Remarks on the Solution of Fractional Ordinary Differential Equations Using Laplace Transform Method

John Ojima Mamman; Gloria Ojima-Ojo Oguche; Usman Akwu

doi:10.21467/ajgr.14.1.21-26

Authors

John Ojima Mamman Department of Mathematics/Statistics/Computer Science, Faculty of Science Federal University of Agriculture Makurdi, Benue State https://orcid.org/0000-0002-8373-328X
Gloria Ojima-Ojo Oguche Department of Mathematics Education, Kogi State University Ayangba, Kogi State https://orcid.org/0009-0001-4804-8543
Usman Akwu Department of Mathematics/Statistics/Computer Science, Faculty of Science Federal University of Agriculture Makurdi, Benue State

DOI:

https://doi.org/10.21467/ajgr.14.1.21-26

Abstract

In this work we used the Laplace transform method to solve linear fractional-order differential equation, fractional ordinary differential equations with constant and variable coefficients. The solutions were expressed in terms of Mittag-Leffler functions, and then written in a compact simplified form. As a special case for simplicity, the order of the derivative determined the order of the solution that was obtained. This paper presented several case studies involving the implementation of Fractional Order calculus-based models, whose results demonstrate the importance of Fractional Order Calculus.

Keywords:

Fractional calculus, Laplace transform, Mittag-Leffler function

Downloads

Download data is not yet available.

References

I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering, Academic Press, San Diego vol. 198. (1999) .http://www.sciepub.com/reference/3051

O. J. Mamman, T. Aboiyar, and T. Tivde "Solution of generalized Abel’s integral equation using orthogonal polynomials", Open J. Math. Anal. 2022, 6(2), 65-73; doi:10.30538/psrp-oma2022.0113

A. Alemnew, M. Benyam, "On Solutions to Fractional Iterative Differential Equations with Caputo Derivative", Journal of Mathematics, vol. 2023, Article ID 5598990, 2023. https://doi.org/10.1155/2023/5598990 DOI: https://doi.org/10.1155/2023/5598990

J. Mamman and T. Aboiyar, “A Numerical Calculation of Arbitrary Integrals of Functions”, Adv. J. Grad. Res., vol. 7, no. 1, pp. 11-17, Oct. (2019). doi:10.21467/ajgr.7.1.11-17 DOI: https://doi.org/10.21467/ajgr.7.1.11-17

S. M. Sivalingam, K. Pushpendra, V. Govindaraj, “A novel numerical scheme for fractional differential equations using extreme learning machine”, Physica A: Statistical Mechanics and its Applications, Volume 622,2023, 128887,ISSN 0378-4371,https://doi.org/10.1016/j.physa.2023.128887. DOI: https://doi.org/10.1016/j.physa.2023.128887

K. Wang, & S. Liu, “He’s fractional derivative and its application for fractional Fornberg-Whitham equation,” Thermal Science. 2016. 54-54. (2016). DOI: https://doi.org/10.1016/j.camwa.2016.03.010

S. Sharma, P. Kumar Sharma, and J. Kaushik, ‘A Review Note on Laplace Transform and Its Applications in Dynamical Systems’, Qualitative and Computational Aspects of Dynamical Systems. IntechOpen, Feb. 08, 2023. doi:10.5772/intechopen.108251. DOI: https://doi.org/10.5772/intechopen.108251

V. Tarasov, “On History of Mathematical Economics: Application of Fractional Calculus,” Mathematics. (2019). 7(6). 509. https://doi.org/10.3390/math7060509 DOI: https://doi.org/10.3390/math7060509

D. Luo, & J. Wang, & M. Feckan, “Applying Fractional Calculus to Analyze Economic Growth Modelling,” Journal of Applied Mathematics, Statistics and Informatics, (2018). 14. 25-36. DOI: https://doi.org/10.2478/jamsi-2018-0003

Y. Chii-Huei "Exact Solution of Linear System of Fractional Differential Equations with Constant Coefficients" International Journal of Mechanical and Industrial Technology ISSN 2348-7593 Vol. 10, Issue 2, pp: (1-7), Month:March 2023, Available at: www.researchpublish.com

A. A. Ebrahem, S. A. Musaad, A. Mounirah, R. E. Essam, E. Abdelhalim, and K. A. Hind, "A Proposed Application of Fractional Calculus on Time Dilation in Special Theory of Relativity" Advances in Fractional Operators and Their Applications in Physical Sciences,(2023)11(15),3343; https://doi.org/10.3390/math11153343 DOI: https://doi.org/10.3390/math11153343

O. K. Wanassi, D.F.M. Torres, “An integral boundary fractional model to the world population growth”, Chaos, Solitons and Fractals, Volume 168, 2023, 113151, ISSN 0960-0779,https://doi.org/10.1016/j.chaos.2023.113151. DOI: https://doi.org/10.1016/j.chaos.2023.113151

V. E. Tarasov, & V. V. Tarasova, "Macroeconomic models with long dynamic memory: Fractional calculus approach," Applied Mathematics and Computation, Elsevier, 2018. vol. 338(C), pages 466-486. DOI: https://doi.org/10.1016/j.amc.2018.06.018

H. Sun, & Y. Zhang, & D. Baleanu, & W. Chen, & Y. Chen, “A new collection of real world applications of fractional calculus in science and engineering,” Communications in Nonlinear Science and Numerical Simulation. (2018). 64. DOI: https://doi.org/10.1016/j.cnsns.2018.04.019

A. Dalal, M. Dalal, A. Hashim "Adomian Decomposition Method for Solving Fractional Time-Klein-Gordon Equations Using Maple" Applied Mathematics Vol.14 No.6, June 6, 2023 DOI: 10.4236/am.2023.146024 DOI: https://doi.org/10.4236/am.2023.146024

K. Diethelm, V. Kiryakova, Y. Luchko, J. A. T. Machado, V. E. Tarasov "Trends, directions for further research, and some open problems of fractional calculus" Nonlinear Dyn (2022) 107:3245–3270 https://doi.org/10.1007/s11071-021-07158-9 DOI: https://doi.org/10.1007/s11071-021-07158-9