A NOVEL ANALYTICAL METHOD FOR TIME FRACTIONAL CONVECTION-DIFFUSION EQUATION THROUGH CLIQUE POLYNOMIALS OF THE COCKTAIL PARTY GRAPH

Mine Aylin Bayrak; Ali Demir; Ahmet Büyük

doi:10.11948/20240168

2025 Volume 15 Issue 1

Article Contents

Article navigation > Journal of Applied Analysis & Computation > 2025 > 15(1): 564-573

Next Article Previous Article

Mine Aylin Bayrak, Ali Demir, Ahmet Büyük. A NOVEL ANALYTICAL METHOD FOR TIME FRACTIONAL CONVECTION-DIFFUSION EQUATION THROUGH CLIQUE POLYNOMIALS OF THE COCKTAIL PARTY GRAPH[J]. Journal of Applied Analysis & Computation, 2025, 15(1): 564-573. doi: 10.11948/20240168

Citation:

Mine Aylin Bayrak, Ali Demir, Ahmet Büyük. A NOVEL ANALYTICAL METHOD FOR TIME FRACTIONAL CONVECTION-DIFFUSION EQUATION THROUGH CLIQUE POLYNOMIALS OF THE COCKTAIL PARTY GRAPH[J]. Journal of Applied Analysis & Computation, 2025, 15(1): 564-573. doi: 10.11948/20240168

A NOVEL ANALYTICAL METHOD FOR TIME FRACTIONAL CONVECTION-DIFFUSION EQUATION THROUGH CLIQUE POLYNOMIALS OF THE COCKTAIL PARTY GRAPH

1.
Department of Mathematics, Kocaeli University, Umuttepe, 41001, Izmit, Kocaeli, Turkey

Author Bio: Email: ademir@kocaeli.edu.tr(A. Demir); Email: ahmetbuyuk89@gmail.com(A. Büyük)
Corresponding author: Email: aylin@kocaeli.edu.tr(M. A. Bayrak)

Abstract

Abstract

This paper is devoted to providing a new approach to solve time fractional convection-diffusion equation (TFCDE) by utilizing Clique polynomials of the Cocktail party graph and collocation points. The main advantage of this method is converting the TFCDE into a system of ordinary fractional differential and algebraic equations. At this stage, Residual power series method (RPSM) is used to determine the unknown functions of the obtained system. Convergence analysis is given to substantiate the importance of the suggested method. Two numerical examples are presented to illustrate the implementation and effectiveness of the proposed method.
- Fractional convection-diffusion equation /
- collocation points /
- Clique polynomials /
- residual power series method
MSC: 35R11, 65L05

FullText(HTML)

References

[1]	R. Amin, B. Alshahrani, M. Mahmoud, A. H. Abdel-Aty, K. Shah and W. Deebani, Haar wavelet method for the solution of distributed order time-fractional differential equations, Alex. Eng. J., 2021, 60(3), 3295–3303. doi: 10.1016/j.aej.2021.01.039 CrossRef Google Scholar
[2]	B. A. Carreras, V. E. Lynch and G. M. Zaslavsky, Anomalous diffusion and exit time distribution of particle tracers in plasma turbulence models, Phys. Plasmas., 2001, 8(12), 5096–5103. doi: 10.1063/1.1416180 CrossRef Google Scholar
[3]	K. Diethelm, The Analysis of Fractional Differential Equations, Springer-Verlag, Berlin, 2010. Google Scholar
[4]	R. Hifler, Applications of Fractional Calculus in Physics, London, World Scientific Publishing Company, 2000. Google Scholar
[5]	M. Izadi, J. Singh and S. Noeiaghdam, Simulating accurate and effective solutions of some nonlinear nonlocal two-point BVPs: Clique and QLM-clique matrix methods, Heliyon, 2023, 9, e22267. doi: 10.1016/j.heliyon.2023.e22267 CrossRef Google Scholar
[6]	M. M. Izadkhah and J. Saberi-Nadjafi, Gegenbauer spectral method for time-fractional convection-diffusion equations with variable coefficients, Math. Meth. Appl. Sci., 2015, 38(15), 3183–3194. doi: 10.1002/mma.3289 CrossRef Google Scholar
[7]	A. Jajarmi, D. Baleanu, S. S. Sajjadi and J. J. Nieto, Analysis and some applications of a regularized ψ-Hilfer fractional derivative, J. Comput. Appl. Math., 2022, 415, 114476. doi: 10.1016/j.cam.2022.114476 CrossRef Google Scholar
[8]	N. Jibenja, B. Yuttanan and M. Razzaghi, An efficient method for numerical solutions of distributed-order fractional differential equations, J. Comput. Nonlinear Dyn., 2018, 13(11), 111003. doi: 10.1115/1.4040951 CrossRef Google Scholar
[9]	A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, in: North-Holland Mathematics Studies, Elsevier Science B.V., Amsterdam, 204, 2006. Google Scholar
[10]	V. S. S. Kumar, The Chebyshev collocation method for a class of time fractional convection-diffusion equation with variable coefficients, Math. Meth. Appl. Sci., 2021, 44, 6666–6678. doi: 10.1002/mma.7215 CrossRef Google Scholar
[11]	O. Kurkcu, E. Aslan and M. Sezer, A novel graph-operational matrix method for solving multi delay fractional differential equations with variable coefficients and a numerical comparative survey of fractional derivative types, Turkish Journal of Mathematics, 2019, 43(1), 373–392. doi: 10.3906/mat-1806-87 CrossRef Google Scholar
[12]	K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, 1993. Google Scholar
[13]	R. Metler and J. Klafter, The random walks guide to anomalous diffusion: A fractional dynamics approach, Phys. Reports, 2000, 339(1), 1–77. doi: 10.1016/S0370-1573(00)00070-3 CrossRef Google Scholar
[14]	A. N. Nirmala and S. Kumbinarasaiah, A novel analytical method for the multi-delay fractional differential equations through the matrix of clique polynomials of the cocktail party graph, Results in Control and Optimization, 2023, 12, 100280. doi: 10.1016/j.rico.2023.100280 CrossRef Google Scholar
[15]	A. N. Nirmala and S. Kumbinarasaiah, A new graph theoretic analytical method for nonlinear distributed order fractional ordinary differential equations by clique polynomial of cocktail party graph, Journal of Umm Al-Qura University for Applied Sciences, 2024. DOI: 10.1007/s43994-023-00116-8. CrossRef Google Scholar
[16]	I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999 Physics and Engineering, Springer, Dordrecht, 2007. Google Scholar
[17]	M. Pourbabaee and A. Saadatmandi, Collocation method based on Chebyshev polynomials for solving distributed order fractional differential equations, Comput. Methods Differ. Equ., 2021, 9(3), 858–873. Google Scholar
[18]	M. Raberto, E. Scalas and F. Mainardi, Waiting-times and returns in high-frequency financial data: An empirical study, Phys. A: Stat. Mech. Appl., 2002, 314(1–4), 749–755. doi: 10.1016/S0378-4371(02)01048-8 CrossRef Google Scholar
[19]	M. Randic, P. J. Hansen and P. C. Jurs, Search for useful graph theoretical invariants of molecular structure, J. Chem. Inf. Comput. Sci., 1988, 28(2), 60–68. doi: 10.1021/ci00058a004 CrossRef Google Scholar
[20]	A. Saadatmandi, M. Dehghan and M. R. Azizi, The Sinc-Legendre collocation method for a class of fractional convection-diffusion equations with variable coefficients, Commun. Nonlinear Sci. Numer. Simul., 2012, 17(11), 4125–4136. doi: 10.1016/j.cnsns.2012.03.003 CrossRef Google Scholar
[21]	L. Sabatelli, S. Keating, J. Dudley and P. Richmond, Waiting time distributions in financial markets, Eur. Phys. J. B., 2002, 27(2), 273–275. Google Scholar
[22]	Y. Shi, M. Dehmer, X. Li and I. Gutman(eds), Graph Polynomials, CRC Press (List of graph polynomials), 2016. Google Scholar
[23]	H. Sun, Y. Zhang, D. Baleanu, W. Chen and Y. Chen, A new collection of real-world applications of fractional calculus in science and engineering, Commun. in Nonlin. Sci. Num. Simul., 2018, 64, 213–231. doi: 10.1016/j.cnsns.2018.04.019 CrossRef Google Scholar
[24]	G. S. Teodoro, J. T. Machado and E. C. De Oliveira, A review of definitions of fractional derivatives and other operators, J. Comput. Phys., 2019, 388, 195–208. doi: 10.1016/j.jcp.2019.03.008 CrossRef Google Scholar
[25]	H. Wang, K. Wang and T. Sircar, A direct O(N log2 N) finite difference method for fractional diffusion equations, J. Comput. Phys., 2010, 229(21), 8095–8104. doi: 10.1016/j.jcp.2010.07.011 CrossRef Google Scholar
[26]	K. Wang and H. Wang, A fast characteristic finite difference method for fractional advection-diffusion equations, Adv. Water Res., 2011, 34(7), 810–816. doi: 10.1016/j.advwatres.2010.11.003 CrossRef Google Scholar
[27]	Y. Xu, Y. Zhang and J. Zhao, Error analysis of the Legendre-Gauss collocation methods for the nonlinear distributed-order fractional differential equation, Appl. Numer. Math., 2019, 142, 122–138. doi: 10.1016/j.apnum.2019.03.005 CrossRef Google Scholar
[28]	G. M. Zaslavsky, D. Stevens and H. Weitzner, Self-similar transport in incomplete chaos, Phys. Rev. E, 1993, 48(3), 1683–1694. doi: 10.1103/PhysRevE.48.1683 CrossRef Google Scholar
[29]	J. Zhang, X. Zhang and B. Yang, An approximation scheme for the time fractional convection-diffusion equation, Appl. Math. Comput., 2018, 335, 305–312. Google Scholar