On the $(k,\psi )$-Generalized Laplace Transforms and Their Applications to Fractional Differential Equations

Adil Mısır; Yasemin Başcı; Emine Cengizhan

doi:10.33434/cams.1739206

On the $(k,\psi )$-Generalized Laplace Transforms and Their Applications to Fractional Differential Equations

Abstract

In this paper, a new generalization of the Laplace transform, called the $(k,\psi )$-generalized Laplace transform, which plays an important role in solving many problem models, is introduced and its special properties are given. In addition, the previously defined integral transforms of some elementary functions and the relations between the new transform, the $(k,\psi )$-generalized Laplace transform, and other generalized Laplace transforms are given. Using the $(k,\psi )$-generalized Laplace transform, solutions to problems of fractional differential equations and fractional tempered differential equations are obtained as applications. Finally, examples are given to show that the $(k,\psi )$-generalized Laplace transform is useful for more general problems.

Keywords

$(k,\psi)$-generalized Laplace transform, Fractional calculus, Integral transforms, Laplace transform

References

[1] P. S. Laplace, Théorie Analytique des Probabilités, Imprimerie Royale, 1812.
[2] G. K. Watugala, 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. https://doi.org/10.1080/0020739930240105
[3] H. K. Zafar, A. K. Waqar, N-transform properties and applications, NUST J. Eng. Sci., 1(1) (2008), 127-133.
[4] T. M. Elzaki, The new integral transform Elzaki transform, Glob. J. Pure Appl. Math., 7(1) (2011), 57-64.
[5] D. Loonker, P. K. Banerji, Applications of natural transform to differential equations, J. Indian Acad. Math., 35(1) (2013), 151-158.
[6] A. Atangana, A. Kilicman, A novel integral operator transform and its application to some FODE and FPDE with some kind of singularities, Math. Probl. Eng., 2013 (2013), Article ID 531984. https://doi.org/10.1155/2013/531984
[7] A. Kashuri, A. Fundo, A new integral transform, Adv. Theor. Appl. Math., 8(1) (2013), 27-43.
[8] K. S. Aboodh, The new integral transform Aboodh transform, Glob. J. Pure Appl. Math., 9(1) (2013), 35-43.
[9] T. Abdeljawad, On conformable fractional calculus, J. Comput. Appl. Math., 279 (2013), 57-66. https://doi.org/10.1016/J.CAM.2014.10.016
[10] H. M. Srivastava, L. Minjie, R. Raina, A new integral transform and its applications, Acta Math. Sci., 35(6) (2015), 1386-1400. https://doi.org/10.1016/S0252-9602(15)30061-8

[11] A. Kamal, H. Sedeeg, The new integral transform Kamal transform, Adv. Theor. Appl. Math., 11(4) (2016), 451-458.
[12] M. A. M. Mahgoub, The new integral transform Mahgoub transform, Adv. Theor. Appl. Math., 11(4) (2016), 391-398.
[13] G. D. Medina, N. R. Ojeda, J. H. Pereira, L. G. Romero, A new $\alpha$-integral Laplace transform, Asian J. Current Eng. Maths., 5(5) (2016), 59-62.
[14] Z. Zafar, ZZ transform method, Int. J. Adv. Eng. Global Tech., 4(1) (2016), 1605-1611.
[15] H. Kim, The intrinsic structure and properties of Laplace-type integral transforms, Math. Probl. Eng., 2017 (2017), Article ID 1762729. https://doi.org/10.1155/2017/1762729
[16] S. L. Shaikh, Introducing a new integral transform: Sadik transform, American International Journal of Research in Science, Technology, Engineering and Mathematics, 22(1) (2018), 100-102.
[17] S. A. P. Ahmadi, H. Hosseinzadeh, A. Y. Cherati, A new integral transform for solving higher order linear ordinary Laguerre and Hermite differential equations, Int. J. Appl. Comput. Math., 5 (2019), Article ID 142. https://doi.org/10.1007/s40819-019-0712-1
[18] S. Maitama, W. Zhao, 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. https://doi.org/10.28924/2291-8639-17-2019-167
[19] M. M. A. Mahgoub, The new integral transform “Mohand transform”, Adv. Theor. Appl. Math., 12(2) (2017), 113-120.
[20] M. M. A. Mahgoub, The new integral transform “Sawi transform”, Adv. Theor. Appl. Math., 14(1) (2019), 81-87.
[21] L. M. Upadhyaya, Introducing the Upadhyaya integral transform, Bull. Pure Appl. Sci., 38(1) (2019), 471-510.
[22] R. Gupta, Propounding a new integral transform: Gupta transform with applications in science and engineering, Int. J. Sci. Res. Multidiscip. Stud., 16(3) (2020), 14-19.
[23] R. Saadeh, A. Qazza, A. Burqan, A new integral transform: ARA transform and its properties and applications, Symmetry, 12(6) (2020), Article ID 925. https://doi.org/10.3390/sym12060925
[24] F. Jarad, T. Abdeljawad, Generalized fractional derivatives and Laplace transform, Discrete Contin. Dyn. Syst., Ser. S, 13(3) (2020), 709-722. https://doi.org/10.3934/dcdss.2020039
[25] H. Jafari, A new general integral transform for solving integral equations, J. Adv. Res., 32 (2021), 133-138. https://doi.org/10.1016/j.jare.2020.08.016
[26] F. M. Sara, K. Emad, S. A. Elaf, Emad-Sara transform a new integral transform, J. Interdiscip. Math., 24(7) (2021), 1985-1994. https://doi.org/10.1080/09720502.2021.1963523
[27] E. Ata, I. O. Kıymaz, A new generalized Laplace transform and its applications to fractional Bagley-Torvik and fractional harmonic vibration problems, Miskolc Math. Notes, 24(2) (2023), 597-610. https://doi.org/10.18514/MMN.2023.4099
[28] R. Z. Saadeh, B. fu’ad Ghaza, A new approach on transforms: Formable integral transform and its applications, Axioms, 10(4) (2021), Article ID 332. https://doi.org/10.3390/axioms10040332
[29] F. S. Khan, M. Khalid, Fareeha transform: A new generalized Laplace transform, Math. Meth. Appl. Sci., 46(9) (2023), 11043-11057. https://doi.org/10.1002/mma.9167
[30] Y. Başcı, A. Mısır, S. Öğrekçi, Generalized derivatives and Laplace transform in ($k,\psi$)-Hilfer form, Math. Meth. Appl. Sci., 46(9) (2023), 10400-10420. https://doi.org/10.1002/mma.9129
[31] R. Diaz, E. Pariguan, On hypergeometric functions and k-Pochhammer symbol, Divulg. Mat., 15(2) (2007), 179-192.
[32] T. R. Prabhakar, A singular integral equation with a generalized Mittag-Leffler function in the kernel, Yokohama Math. J., 19(1) (1971), 7-15.
[33] A. Mısır, E. Cengizhan, Y. Başcı, Ulam type stability of $\psi$-Riemann-Liouville fractional differential equations using ($k,\psi $)-generalized Laplace transform, J. Nonlinear Sci. Appl., 17(2) (2024), 100-114. https://doi.org/10.22436/jnsa.017.02.03
[34] R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific Publishing Company, 2000. https://doi.org/10.1142/3779
[35] J. V. C. Sousa, E. C. Oliveira, On the $\psi$-Hilfer fractional derivative, Commun. Nonlinear Sci. Numer. Simul., 60 (2018), 72-91. https://doi.org/10.1016/j.cnsns.2018.01.005
[36] K. D. Kucche, A. D. Mali, On the nonlinear ($k,\psi $)-Hilfer fractional differential equations, Chaos Solitons Fractals, 152 (2021), Article ID 111335. https://doi.org/10.1016/j.chaos.2021.111335
[37] K. D. Kucche, A. D. Mali, A. Fernandez, H. M. Fahad, On tempered Hilfer fractional derivatives with respect to functions and the associated fractional differential equations, Chaos Solitons Fractals, 163 (2022), Article ID 112547. https://doi.org/10.1016/j.chaos.2022.112547
[38] A. D. Mali, K. D. Kucche, A. Fernandez, H. M. Fahad, On tempered fractional calculus with respect to functions and the associated fractional differential equations, Math. Meth. Appl. Sci., 45(17) (2022), 11134-11157. https://doi.org/10.1002/mma.8441
[39] A. Salima, M. Benchohra, A Study on tempered ($k,\psi $)-Hilfer fractional operator, Lett. Nonlinear Anal. Appl., 1(3) (2023), 101-121. https://doi.org/10.5281/zenodo.8361961
[40] M. Benchohra, E. Karapınar, J. E. Lazreg, A. Salim, Fractional Differential Equations: New Advancements for Generalized Fractional Derivative, Springer, Cham, 2023.
[41] M. Benchohra, E. Karapınar, J. E. Lazreg, A. Salim, Advanced Topics in Fractional Differential Equations: A Fixed Point Approach, Springer, Cham, 2023.
[42] C. Li, W. Deng, L. Zhao, Well-posedness and numerical algorithm for the tempered fractional differential equations, Discrete Contin. Dyn. Syst. Ser. B, 24(4) (2019), 1989-2015. https://doi.org/10.3934/dcdsb.2019026

Details

Primary Language

English

Subjects

Ordinary Differential Equations, Difference Equations and Dynamical Systems, Applied Mathematics (Other)

Journal Section

Research Article

Authors

Adil Mısır ^*
0000-0002-4552-0769
Türkiye

Yasemin Başcı
0000-0003-3151-6467
Türkiye

Emine Cengizhan
0000-0001-6740-1448
Türkiye

Early Pub Date

December 3, 2025

Publication Date

December 8, 2025

Submission Date

July 10, 2025

Acceptance Date

November 4, 2025

Published in Issue

Year 2025 Volume: 8 Number: 4

DOI

https://doi.org/10.33434/cams.1739206

IZ

https://izlik.org/JA64PM47CD

APA

Mısır, A., Başcı, Y., & Cengizhan, E. (2025). On the $(k,\psi )$-Generalized Laplace Transforms and Their Applications to Fractional Differential Equations. Communications in Advanced Mathematical Sciences, 8(4), 210-224. https://doi.org/10.33434/cams.1739206

AMA

1.Mısır A, Başcı Y, Cengizhan E. On the $(k,\psi )$-Generalized Laplace Transforms and Their Applications to Fractional Differential Equations. Communications in Advanced Mathematical Sciences. 2025;8(4):210-224. doi:10.33434/cams.1739206

Chicago

Mısır, Adil, Yasemin Başcı, and Emine Cengizhan. 2025. “On the $(k,\psi )$-Generalized Laplace Transforms and Their Applications to Fractional Differential Equations”. Communications in Advanced Mathematical Sciences 8 (4): 210-24. https://doi.org/10.33434/cams.1739206.

EndNote

Mısır A, Başcı Y, Cengizhan E (December 1, 2025) On the $(k,\psi )$-Generalized Laplace Transforms and Their Applications to Fractional Differential Equations. Communications in Advanced Mathematical Sciences 8 4 210–224.

IEEE

[1]A. Mısır, Y. Başcı, and E. Cengizhan, “On the $(k,\psi )$-Generalized Laplace Transforms and Their Applications to Fractional Differential Equations”, Communications in Advanced Mathematical Sciences, vol. 8, no. 4, pp. 210–224, Dec. 2025, doi: 10.33434/cams.1739206.

ISNAD

Mısır, Adil - Başcı, Yasemin - Cengizhan, Emine. “On the $(k,\psi )$-Generalized Laplace Transforms and Their Applications to Fractional Differential Equations”. Communications in Advanced Mathematical Sciences 8/4 (December 1, 2025): 210-224. https://doi.org/10.33434/cams.1739206.

JAMA

1.Mısır A, Başcı Y, Cengizhan E. On the $(k,\psi )$-Generalized Laplace Transforms and Their Applications to Fractional Differential Equations. Communications in Advanced Mathematical Sciences. 2025;8:210–224.

MLA

Mısır, Adil, et al. “On the $(k,\psi )$-Generalized Laplace Transforms and Their Applications to Fractional Differential Equations”. Communications in Advanced Mathematical Sciences, vol. 8, no. 4, Dec. 2025, pp. 210-24, doi:10.33434/cams.1739206.

Vancouver

1.Adil Mısır, Yasemin Başcı, Emine Cengizhan. On the $(k,\psi )$-Generalized Laplace Transforms and Their Applications to Fractional Differential Equations. Communications in Advanced Mathematical Sciences. 2025 Dec. 1;8(4):210-24. doi:10.33434/cams.1739206

The published articles in CAMS are licensed under a Creative Commons Attribution-NonCommercial 4.0 International License..

On the $(k,\psi )$-Generalized Laplace Transforms and Their Applications to Fractional Differential Equations

Abstract

Keywords

References

Details

Primary Language

Subjects

Journal Section

Authors

Early Pub Date

Publication Date

Submission Date

Acceptance Date

Published in Issue

DOI

IZ

Cite