Application of quadruple Shehu transforms to Caputo fractional partial differential equations

PAWAR, D.D.; Kadam, R. D.; Ahmed, Wagdi F. S.

doi:10.47974/JIM-2220

Open Access Research Article

Application of quadruple Shehu transforms to Caputo fractional partial differential equations

D.D. PAWARDYPAWAR@YAHOO.COMSchool of Mathematical Sciences Swami Ramanand Teerth Marathwada UniversitySCHOOL OF MATHEMATICAL SCIENCES, SWAMI RAMANAND TEERTH MARATHWADA UNIVERSITY, VISHNUPURI NANDED-431 606, INDIVishnupuri, Nanded, Maharashtra, 431606, India0000-0001-8986-5243View full profile → , ^*R. D. KadamCorresponding authorkadamram3113@gmail.comSchool of Mathematical Sciences Swami Ramanand Teerth Marathwada UniversityNanded, Maharashtra, 431606, India0009-0007-3387-1776View full profile → , Wagdi F. S. Ahmedwagdialakel@gmail.comDepartment of Mathematics Lahej University P. O. Box 125School of Mathematical Sciences, Swami Raman and Teerth Marathwada University-Nanded, Maharashtra, 431606, India0000-0002-5115-4355View full profile →

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pp. 793–807Vol. 29Issue 4April 2026DOI: 10.47974/JIM-2220XML

Received:: 10 Jan 2024
Published Online:: 07 Nov 2025
Article type:: Research Article
Language:: EN
Article no.:: JIM-2220
Pages:: 793–807

Abstract

This article introduces an expansion of the triple Shehu transformation concept, extending it to the fourth order. Our approach begins by establishing the theoretical framework for these extended Shehu transformations, which we subsequently apply to obtain solutions for the fractional order partial differential equations within a three-dimensional context. It is important to note that the fractional derivatives in this study are considered in the Caputo sense. To illustrate the utility of our approach, we specifically focus on the fractional-order of the three-dimensional homogeneous heat equation. To present the resulting solution in a concise and elegant manner, we leverage the principles of Fox-function theory.

Keywords

Caputo derivative Fractional heat equation Fractional integral Quadruple Shehu transform

Subject Classifications

35A2226A3335K05

References

[1] M. I. Abbas, “Ulam Stability and existence results for fractional differential equations with hybrid proportional-Caputo derivatives,” Journal of Interdisciplinary Mathematics, vol. 25, no. 2, pp. 213–231 (2022).
[2] W. F. S. Ahmed, D. D. Pawar, and W. D. Patil, “Fractional kinetic equations involving generalized V-function via Laplace transform,” Advances in Mathematics: Scientific Journal, vol. 10, no. 5, pp. 2593–2610 (2021).
[3] W. F. S. Ahmed and D. D. Pawar, “Application of Sumudu Transform on Fractional Kinetic Equation Pertaining to the Generalized k-Wright Function,” Advances in Mathematics: Scientific Journal, vol. 9, no. 10, pp. 8091–8103 (2020).
[4] W. F. S. Ahmed, D. D. Pawar, and A. Y. A. Salamooni, “On the Solution of Kinetic Equation for Katugampola Type Fractional Differential Equations,” Journal of Dynamical Systems and Geometric Theories, vol. 19, no. 1, pp. 125–134 (2021), doi: 10.1080/1726037X.2021.1966946.
[5] W. F. S. Ahmed, A. Y. A. Salamooni, and D. D. Pawar, “Solution of fractional Kinetic Equation For Hadamard type fractional integral Via Mellin Transform,” Gulf Journal of Mathematics, vol. 12, no. 1, pp. 15–27 (2022).
[6] S. Alfaqeih and E. Mısırlı, “On double Shehu transform and its properties with applications,” International Journal of Analysis and Applications, vol. 18, no. 3, pp. 381–395 (2020).
[7] S. Alfaqeih, G. Bakıcıerler, and E. Mısırlı, “Application of Double Shehu Transforms to Caputo Fractional Partial Differential Equations,” Punjab University Journal of Mathematics, vol. 54, no. 1, pp. 1–13 (2022).
[8] S. R. Alkaleeli, A. A. H. Mtawal, and M. S. Hmad, “Triple Shehu transform and its properties with applications,” African Journal of Mathematics and Computer Science Research, vol. 14, no. 1, pp. 4–12 (Jan.–Jun. 2021).
[9] A. M. O. Anwar, F. Jarad, D. Baleanu, and F. Ayaz, “Fractional Caputo heat equation within the double Laplace transform,” Romanian Journal of Physics, vol. 58, no. 1-2, pp. 15–22 (2013).
[10] M. Awadalla, T. Ozis, and S. Alfaqeih, “On System of Nonlinear Fractional Differential Equations Involving Hadamard Fractional Derivative with Nonlocal Integral Boundary Conditions,” Progress in Fractional Differentiation and Applications, vol. 5, no. 3, pp. 225–232 (2019).
[11] D. Baleanu, K. Diethelm, E. Scalas, and J. J. Trujillo, Fractional Calculus: Models and Numerical Methods, World Scientific (2012).
[12] R. Belgacem, D. Baleanu, and A. Bokhari, “Shehu Transform and Applications to Caputo-Fractional Differential Equations,” International Journal of Analysis and Applications, vol. 17, no. 6, pp. 917–927 (2019).
[13] A. Bokhari, D. Baleanu, and R. Belgacem, “Application of Shehu transform to Atangana-Baleanu derivatives,” Journal of Mathematics and Computer Science, vol. 20, pp. 101–107 (2019).
[14] R. R. Dhunde and G. L. Waghmare, “Solving partial integro-differential equations using double Laplace transform method,” American Journal of Computational and Applied Mathematics, vol. 5, no. 1, pp. 7–10 (2015).
[15] R. R. Dhunde and G. L. Waghmare, “Double Laplace Transform Method for Solving Space and Time Fractional Telegraph Equations,” International Journal of Mathematics and Mathematical Sciences, vol. 2016, Article ID 1414595, 7 pages (2016).
[16] D. G. Duffy, Transform Methods for Solving Partial Differential Equations, Boca Raton, FL, USA: CRC Press (2004).
[17] H. Eltayeb and A. Kilicman, “On double Sumudu transform and double Laplace transform,” Malaysian Journal of Mathematical Sciences, vol. 4, no. 1, pp. 17–30 (2010).
[18] T. Elzaki, “Double Laplace variational iteration method for solution of nonlinear convolution partial differential equations,” Archives des Sciences, vol. 65, no. 12, pp. 588–593 (2012).
[19] T. A. Estrin and T. J. Higgins, “The solution of boundary value problems by multiple Laplace transformations,” Journal of the Franklin Institute, vol. 252, no. 2, pp. 153–167 (1951).
[20] T. Khan, M. M. Rashid, S. Ahmad, M. Sohail, and A. Khan, “On fractional order multiple integral transforms technique to handle three dimensional heat equation,” Boundary Value Problems, vol. 2022, no. 1, p. 16 (2022).
[21] A. Khan and A. Khan, “Extension of Triple Laplace Transform for Solving Fractional Differential Equations,” Discrete and Continuous Dynamical Systems - Series S, vol. 13, no. 3 (Mar. 2020).
[22] A. A. Kilbas, O. I. Marichev, and S. G. Samko, Fractional Integrals and Derivatives: Theory and Applications, Switzerland: Gordon and Breach (1993).
[23] A. Kilicman and H. Gadain, “An application of double Laplace transform and double Sumudu transform,” Lobachevskii Journal of Mathematics, vol. 30, no. 3, pp. 214–223 (2009).
[24] S. Maitama and W. Zhao, “New integral transform: Shehu transform, a generalization of Sumudu and Laplace transform for solving differential equations,” International Journal of Analysis and Applications, vol. 17, no. 2, pp. 167–190 (2019).
[25] D. D. Pawar and W. F. S. Ahmed, “Solution of Fractional Kinetic Equations Involving generalized q-Bessel function,” Results in Nonlinear Analysis, vol. 5, no. 1, pp. 87–95 (2022).
[26] D. D. Pawar, G. G. Bhuttampalle, S. B. Chavhan, W. F. Ahmed, and R. D. Kadam, “Quadruple Shehu Transform and its Applications,” arXiv preprint arXiv:2211.17265 (Nov. 2022).
[27] I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, vol. 198, San Diego, CA, USA: Academic Press (1999).
[28] Z. D. Ridha and M. F. A. Mohommed, “Solving linear ordinary differential equations by using Shehu transform with variable coefficient,” Journal of Interdisciplinary Mathematics, vol. 24, no. 8, pp. 2391–2400 (2021).
[29] L. Romero and R. Cerutti, “Fractional Fourier transform and special k-function,” International Journal of Contemporary Mathematical Sciences, vol. 7, pp. 693–704 (2012).
[30] S. T. M. Thabet, W. F. S. Ahmed, and D. D. Pawar, “On Fractional Kinetic Equation Relating to the Generalized k-Bessel Function by the Mellin Transform,” Journal of Fractional Calculus and Nonlinear Systems, vol. 3, no. 2, pp. 1–12 (2022).
[31] S. T. M. Thabet, W. F. S. Ahmed, D. D. Pawar, and A. Y. A. Salamooni, “Generalized fractional Sturm-Liouville and Langevin equations involving Caputo derivative with nonlocal conditions,” Progress in Fractional Differentiation and Applications, vol. 6, no. 3, pp. 225–237 (2020).
[32] G. K. Watugala, “The Sumudu transform for functions of two variables,” Mathematics in Engineering, Science and Aerospace (MESA), vol. 8, pp. 293–302 (2001).

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