# Gegenbauer wavelet operational matrix method for solving variable-order non-linear reaction–diffusion and Galilei invariant advection–diffusion equations

Article

## Abstract

In this article, a variable-order operational matrix of Gegenbauer wavelet method based on Gegenbauer wavelet is applied to solve a space–time fractional variable-order non-linear reaction–diffusion equation and non-linear Galilei invariant advection diffusion equation for different particular cases. Operational matrices for integer-order differentiation and variable-order differentiation have been derived. Applying collocation method and using the said matrices, fractional-order non-linear partial differential equation is reduced to a system of non-linear algebraic equations, which have been solved using Newton iteration method. The salient feature of the article is the stability analysis of the proposed method. The efficiency, accuracy and reliability of the proposed method have been validated through a comparison between the numerical results of six illustrative examples with their existing analytical results obtained from literature. The beauty of the article is the physical interpretation of the numerical solution of the concerned variable-order reaction–diffusion equation for different particular cases to show the effect of reaction term on the pollution concentration profile.

## Keywords

Fractional PDE Variable-order diffusion equation Operational matrix Gegenbauer wavelet Collocation method

## Mathematics Subject Classification

35R11 34A08 41A10

## References

1. Abd-Elkawy MA, Alqahtani RT (2017) Space-time spectral collocation algorithm for the variable-order galilei invariant advection diffusion equations with a nonlinear source term. Math Model Anal 22(1):1–20
2. Anh VV, Angulo JM, Ruiz-Medina MD (2005) Diffusion on multifractals. Nonlinear Anal Theory Methods Appl 63(5–7):e2043–e2056
3. Chechkin AV, Gorenflo R, Sokolov IM (2005) Fractional diffusion in inhomogeneous media. J Phys A Math Gen 38(42):L679
4. Chechkin A, Gonchar VY, Gorenflo R, Korabel N, Sokolov I (2008) Generalized fractional diffusion equations for accelerating subdiffusion and truncated lévy flights. Phys Rev E 78(2):021111
5. Coimbra CF (2003) Mechanics with variable-order differential operators. Ann Phys 12(11–12):692–703
6. Couteron P, Lejeune O (2001) Periodic spotted patterns in semi-arid vegetation explained by a propagation-inhibition model. J Ecol 89(4):616–628
7. Dabiri A, Moghaddam BP, Machado JT (2018) Optimal variable-order fractional pid controllers for dynamical systems. J Comput Appl Math 339:40–48
8. Darania P, Ebadian A (2007) A method for the numerical solution of the integro-differential equations. Appl Math Comput 188:657–668
9. Das S, Singh A, Ong SH (2018) Numerical solution of fractional order advection-reaction-diffusion equation. Therm Sci 22:S309–S316
10. Das S, Vishal K, Gupta P (2011) Solution of the nonlinear fractional diffusion equation with absorbent term and external force. Appl Math Model 35(8):3970–3979
11. De Villiers J (2012) Mathematics of approximation, vol 1. Springer Science and Business Media, New York
12. Diethelm K, Ford NJ, Freed AD (2002) A predictor-corrector approach for the numerical solution of fractional differential equations. Nonlinear Dyn 29(1–4):3–22
13. Elgindy KT, Smith-Miles KA (2013) Solving boundary value problems, integral, and integro-differential equations using Gegenbauer integration matrices. J Comput Appl Math 237(1):307–325
14. Gasca M, Sauer T (2001) On the history of multivariate polynomial interpolation. In: Numerical analysis: historical developments in the 20th century. Elsevier, pp 135–147Google Scholar
15. Gürbüz B, Sezer M (2016) Laguerre polynomial solutions of a class of initial and boundary value problems arising in science and engineering fields. Acta Phys Pol A 130(1):194–197
16. Hajipour M, Jajarmi A, Baleanu D, Sun H (2019) On an accurate discretization of a variable-order fractional reaction-diffusion equation. Commun Nonlinear Sci Numer Simul 69:119–133
17. Hashim I, Abdulaziz O, Momani S (2009) Homotopy analysis method for fractional IVPS. Commun Nonlinear Sci Numer Simul 14(3):674–684
18. Jafari H, Yousefi S, Firoozjaee M, Momani S, Khalique CM (2011) Application of legendre wavelets for solving fractional differential equations. Comput Math Appl 62(3):1038–1045
19. Jaiswal S, Chopra M, Das S (2018) Numerical solution of two-dimensional solute transport system using operational matrices. Transp Porous Media 122(1):1–23
20. Keshi FK, Moghaddam BP, Aghili A (2018) A numerical approach for solving a class of variable-order fractional functional integral equations. Comput Appl Math 37(4):4821–4834
21. Kilbas A, Srivastava H, Trujillo JJ (2006) Theory and applications of the fractional differential equations, vol 204. Elsevier (North-Holland), Amsterdam
22. Kondo S (2009) How animals get their skin patterns: fish pigment pattern as a live turing wave. In: Systems biology. Springer, Berlin, pp 37–46Google Scholar
23. Kondo S, Asai R (1995) A reaction-diffusion wave on the skin of the marine angelfish pomacanthus. Nature 376(6543):765
24. Li Y, Sun N (2011) Numerical solution of fractional differential equations using the generalized block pulse operational matrix. Comput Math Appl 62(3):1046–1054
25. Li X, Wu B (2018) Iterative reproducing kernel method for nonlinear variable-order space fractional diffusion equations. Int J Comput Math 95(6–7):1210–1221
26. Li Y, Zhao W (2010) Haar wavelet operational matrix of fractional order integration and its applications in solving the fractional order differential equations. Appl Math Comput 216(8):2276–2285
27. Lin R, Liu F, Anh V, Turner I (2009) Stability and convergence of a new explicit finite-difference approximation for the variable-order nonlinear fractional diffusion equation. Appl Math Comput 212(2):435–445
28. Lv C, Xu C (2016) Error analysis of a high order method for time-fractional diffusion equations. SIAM J Sci Comput 38(5):A2699–A2724
29. Machado JT, Kiryakova V, Mainardi F (2011) Recent history of fractional calculus. Commun Nonlinear Sci Numer Simul 16(3):1140–1153
30. Machado JAT, Moghaddam BP (2018) A robust algorithm for nonlinear variable-order fractional control systems with delay. Int J Nonlinear Sci Numer Simul 19(3–4):231–238
31. Malesza W, Macias M, Sierociuk D (2019) Analytical solution of fractional variable order differential equations. J Comput Appl Math 348:214–236
32. Milici C, Draganescu G, Machado JT (2019) Introduction to fractional differential equations. Nonlinear systems and complexity. Springer, Switzerland.
33. Moghaddam BP, Machado JAT (2017a) A computational approach for the solution of a class of variable-order fractional integro-differential equations with weakly singular kernels. Fract Calculus Appl Anal 20(4):1023–1042
34. Moghaddam BP, Machado JAT (2017b) Extended algorithms for approximating variable order fractional derivatives with applications. J Sci Comput 71(3):1351–1374
35. Moghaddam BP, Machado JAT (2017c) Sm-algorithms for approximating the variable-order fractional derivative of high order. Fundam Inf 151(1–4):293–311
36. Moghaddam BP, Mostaghim ZS (2017) Modified finite difference method for solving fractional delay differential equations. Boletim da Sociedade Paranaense de Matemática 35(2):49–58
37. Moghaddam B, Machado J, Behforooz H (2017) An integro quadratic spline approach for a class of variable-order fractional initial value problems. Chaos Solitons Fractals 102:354–360
38. Moghaddam B, Dabiri A, Lopes AM, Machado JT (2019) Numerical solution of mixed-type fractional functional differential equations using modified lucas polynomials. Comput Appl Math 38(2):46
39. Murray JD (1981) A pre-pattern formation mechanism for animal coat markings. J Theor Biol 88(1):161–199
40. Odibat Z (2011) On legendre polynomial approximation with the vim or ham for numerical treatment of nonlinear fractional differential equations. J Comput Appl Math 235(9):2956–2968
41. Ortigueira MD, Valério D, Machado JT (2019) Variable order fractional systems. Commun Nonlinear Sci Numer Simul 71:231–243. . http://www.sciencedirect.com/science/article/pii/S1007570418303782
42. Podlubny I (1998) Fractional differential equations, to methods of their solution and some of their applications. Fractional differential equations: an introduction to fractional derivatives. Academic Press, San DiegoGoogle Scholar
43. Rehman MU, Saeed U (2015) Gegenbauer wavelets operational matrix method for fractional differential equations. J Korean Math Soc 52:1069–1096
44. Samko SG, Ross B (1993) Integration and differentiation to a variable fractional order. Integr Transforms Spec Funct 1(4):277–300
45. Shen S, Liu F, Chen J, Turner I, Anh V (2012) Numerical techniques for the variable order time fractional diffusion equation. Appl Math Comput 218(22):10861–10870
46. Soon CM, Coimbra CF, Kobayashi MH (2005) The variable viscoelasticity oscillator. Ann Phys 14(6):378–389
47. Suarez L, Shokooh A (1997) An eigenvector expansion method for the solution of motion containing fractional derivatives. ASME J Appl Mech 64:629–635
48. Sun H, Chen W, Wei H, Chen Y (2011) A comparative study of constant-order and variable-order fractional models in characterizing memory property of systems. Eur Phys J Spec Top 193(1):185
49. Tavares D, Almeida R, Torres DF (2016) Caputo derivatives of fractional variable order: numerical approximations. Commun Nonlinear Sci Numer Simul 35:69–87
50. Tripathi NK, Das S, Ong SH, Jafari H, Al Qurashi M (2016) Solution of higher order nonlinear time-fractional reaction diffusion equation. Entropy 18(9):329
51. Valério D, Sá da CJ (2013) Variable order fractional controllers. Asian J Control 15(3):648–657
52. Xiang M, Zhang B, Yang D (2019) Multiplicity results for variable-order fractional Laplacian equations with variable growth. Nonlinear Anal 178:190–204
53. Yuanlu L (2010) Solving a nonlinear fractional differential equation using Chebyshev wavelets. Commun Nonlinear Sci Numer Simul 15(9):2284–2292
54. Zayernouri M, Karniadakis GE (2015) Fractional spectral collocation methods for linear and nonlinear variable order FPDES. J Comput Phys 293:312–338