Application of the Upadhyaya transform to Volterra integral equations of the first kind ∗

Adil Mousa

doi:10.48165/

Authors

Adil Mousa Department of Mathematics, Faculty of Science and Technology, Omdurman Islamic University, Khartoum, Sudan.

DOI:

https://doi.org/10.48165/

Keywords:

Upadhyaya transform, Inverse Upadhyaya transform, Volterra integral equation, Convolution theorem, Bessel function

Abstract

The application of the methods of integral transforms has tremendously increased during the past many years for finding the exact solutions and the numeri cal solutions of many new problems arising in the fields of engineering and applied sci ences. Some two years back the Upadhyaya transform (https://www.researchgate. net/publication/334033797) was introduced as the most powerful generalization and the most powerful unification of a number of existing variants of the classical Laplace transform occurring in the mathematics research literature. In this paper we apply the Upadhyaya transform for solving the Volterra integral equations of the first kind. Our analysis leads to the conclusion that the Upadhyaya transform is a very powerful tool for finding the solutions to the Volterra integral of the first kind.

References

Hendry, W.L. (1976). A Volterra integral equation of the first kind, J. Math. Anal. Appl., 54, 266–278.

Ahmad, Mumtaz and Tahir, Bushra (2018). A novel approach to estimate solution of Volterra integral equations, Communications in Mathematics and Applications, 9(3), 327–338. DOI: 10.26713/cma.v9i3.799

Berenguer, M.I., Gámez, D., Garralda-Guillem, A.I. and Serrano Pérez, M.C. (2010). Nonlinear Volterra integral equation of the second kind and biorthogonal systems, Abstract and Applied Analysis, Volume 2010, Article ID 135216, 11 pages. doi:10.1155/2010/135216

Babolian, E. and Masouri, Z. (2008). Direct method to solve Volterra integral equation of the first kind using operational matrix with block-pulse functions, Journal of Computational and Applied Mathematics, 220, 51–57.

Armand, A. and Gouyandeh, Z. (2014). Numerical solution of the system of Volterra integral equations of the first kind, Int. J. Industrial Mathematics, 6(1), Article ID IJIM-00310, 9 pages. [6] Brunner, Hermann (1977). Discretization of Volterra integral equations of the first kind, Math ematics of Computation, 31(139), 708–716. https://doi.org/10.2307/2006002; https://www. jstor.org/stable/2006002

Markova, E.V. and Sidorov, D.N. (2012). Volterra integral equations of the first kind with piecewise continuous kernels in the theory of evolving systems modeling, The Bulletin of Irkutsk State University. Series Mathematics, 5(2), 31–45. http://mi.mathnet.ru/eng/iigum64; http://mi. mathnet.ru/eng/iigum/v5/i2/p31

Gnanavel, M.G., Saranya, C. and Viswanathan, A. (2019). Applications of linear Volterra integral equations of first kind by using Tarig transform, International Journal of Innovative Technology and Exploring Engineering (IJITEE), 8(10), 1478–1480. DOI: 10.35940/ijitee.A1020.0881019

Mirzaee, Farshid (2012). Numerical solution for Volterra integral equations of the first kind via quadrature rule, Applied Mathematical Sciences, 6(20), 969–974.

Davies, Penny J. and Duncan, Dugald B.(2017). Numerical approximation of first kind Volterra convolution integral equations with discontinuous kernels, J. Integral Equations Applications, 29(1), 41–73. https://doi.org/10.1216/JIE-2017-29-1-41

Aggarwal, Sudhanshu, Gupta, Anjana Rani and Sharma, Swarg Deep (2019). A new application of Shehu transform for handling Volterra integral equations of first kind, International Journal of Research in Advent Technology, 7(4), 438–445.

Ngarasta, Ngarkodje (2009). Solving integral equations of the first kind by decomposition method, Kybernetes, 38(5), 733–743. https://doi.org/10.1108/03684920910962632

Taylor, P.J. (1976). The solution of Volterra integral equations of the first kind using inverted differentiation formulae, BIT Numerical Mathematics, 16, 416–425. https://doi.org/10.1007/ BF01932725

Sidorov, Nikolai A., Falaleev, Michail V. and Sidorov, Denis N. (2006). Generalized solutions of Volterra integral equations of the first kind, Bull. Malays. Math. Sci. Soc. (2), 29(1), 101–109. [15] Brunner, Hermann (2017). Volterra Integral Equations- An Introduction to Theory and Applica tions, Cambridge University Press, Cambridge. https://doi.org/10.1017/9781316162491 [16] Weisstein, Eric W. Volterra Integral Equation of the First Kind. From MathWorld –A Wolfram Web Resource. https://mathworld.wolfram.com/VolterraIntegralEquationoftheFirstKind. html

Upadhyaya, Lalit Mohan (2019). Introducing the Upadhyaya integral transform, Bull. Pure Appl. Sci. Sect. E Math. Stat., 38E(1), 471–510. doi 10.5958/2320-3226.2019.00051.1 https: //www.bpasjournals.com/, https://www.researchgate.net/publication/334033797

Upadhyaya, Lalit Mohan, Shehata, Ayman and Kamal, A. (2021). An update on the Upadhyaya transform, Bull. Pure Appl. Sci. Sect. E Math. Stat., 40E(1), 26–44. doi 10.5958/2320-3226.2021.00004.7 https://www.researchgate.net/publication/353599884, https://www.bpasjournals.com/

Chauhan, R. and Aggarwal, S. (2019). Laplace transform for convolution type linear Volterra integral equation of second kind, Journal of Advanced Research in Applied Mathematics and Statistics, 4(3 and 4), 1–7.

Sharma, N. and Aggarwal, S. (2019). Laplace transform for the solution of Abel’s integral equation, Journal of Advanced Research in Applied Mathematics and Statistics, 4(3 and 4), 8–15. [21] Aggarwal, S. and Sharma, N. (2019). Laplace transform for the solution of first kind linear Volterra integral equation, Journal of Advanced Research in Applied Mathematics and Statistics, 4(3 and 4), 16–23.

Aggarwal, S., Chauhan, R. and Sharma, N. (2018). A new application of Kamal transform for solv ing linear Volterra integral equations, International Journal of Latest Technology in Engineering, Management and Applied Science, 7(4), 138–140.

Aggarwal, S., Sharma, N. and Chauhan, R. (2018). Application of Kamal transform for solv ing linear Volterra integral equations of first kind, International Journal of Research in Advent Technology, 6(8), 2081–2088.

Aggarwal, S., Chauhan, R. and Sharma, N. (2018). A new application of Mahgoub transform for solving linear Volterra integral equations, Asian Resonance, 7(2), 46–48.

Aggarwal, S., Sharma, N. and Chauhan, R. (2018). Application of Mahgoub transform for solv ing linear Volterra integral equations of first kind, Global Journal of Engineering Science and Researches, 5(9), 154–161.

Aggarwal, S., Sharma, N. and Chauhan, R. (2018). Solution of linear Volterra integral equations of second kind using Mohand transform, International Journal of Research in Advent Technology, 6(11), 3098–3102.

Aggarwal, S., Sharma, S.D. and Gupta, A.R. (2019). A new application of Mohand transform for handling Abel’s integral equation, Journal of Emerging Technologies and Innovative Research, 6(3), 600–608.

Aggarwal, S., Sharma, N. and Chauhan, R. (2018). A new application of Aboodh transform for solving linear Volterra integral equations, Asian Resonance, 7(3), 156–158.

Aggarwal, S., Sharma, N. and Chauhan, R. (2018). Application of Aboodh transform for solv ing linear Volterra integral equations of first kind, International Journal of Research in Advent Technology, 6(12), 3745–3753.

Aggarwal, S. and Sharma, S.D. (2019). Solution of Abel’s integral equation by Aboodh transform method, Journal of Emerging Technologies and Innovative Research, 6(4), 317–325. [31] Aggarwal, S., Chauhan, R. and Sharma, N. (2018). Application of Elzaki transform for solv ing linear Volterra integral equations of first kind, International Journal of Research in Advent Technology, 6(12), 3687–3692.

Aggarwal, S. and Gupta, A.R. (2019). Sumudu transform for the solution of Abel’s integral equa tion, Journal of Emerging Technologies and Innovative Research, 6(4), 423–431. [33] Aggarwal, S. and Gupta, A.R. (2019). Shehu transform for solving Abel’s integral equation, Jour nal of Emerging Technologies and Innovative Research, 6(5), 101–110.

Adil Mousa

Aggarwal, S. and Bhatnagar, K. (2019). Solution of Abel’s integral equation using Sadik transform, Asian Resonance, 8(2), (Part-1), 57–63.

Higazy, M., Aggarwal, S. and Nofal, T.A. (2020). Sawi decomposition method for Volterra integral equation with application, Journal of Mathematics, Article ID 6687134, 13 pages. https://doi.org/10.1155/2020/6687134

Aggarwal, S., Vyas, A. and Sharma, S.D. (2020). Primitive of second kind linear Volterra inte gral equation using Shehu transform, International Journal of Latest Technology in Engineering, Management and Applied Science, 9(8), 26–32.

Adil Mousa

Aggarwal, S. and Bhatnagar, K. (2019). Solution of Abel’s integral equation using Sadik transform, Asian Resonance, 8(2), (Part-1), 57–63.

Higazy, M., Aggarwal, S. and Nofal, T.A. (2020). Sawi decomposition method for Volterra integral equation with application, Journal of Mathematics, Article ID 6687134, 13 pages. https://doi.org/10.1155/2020/6687134

Aggarwal, S., Vyas, A. and Sharma, S.D. (2020). Primitive of second kind linear Volterra inte gral equation using Shehu transform, International Journal of Latest Technology in Engineering, Management and Applied Science, 9(8), 26–32.