Khalouta transform via different fractional derivative operators

A. Khalouta

Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, Tome 28 (2024) no. 3, pp. 407-425.

Résumé

Recently, the author defined and developed a new integral transform namely the Khalouta transform, which is a generalization of many well-known integral transforms. The purpose of this paper is to extend this new integral transform to include different fractional derivative operators. The fractional derivatives are described in the sense of Riemann–Liouville, Liouville–Caputo, Caputo–Fabrizio, Atangana–Baleanu–Riemann–Liouville, and Atangana–Baleanu–Caputo. Theorems dealing with the properties of the Khalouta transform for solving fractional differential equations using the mentioned fractional derivative operators are proven. Several examples are presented to verify the reliability and effectiveness of the proposed technique. The results show that the Khalouta transform is more efficient and useful in dealing with fractional differential equations.

Détail

Keywords: fractional differential equations, Riemann–Liouville derivative, Liouville–Caputo derivative, Caputo–Fabrizio derivative, Atangana–Baleanu–Riemann–Liouville derivative, Atangana–Baleanu–Caputo derivative
Mots-clés : Khalouta transform, exact solution

Bibliographie

[1] Chen Y., Moore K. L., “Analytical stability bound for a class of delayed fractional-order dynamic systems”, Nonlinear Dyn., 29 (2002), 191–200 | DOI | MR | Zbl

[2] Friedrich C., “Relaxation and retardation functions of the Maxwell model with fractional derivatives”, Rheol. Acta, 30 (1991), 151–158 | DOI

[3] Khalouta A., “The existence and uniqueness of solution for fractional Newel–Whitehead–Segel equation within Caputo–Fabrizio fractional operator”, Appl. Appl. Math., 16:2 (2021), 894–909 https://digitalcommons.pvamu.edu/aam/vol16/iss2/7/ | MR | Zbl

[4] Khalouta A., “A novel representation of numerical solution for fractional Bratu-type equation”, Adv. Stud.: Euro-Tbil. Math. J., 15:1 (2022), 93–109 | DOI | MR | Zbl

[5] Magin R. L., Ingo C., Colon-Perez L., et al., “Characterization of anomalous diffusion in porous biological tissues using fractional order derivatives and entropy”, Microporous Mesoporous Mater., 178 (2013), 39–43 | DOI

[6] Watugula G. K., “Sumudu transform: A new integral transform to solve differential equations and control engineering problems”, Int. J. Math. Educ. Sci. Technol., 24:1 (1993), 35–43 | DOI | MR | Zbl

[7] Khan Z. H., Khan W. A., “N-transform – properties and applications”, NUST J. Eng. Sci, 1:1 (2008), 127–133

[8] Elzaki T. M., “The new integral transform “Elzaki transform””, Glob. J. Pure Appl. Math., 7:1 (2011), 57–64 https://www.ripublication.com/gjpamv7/gjpamv7n1_7.pdf

[9] Atangana A., Kiliçman A., “A novel integral operator transform and its application to some FODE and FPDE with some kind of singularities”, Math. Probl. Eng., 2013, 531984 | DOI | MR | Zbl

[10] Srivastava H. M., Luo M., Raina R. K., “A new integral transform and its applications”, Acta Math. Sci., Ser. B, Engl. Ed., 35:6 (2015), 1386–1400 | DOI | MR | Zbl

[11] Zafar Z. U. A., “ZZ transform method”, Int. J. Adv. Eng. Glob. Technol., 4:1 (2016), 1605–1611

[12] Ramadan M., Raslan K. R., El-Danaf T., Hadhoud A., “On a new general integral transform: Some properties and remarks”, J. Math. Comput. Sci., 6:1 (2016), 103–109 https://scik.org/index.php/jmcs/article/view/2392

[13] Barnes B., “Polynomial integral transform for solving differential equations”, Eur. J. Pure Appl. Math., 9:2 (2016), 140–151 http://www.ejpam.com/index.php/ejpam/article/view/2531 | MR | Zbl

[14] Yang X. J., “A new integral transform method for solving steady heat-transfer problem”, Thermal Science, 20, Suppl. 3 (2016), S639–S642 | DOI

[15] Aboodh K. S., Abdullahi I., Nuruddeen R. I., “On the Aboodh transform connections with some famous integral transforms”, Int. J. Eng. Inf. Syst., 1:9 (2017), 143–151 http://ijeais.org/wp-content/uploads/2017/11/IJEAIS171116.pdf

[16] Rangaig N., Minor N., Penonal G., et al., “On another type of transform called Rangaig transform”, Int. J. Partial Differ. Equ. Appl., 5:1 (2017), 42–48 | DOI

[17] Maitama S., Zhao W., “New integral transform: Shehu transform a generalization of Sumudu and Laplace transform for solving differential equations”, Int. J. Anal. Appl., 17:2 (2019), 167–190, arXiv: [math.GM] 1904.11370 | DOI | MR | Zbl

[18] Khalouta A., “A new exponential type kernel integral transform: Khalouta transform and its applications”, Math. Montisnigri, 57 (2023), 5–23 | DOI | MR | Zbl

[19] Kilbas A. A., Srivastava H. M., Trujillo J. J., Theory and Applications of Fractional Differential Equations, North Holland Mathematics Studies, 204, Elsevier, Amsterdam, 2006, xv+523 pp | DOI | MR | Zbl

[20] Caputo M., Fabrizio M., “A new definition of fractional derivative without singular kernel”, Progr. Fract. Differ. Appl., 1:2 (2015), 73–85 https://www.naturalspublishing.com/files/published/0gb83k287mo759.pdf

[21] Losada J., Nieto J. J., “Properties of a new fractional derivative without singular kernel”, Progr. Fract. Differ. Appl., 1:2 (2015), 87–92 https://www.naturalspublishing.com/files/published/2j1ns3h8o2s789.pdf

[22] Atangana A., Baleanu D., “New fractional derivatives with non-local and non-singular kernel: Theory and application to heat transfer model”, Thermal Science, 20:2 (2016), 763–769, arXiv: [math.GM] 1602.03408 | DOI | MR

[23] Rawashdeh M. S., Al-Jammal H., “Theories and applications of the inverse fractional natural transform method”, Adv. Differ. Equ., 2018 (2018), 222 | DOI | MR | Zbl

[24] Bodkhe D. S., Panchal S. K., “On Sumudu transform of fractional derivatives and its applications to fractional differential equations”, Asian J. Math. Comp. Res., 11:1 (2016), 69–77 https://ikprress.org/index.php/AJOMCOR/article/view/380

[25] Aruldoss R., Devi R. A., “Aboodh transform for solving fractional differential equations”, Glob. J. Pure Appl. Math., 16:2 (2020), 145–153 https://www.ripublication.com/gjpam20/gjpamv16n2_01.pdf | MR

[26] Belgacem R., Baleanu D., Bokhari A., “Shehu transform and applications to Caputo–Fractional differential equations”, Int. J. Anal. Appl., 17:6 (2019), 917–927 | DOI

[27] Toprakseven Ş., “The existence and uniqueness of initial-boundary value problems of the fractional Caputo–Fabrizio differential equations”, Univers. J. Math. Appl., 2:2 (2019), 100–106 | DOI

[28] Akgul A., Özturk G., “Application of the Sumudu transform to some equations with fractional derivatives”, Sigma J. Eng. Nat. Sci., 41:6 (2023), 1132–1143 | DOI

[29] Bokhari A., Baleanub D., Belgacem R., “Application of Shehu transform to Atangana–Baleanu derivatives”, J. Math. Comput. Sci., 20:2 (2020), 101–107 | DOI

[30] Jena R. M., Chakraverty S., Baleanu D., Alqurashi M. M., “New aspects of ZZ transform to fractional operators with Mittag–Leffler kernel”, Front. Phys., 8 (2020), 352 | DOI