The Rishi Transform method for solving multi-high order fractional differential equations with constant coefficients

Ali Turab; Hozan Hilmi; Juan L.G. Guirao; Shabaz Jalil; Nejmeddine Chorfi; Pshtiwan Othman Mohammed; Ali Turab; Hozan Hilmi; Juan L.G. Guirao; Shabaz Jalil; Nejmeddine Chorfi; Pshtiwan Othman Mohammed

doi:10.3934/math.2024187

AIMS Mathematics

2024, Volume 9, Issue 2: 3798-3809. doi: 10.3934/math.2024187

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The Rishi Transform method for solving multi-high order fractional differential equations with constant coefficients

1.
School of Software, Northwestern Polytechnical University, 127 West Youyi Road, Beilin District, Xi'an 710072, China
2.
Department of Mathematics, College of Science, University of Sulaimani, Sulaymani 46001, Kurdistan Region, Iraq
3.
Department of Applied Mathematics and Statistics, Technical University of Cartagena, Hospital de Marina, Cartagena 30203, Spain
4.
Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia
5.
Department of Mathematics, College of Education, University of Sulaimani, Sulaimani 46001, Kurdistan Region, Iraq

Received: 01 November 2023 Revised: 20 December 2023 Accepted: 22 December 2023 Published: 11 January 2024

In this paper, we suggest the Rishi transform, which may be used to find the analytic (exact) solution to multi-high-order linear fractional differential equations, where the Riemann-Liouville and Caputo fractional derivatives are used. We first developed the Rishi transform of foundational mathematical functions for this purpose and then described the important characteristics of the Rishi transform, which may be applied to solve ordinary differential equations and fractional differential equations. Following that, we found an exact solution to a particular example of fractional differential equations. We looked at four numerical problems and solved them all step by step to demonstrate the value of the Rishi transform. The results show that the suggested novel transform, "Rishi Transform, " yields exact solutions to multi-higher-order fractional differential equations without doing complicated calculation work.
Citation: Ali Turab, Hozan Hilmi, Juan L.G. Guirao, Shabaz Jalil, Nejmeddine Chorfi, Pshtiwan Othman Mohammed. The Rishi Transform method for solving multi-high order fractional differential equations with constant coefficients[J]. AIMS Mathematics, 2024, 9(2): 3798-3809. doi: 10.3934/math.2024187

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Abstract

In this paper, we suggest the Rishi transform, which may be used to find the analytic (exact) solution to multi-high-order linear fractional differential equations, where the Riemann-Liouville and Caputo fractional derivatives are used. We first developed the Rishi transform of foundational mathematical functions for this purpose and then described the important characteristics of the Rishi transform, which may be applied to solve ordinary differential equations and fractional differential equations. Following that, we found an exact solution to a particular example of fractional differential equations. We looked at four numerical problems and solved them all step by step to demonstrate the value of the Rishi transform. The results show that the suggested novel transform, "Rishi Transform, " yields exact solutions to multi-higher-order fractional differential equations without doing complicated calculation work.

References

[1]	J. T. Machado, V. Kiryakova, F. Mainardi, Recent History of Fractional Calculus, Sci. Numer. Simul., 16 (2011), 1140–1153. https://doi.org/10.1016/j.cnsns.2010.05.027 doi: 10.1016/j.cnsns.2010.05.027
[2]	K. B. Oldham, J. Spanier, The Fractional Calculus Theory and Applications of Differentiation and Integration to Arbitrary Order, Academic press, INC: New York, (1974), ISBN: 9780080956206.
[3]	M. Y. Almusawa, P. O. Mohammed, Approximation of sequential fractional systems of Liouville-Caputo type by discrete delta difference operators, Chaos Soliton. Fract., 176 (2023), 114098. https://doi.org/10.1016/j.chaos.2023.114098 doi: 10.1016/j.chaos.2023.114098
[4]	P. O. Mohammed, D. Baleanu, E. Al-Sarairah, T. Abdeljawad, N. Chorfi, Theoretical and numerical computations of convexity analysis for fractional differences using lower boundedness, Fractals, 31 (2023), 2340183. https://doi.org/10.1142/S0218348X23401837 doi: 10.1142/S0218348X23401837
[5]	R. Hilfer, Applications of Fractional Calculus in Physics, Appl. Fract. Calc. Phys., 90 (2000), https://doi.org/10.1142/3779 doi: 10.1142/3779
[6]	Y. Povstenko, Essentials of fractional calculus, in Fractional Thermoelasticity, Springer, 219 (2015), 1–11. https://doi.org/10.1007/978-3-319-15335-3_1
[7]	K. S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John-Wily Sons, (1993), ISBN-100471588849.
[8]	H. HamaRashid, H. M. Srivastava, M. Hama, P. O. Mohammed, E. Al-Sarairah, M. Y. Almusawa, New numerical results on existence of Volterra-Fredholm Integral equation of nonlinear boundary integro-differential type, Symmetry, 15 (2023), 1144. https://doi.org/10.3390/sym15061144 doi: 10.3390/sym15061144
[9]	A. Jan, H. M. Srivastava, A. Khan, P. O. Mohammed, R. Jan, Y. S. Hamed, In Vivo HIV Dynamics, Modeling the Interaction of HIV and Immune System via Non-Integer Derivatives, Fractal Fract., 7 (2023), 361. https://doi.org/10.3390/fractalfract7050361 doi: 10.3390/fractalfract7050361
[10]	P. O. Mohammed, R. Dahal, C. S. Goodrich, Y. S. Hamed, D. Baleanu, Analytical and numerical negative boundedness of fractional differences with Mittag--Leffler kernel, AIMS Math., 8 (2023), 5540–5550. https://doi.org/10.3934/math.2023279 doi: 10.3934/math.2023279
[11]	R. kumar, J. Chandel, S. Aggarwal, A new integral transform "Rishi Transform" with application, J. Sci. Res., 14 (2022), 521–532. https://doi.org/doi:10.3329/jsr.v2i3.4899
[12]	S. A. Ahmad, S. K. Rafiq, H. D. Hilmi, H. U. Ahmed, Mathematical modeling techniques to predict the compressive strength of pervious concrete modified with waste glass powders, Asian J. Civ. Eng., 8 (2023). https://doi.org/10.1007/s42107-023-00811-1 doi: 10.1007/s42107-023-00811-1
[13]	V. R. Hosseini, A. A. Mehrizi, H. Karimi-Maleh, M. Naddafi, A numerical solution of fractional reaction-convection-diffusion for modeling PEM fuel cells based on a meshless approach, Eng. Anal. Bound. Elem., 155 (2023), 707–716. https://doi.org/ 10.1016/j.enganabound.2023.06.016 doi: 10.1016/j.enganabound.2023.06.016
[14]	V. R. Hosseini, W. Zou, The peridynamic differential operator for solving time-fractional partial differential equations, Nonlinear Dyn., 109 (2022), 1823–1850. https//doi.org/10.1007/s11071-022-07424-4 doi: 10.1007/s11071-022-07424-4
[15]	R. Khandelwal, P. Choudhary, Y. Khandelwal, Solution of fractional ordinary differential equation by Kamal transform, Int. J. Stat. Appl. Math., 3 (2018), 279–284.
[16]	K. S. Aboodh, The new integral transform "Aboodh Transform," Glob. J. Pure Appl. Math., 9 (2013), 35–43.
[17]	V. Daftardar-Gejji, Fractional Calculus and Fractional Differential Equations, Eds., Birkhauser, (2019). https://doi.org/10.1007/978-981-13-9227-6
[18]	M. F. Kazem, A. Al-Fayadh, Solving Fredholm Integro-Differential Equation of Fractional Order by Using Sawi Homotopy Perturbation Method, J. Phys. Conf. Ser., (2022). https://doi.org/10.1088/1742-6596/2322/1/012056 doi: 10.1088/1742-6596/2322/1/012056
[19]	S. Aggarwal, R. kumar, J. Chandel, Solution of linear Volterra integral equation of second kind via Rishi transform, J. Sci. Res., 15 (2023), 111–119. https://doi.org/10.3329/jsr.v2i3.4899 doi: 10.3329/jsr.v2i3.4899
[20]	K. Diethelm, The analysis of fractional differential equations, Springer, (2004), https://doi.org/10.1007/978-3-642-14574-2
[21]	T. Abdeljawad, D. Baleanu, On fractional derivatives with generalized Mittag-Leffler kernels, Adv. Differ. Equ., 468 (2018). https://doi.org/10.1186/s13662-018-1914-2 doi: 10.1186/s13662-018-1914-2
[22]	G. Doetsch, Introduction to the Theory and Application of the Laplace Transformation, Springer, Berlin, (2012). https://doi.org/10.1007/978-3-642-65690-3
[23]	S. Ahmed, Solving Fractional Differential Equations by Using Conformable Fractional Derivatives Definition, (2016).
[24]	K. jwamer, H. Hilmi, Asymptotic Behavior of Eigenvalues and Eigenfunctions of T, Regge Fractional Problem, J. Al-Qadisiyah Comput. Sci. Math., 14 (2022), 89–100. https://doi.org/10.29304/jqcm.2022.14.3.1031 doi: 10.29304/jqcm.2022.14.3.1031
[25]	H. Hilmi, K. jwamer, Existence and Uniqueness Solution of Fractional Order Regge Problem, J. Babylon. Res., 30 (2022), 80–96. https://doi.org/10.29196/jubpas.v30i2.4186 doi: 10.29196/jubpas.v30i2.4186
[26]	T. Abdeljawad, Fractional operators with exponential kernels and a Lyapunov type inequality, Adv. Differ. Equ., 313 (2017). https://doi.org/10.1186/s13662-017-1285-0 doi: 10.1186/s13662-017-1285-0

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