, University of Minnesota, Minneapolis, U.S.A. (https://www.ima.umn.edu/sites/default/files/1943.pdf http://www.ima.umn.edu/preprints/abstracts/1943ab.pdf

http://www.ima.umn.edu/preprints/nov2003/1943.pdf

http://hdl.handle.net/11299/3955

https://zbmath.org/?q=an:1254.33008

https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.192.2172&rank=51

https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.192.2172&rep=rep1&type=pdf (Retrieved from the University of Minnesota Digital Conservancy,

http://hdl.handle.net/11299/3955).

. Elzaki, Tarig M., Elzaki, Salih M. and Elnour, Elsayed A. (2012). On the New Integral Transform “ELzaki Transform” Fundamental Properties Investigations and Applications, Global Journal of Mathematical Sciences: Theory and Practical, Vol. 4, No. 1, 1-13.

. Deakin, Michael A.B. (1997). The ‘Sumudu transform’ and the Laplace transform, Letters to the Editor (Received 24.03.1995), Int. J. Math. Educ. Sci. Technol., Vol. 28, No. 1, 159-160. [18]. Deakin M.A.B. (1992). Arch. Hist. Ex. Sci., 44, 265.

. Ditkin, V.A. and Prudnikov, A.P. (1965). Integral Transforms and Operational Calculus, Oxford : Pergamon.

. Watugala, G.K. (1993). Sumudu transform: a new integral transform to solve differential equations and

control engineering problems, International Journal of Mathematical Education in Science and Technology, 24(1), 35-43.

. Weerakoon, S. (1994). Application of Sumudu transform to partial differential equations, International Journal of Mathematical Education in Science and Technology, 25, 277-283.

. Balser, W. (1994). From Divergent Power Series to Analytic Functions, Lecture Notes in Mathematics 1582, Berlin: Springer.

. Al-Omari, Shrideh K. Qasem and Agarwal, Praveen (2016). Some general properties of a fractional Sumudu transform in the class of Boehmians, Kuwait J. Sci., 43(2), 16-30. MR3524772 [24]. Belgacem, F.B.M., Karaballi, A.A. and Kalla, S.L. (2003). Analytical investigations of the Sumudu transform and applications to integral production equations, Mathematical Problems in Engineering, 3-4, 103-118.

. Belgacem, Fethi Bin Muhammad and Silambarasan, Rathinavel (2017). A distinctive Sumudu treatment of trigonometric functions, J. Comput. Appl. Math. 312, 74-81. Zbl 1351.44001 [26]. Laplace, P.S. (1820). Théorie Analytique des Probabilitiés, Vol. I, Part 2, Lerch, Paris. [27]. Nuruddeen, R.I., Muhammad, Lawal, Nass, A.M. and Sulaiman, T.A. (2018). A Review of the Integral Transforms-Based Decomposition Methods and their Applications in Solving Nonlinear PDEs, Palestine Journal of Mathematics, Vol. 7(1), 262-280.

. Nuruddeen, R.I., Nass, A.M. and Muhammad, Lawal (2016). A More Recent Review of the Integral Transforms and their Applications in Solving Differential Equations, Technical Report – July 2016, DOI: 10.13140/RG.2.2.13024.69128

. Khan, Z.H. and Khan, W.A. (2008). Natural transform-properties and applications, NUST Journal of Engineering Sciences, 1, 127-133.

. Maitama, Shehu and Zhao, Weidong (2019). New Integral Transform: Shehu Transform a Generalization of Sumudu and Laplace transform for solving Differential Equations, International Journal of Analysis and Applications, Volume 17, Number 2, 167-190. DOI: 10.28924/2291-8639-17- 2019-167; arXiv:1904.11370v1 [math.GM] 19 Apr 2019.

. Rawashdeh, Mahmoud S. and Maitama, Shehu (2015). Solving Nonlinear Ordinary Differential Equations Using the NDM, Journal of Applied Analysis and Computation, Volume 5, Number 1, 77- 88. doi:10.11948/2015007

. Elzaki T.M. (2011). The new integral transforms "Elzaki transform", Global Journal of Pure and Applied Mathematics, 7(1), 57-64.

. Aboodh, Khalid Suliman (2013). The New Integral Transform ''Aboodh Transform'', Global Journal of Pure and Applied Mathematics, Volume 9, Number 1, 35-43.

. Kashuri, A. and Fundo, A. (2013). A new integral transform, Advances in Theoretical and Applied Mathematics, 8 (1), 27-43.

. Zafar, Zain Ul Abadin (2016). ZZ Transform Method, International Journal of Advanced Engineering and Global Technology, Vol. 4, Issue 1, 1605-1611.

. Raslan, R.K., Ramadan, M.A., El-Danaf, T.S. and Hadhoud, A.R. (2016). On a new general integral transform: some properties and remarks, J. Math. Comput. Sci., 6(1), 103-109.

. Mahgoub, Mohand M. Abdelrahim (2016). The New Integral Transform “Mahgoub Transform”, Advances in Theoretical and Applied Mathematics, Volume 11, Number 4, 391-398. [38]. Sedeeg, Abdelilah Kamal. H.(2016). The New Integral Transform ''Kamal Transform'', Advances in Theoretical and Applied Mathematics, Volume 11, Number 4, 451-458.

. Mahgoub, Mohand M. Abdelrahim (2017). The New Integral Transform ''Mohand Transform'', Advancesin Theoretical and Applied Mathematics, Volume 12, Number 2, 113-120.

. Elzaki, Tarig M. and Elzaki, Salih M. (2011). On the new integral Transform 'Tarig Transform' and Systems of Ordinary Differential Equations, Applied Mathematics, Elixir Appl. Math. 36, 3226-3229. [41]. Elzaki, Tarig M. and Elzaki, Salih M. (2011). On the relationship between Laplace Transform and new integral transform Tarig Transform', Applied Mathematics, Elixir Appl. Math., 36 , 3230-3233. [42]. Elzaki, Tarig M. and Salih M. Elzaki, (2011), On the Tarig Transform and ordinary differential equation with variable coefficients, Applied Mathematics, Elixir Appl. Math., 38, 4250- 4252. [43]. Elzaki , Tarig. M. and Elzaki, Salih M. (2013). New Integral Transform "Tarig Transform" and System of Integro-Differential Equations; Applied Mathematics; Elixir Appl. Math., 57 (2013) 13982-13985. [44]. Shaikh, Sadikali Latif. (2018). Introducing a New Integral Transform: Sadik Transform, American International Journal of Research in Science, Technology, Engineering & Mathematics (AIJRSTEM), 22(1), 100-102.

. Yang, Xiao-Jun (2016). A New Integral Transform Method for Solving Steady Heat-Transfer Problem, Thermal Science, Vol. 20, Suppl. 3, S639-S642.

. Mahgoub, Mohand M. Abdelrahim (2019). The New Integral Transform ''Sawi Transform'', Advances in Theoretical and Applied Mathematics, Volume 14, Number 1, 81-87.

. Barnes, Benedict (2016). Polynomial Integral Transform for Solving Differential Equations, European Journal of Pure and Applied Mathematics, Vol. 9, No. 2, 140-151.

. Atangana, Abdon and Kilicman, Adem (2013). A Novel Integral Operator Transform and Its Application to Some FODE and FPDE with Some Kind of Singularities, Mathematical Problems in Engineering, Volume 2013, Article ID 531984, 7 pages. http://dx.doi.org/10.1155/2013/531984

. Makarov, A.M. (1970). Application of the Laplace-Carson method of integral transformation to the theory of unsteady visco-plastic flows, J. Engrg. Phys. Thermophys, 19, 94-99.

. Taha, Nidal E. Hassan, Nuruddeen, R. I. and Sedeeg, Abdelilah Kamal H. (2017). Dualities between ''Kamal & Mahgoub Integral Transforms'' and ''Some Famous Integral Transforms'', British Journal of Applied Science & Technology, 20(3): 1-8, 2017; Article no.BJAST.32380.

. Belgacem, Fethi Bin Muhammed and Karaballi, Ahmed Abdullatif (2006). Sumudu Transform Fundamental Properties Investigations and Applications, Journal of Applied Mathematics and Stochastic Analysis, Volume 2006, Article ID 91083, Pages 1–23. DOI 10.1155/JAMSA/2006/91083

. Yürekli, O. and Sadek, I. (1991). A Parseval-Goldstein type theorem on the Widder potential transform and its applications, Int. J. Math. Ed. Sci. Tech., 14, No. 3, 517–524.

. Aylikçi, Fatih and Dernek, Neişe (2018). Some relations on the double L22 −integral transform and their applications, Mediterr. J. Math., 15 (2018), No. 2, Art.37, 21 pp. MR3765932

. Mathai, A.M. (1993). Hypergeometric Functions of Several Matrix Arguments (A Preliminary Report), Publication No. 25, Center for Mathematical Sciences, Vazhuthacaud, Trivandrum, Kerala, India. [55]. Atangana, Abdón and Alkaltani, Badr Saad T. (2016). A novel double integral transform and its applications, J. Nonlinear Sci. Appl. 9 (2016), 424-434.

. Aboodh, K.S., Farah, R.A., Almardy, I.A. and ALmostafa, F.A. (2017). Solution of Telegraph Equation by Using Double Aboodh Transform, Elixir Appl. Math. 110 , 48213-48217. [57]. Aboodh, K.S., Farah, R.A., Almardy, I.A. and ALmostafa, F.A. (2017). Solution of Partial Integro Differential Equations by using Aboodh and Double Aboodh Transform Methods, Global Journal of Pure and Applied Mathematics, Volume 13, Number 8, 4347-4360.

. Thangavelu, K., Pradeep, M. and Vinothini, K. (2018). Solution of Telegraph Equation by Using Double Mahgoub Transform, South East Asian J. of Math. & Math. Sci., Vol. 14, No.2, 15-20. [59]. Elzaki, Tarig M., and Hilal, Eman M.A. (2012). Solution of Telegraph Equation by Modified of

Double Sumudu Transform "Elzaki Transform", Mathematical Theory and Modeling, Vol.2, No.4, 95 – 103.

. Hassaballa, Abaker A. and Salih, Yagoub A. (2015). On Double Elzaki Transform and Double Laplace Transform, IOSR Journal of Mathematics (IOSR-JM), Vol. 11, Issue 1, Ver. VI (Jan. –Feb. 2015), 35- 41.

. Idrees, Muhammad Imran, Ahmed, Zulfiqar, Awais, Muhammad and Perveen, Zahida (2018). On the Convergence of Double Elzaki Transform, International Journal of Advanced and Applied Sciences, 5(6), 19-24.

. Kashuri, Artion, Fundo, Akli and Liko, Rozana (2013). On Double New Integral Transform and Double Laplace Transform, European Scientific Journal, November 2013 edition Vol.9, No.33, 82-90. [63]. Watugala, G.K., (2002). The Sumudu transform for functions of two variables, Mathematical Engineering in Industry, 8, No. 4, 293-302.

. Tchuenche, Jean M. and Mbare, Nyimvua S. (2007). An Application of the Double Sumudu Transform, Applied Mathematical Sciences, Vol. 1, No. 1, 31 – 39.

. Eltayeb, Hassan and Kiliçman, Adem (2010). On Double Sumudu Transform and Double Laplace Transform, Malaysian Journal of Mathematical Sciences, 4(1), 17-30.

. Jarad, F. and Tas, K. (2012). Application of Sumudu and Double Sumudu transforms to Caputo – Fractional Differential Equations, Journal of Computational Analysis and Applications, Vol. 14, No. 3, 475-483.

. Kiliçman, Adem and Omran, Maryam (2017). On double Natural transform and its applications, J. Nonlinear Sci. Appl., 10, 1744–1754.

. Atangana, Abdon (2013). A Note on the Triple Laplace Transform and Its Applications to Some Kind of Third-Order Differential Equation, Abstract and Applied Analysis, Volume 2013, Article ID 769102, 10 pages. http://dx.doi.org/10.1155/2013/769102

. Khan, Tahir, Shah, Kamal, Khan, Amir and Khan, Rahmat Ali (2018). Solution of fractional order heat equation via triple Laplace transform in 2 Dimensions, Mathematical Methods in the Applied Sciences, Vol. 41, Issue2, (30 January 2018), 818-825. (First published: 27 October 2017)

https://doi.org/10.1002/mma.4646

. Elzaki, Tarig M. and Mousa, Adil (2019). On the convergence of triple Elzaki transform, SN Applied Sciences (2019) 1:275 | https://doi.org/10.1007/s42452-019-0257-2

. Kim, YunJae, Kim, Byung Moon, Jang, Lee-Chae and Kwon, Jongkyum (2018). A Note on Modified Degenerate Gamma and Laplace Transformation, Symmetry 2018, 10, 471, 8pp. doi:10.3390/sym10100471

. Mathai A.M. (1993). Lauricella Functions of Real Symmetric Positive Definite Matrices, Indian J. Pure Appl. Math., 24(9), 513-531.

. Upadhyaya, Lalit Mohan (2017). On Exton’s Triple Hypergeometric Functions of Matrix Arguments –II, Bulletin of Pure and Applied Sciences, Section-E, Mathematics & Statistics, Vol. 36(E), No.2, 207-217. (Article DOI: 10.5958/2320-3226.2017.00023.6)

https://web.a.ebscohost.com/abstract?direct=true&profile=ehost&scope=site&authtype=crawler&jrnl= 09706577&AN=128769173&h=VKjrBgoGK8zPkmFuWAH0mPQ9

http://www.indianjournals.com/ijor.aspx?target=ijor:bpasms&volume=36e&issue=2&article=013 http://www.ijour.net/ijor.aspx?target=ijor:bpasms&volume=36e&issue=2&article=013 [74]. Rehman, H.U., Iftikhar, M., Saleem, S., Younis, M. and Mueed, A. (2014) A Computational Quadruple Laplace Transform for the Solution of Partial Differential Equations, Applied Mathematics, 5, 33723382.

http://dx.doi.org/10.4236/am.2014.521314

. Gasper, George and Rahman, Mizan (2004). Basic Hypergeometric Series, Second Edition, Encyclopedia of Mathematics and its Applications, Volume 96, Cambridge University Press, Cambridge, New York.

. Ramadan, Mohamed A. and Hadhoud, Adel R. (2018). Double Ramadan Group Integral Transform: Definition and Properties with Applications to Partial Differential Equations, Appl. Math. Inf. Sci., 12, No. 2, 389-396. http://dx.doi.org/10.18576/amis/120213