AIMS Mathematics, 2020, 5(4): 3201-3222. doi: 10.3934/math.2020206

Research article

Export file:


  • RIS(for EndNote,Reference Manager,ProCite)
  • BibTex
  • Text


  • Citation Only
  • Citation and Abstract

A comparative study for fractional chemical kinetics and carbon dioxide CO2 absorbed into phenyl glycidyl ether problems

1 Department of Mathematics, National Institute of Technology, Jamshedpur, 831014, Jharkhand, India
2 Department of Mathematics, JECRC University, Jaipur-303905, Rajasthan, India
3 Department of Basic Engineering Sciences, College of Engineering, Imam Abdulrahman Bin Faisal University, P. O. Box 1982, Dammam 31441, Saudi Arabia

The essential objective of this work is to implement Adam Bashforth’s Moulton (ABM) and Haar wavelet method (HWM) to solve fractional chemical kinetics and another problem that relates the condensations of carbon dioxide (CO2) and phenyl glycidyl ether (PGE) with two variety of Drichlet and a mixed set of Neumann boundary and Drichlet type conditions respectively. We have been solved the above system of differential equations by Adam Bashforth’s Moulton and Haar wavelet operational method where this technique is to convert the system of differential equations into the system of algebraic equation which can be solved easily. This work is expects to contribute the vast advantage of Haar wavelets in chemical science. The Adam Bashforth’s Moulton and Haar wavelet method is impressive and convenient for obtaining numerical solutions of chemical engineering type problems. A complete agreement is acheived between Adam Bashforth’s Moulton solution and Haar wavelet solution. To manifest about the performance and applicability of the method, two test examples are deliberated.
  Article Metrics


1. M. A. Al-Jawary, R. K. Raham, A semi-analytical iterative technique for solving chemistry problems, J. King Saud Univ. Sci., 29 (2017), 320-332.    

2. S. Abbasbandy, A. Shirzadi, Homotopy analysis method for a nonlinear chemistry problem, Studies in Nonlinear Sciences, 1 (2010), 127-132.

3. M. AL-Jawary, R. Raham, G. Radhi, An iterative method for calculating carbon dioxide absorbed into phenyl glycidyl ether, Journal of Mathematical and Computational Science, 6 (2016), 620-632.

4. Y. S. Choe, S. W. Park, D. W. Park, et al. Reaction kinetics of carbon dioxide with phenyl glycidyl ether by TEA-CP-MS41 catalyst, J. JPN Petrol. Inst., 53 (2010), 160-166.    

5. R. Singha, A. M. Wazwaz, Steady-state concentrations of carbon dioxide absorbed into phenyl glycidyl ether: An optimal homotopy analysis method, Match-Commun Math. Co., 81 (2019), 800-812.

6. H. Robertson, The solution of a set of reaction rate equations, Numerical analysis: an introduction, 178182.

7. M. Matinfar, M. Saeidy, B. Gharahsuflu, et al. Solutions of nonlinear chemistry problems by homotopy analysis, Comput. Math. Model., 25 (2014), 103-114.    

8. D. Ganji, M. Nourollahi, E. Mohseni, Application of he's methods to nonlinear chemistry problems, Comput. Math. Appl., 54 (2007), 1122-1132.    

9. A. Dokoumetzidis, R. Magin, P. Macheras, Fractional kinetics in multi-compartmental systems, J. Pharmacokinet. Phar., 37 (2010), 507-524.    

10. F. Mainardi, Fractional calculus and waves in linear viscoelasticity: an introduction to mathematical models, World Scientific, 2010.

11. I. Podlubny, Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Vol. 198, Elsevier, 1998.

12. K. Diethelm, The analysis of fractional differential equations: An application-oriented exposition using differential operators of Caputo type, Springer Science & Business Media, 2010.

13. A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, Vol. 204, Elsevier Science Limited, 2006.

14. S. Das, Functional fractional calculus for system identification and controls, Springer-Verlag, 2008.

15. J. S. Walker, A primer on wavelets and their scientific applications, CRC press, 2002.

16. J. J. Benedetto, Wavelets: mathematics and applications, Vol. 13, CRC press, 1993.

17. C. K. Chui, Wavelets: a mathematical tool for signal analysis, Vol. 1, Siam, 1997.

18. S. Gopalakrishnan, M. Mitra, Wavelet methods for dynamical problems: with application to metallic, composite, and nano-composite structures, CRC Press, 2010.

19. Y. Wang, Q. Fan, The second kind chebyshev wavelet method for solving fractional differential equations, Appl. Math. Comput., 218 (2012), 8592-8601.

20. J.-L. Wu, A wavelet operational method for solving fractional partial differential equations numerically, Appl. Math. Comput., 214 (2009), 31-40.

21. M. Heydari, M. Hooshmandasl, F. Mohammadi, et al. Wavelets method for solving systems of nonlinear singular fractional volterra integro-differential equations, Commun. Nonlinear Sci., 19 (2014), 37-48.    

22. S. Balaji, Legendre wavelet operational matrix method for solution of fractional order riccati differential equation, Journal of the Egyptian Mathematical Society, 23 (2015), 263-270.    

23. M. ur Rehman, R. A. Khan, The legendre wavelet method for solving fractional differential equations, Commun. Nonlinear Sci., 16 (2011), 4163-4173.    

24. M. Hosseininia, M. Heydari, F. M. Ghaini, et al. A wavelet method to solve nonlinear variableorder time fractional 2d klein-gordon equation, Comput. Math. Appl., 78 (2019), 3713-3730.    

25. P. Pirmohabbati, A. R. Sheikhani, H. S. Najafi, et al. Numerical solution of fractional mathieu equations by using block-pulse wavelets, Journal of Ocean Engineering and Science, 4 (2019), 299-307.    

26. M. Hosseininia, M. Heydari, R. Roohi, et al. A computational wavelet method for variable-order fractional model of dual phase lag bioheat equation, J. Comput. Phys., 395 (2019), 1-18.    

27. M. H. Heydari, Z. Avazzadeh, Legendre wavelets optimization method for variable-order fractional poisson equation, Chaos, Solitons & Fractals, 112 (2018), 180-190.

28. S. Kumar, R. Kumar, J. Singh, et al. An efficient numerical scheme for fractional model of hiv-1 infection of cd4+ t-cells with the effect of antiviral drug therapy, Alexandria Engineering Journal, 2020.

29. S. S. Ray, A. Patra, Numerical simulation for fractional order stationary neutron transport equation using haar wavelet collocation method, Nucl. Eng. Des., 278 (2014), 71-85.    

30. A. Patra, S. S. Ray, A numerical approach based on haar wavelet operational method to solve neutron point kinetics equation involving imposed reactivity insertions, Ann. Nucl. Energy, 68 (2014), 112-117.    

31. S. S. Ray, A. Patra, Haar wavelet operational methods for the numerical solutions of fractional order nonlinear oscillatory van der pol system, Appl. Math. Comput., 220 (2013), 659-667.

32. S. S. Ray, On haar wavelet operational matrix of general order and its application for the numerical solution of fractional bagley torvik equation, Appl. Math. Comput., 218 (2012), 5239-5248.

33. I. Aziz, F. Haq, et al. A comparative study of numerical integration based on haar wavelets and hybrid functions, Comput. Math. Appl., 59 (2010), 2026-2036.    

34. I. Aziz, A. Al-Fhaid, et al. An improved method based on haar wavelets for numerical solution of nonlinear integral and integro-differential equations of first and higher orders, J. Comput. Appl. Math., 260 (2014), 449-469.    

35. I. Aziz, A. Al-Fhaid, A. Shah, et al. A numerical assessment of parabolic partial differential equations using haar and legendre wavelets, Appl. Math. Model., 37 (2013), 9455-9481.    

36. R. Jiwari, A haar wavelet quasilinearization approach for numerical simulation of burgers' equation, Comput. Phys. Commun., 183 (2012), 2413-2423.    

37. Ü. Lepik, Solving fractional integral equations by the haar wavelet method, Appl. Math. Comput., 214 (2009), 468-478.

38. Z. Shi, Y.-y. Cao, A spectral collocation method based on haar wavelets for poisson equations and biharmonic equations, Math. Comput. Model., 54 (2011), 2858-2868.    

39. H. Hein, L. Feklistova, Computationally efficient delamination detection in composite beams using haar wavelets, Mech. Syst. Signal Pr., 25 (2011), 2257-2270.    

40. Z. Gao, X. Liao, Discretization algorithm for fractional order integral by haar wavelet approximation, Appl. Math. Comput., 218 (2011), 1917-1926.

41. C. Chen, C. Hsiao, Haar wavelet method for solving lumped and distributed-parameter systems, IEE Proceedings-Control Theory and Applications, 144 (1997), 87-94.    

42. Y. Chen, M. Yi, C. Yu, Error analysis for numerical solution of fractional differential equation by haar wavelets method, J. Comput. Sci., 3 (2012), 367-373.    

43. X. Xie, G. Jin, Y. Yan, et al. Free vibration analysis of composite laminated cylindrical shells using the haar wavelet method, Composite Structures, 109 (2014), 169-177.    

44. U. Saeed, M. ur Rehman, Haar wavelet-quasilinearization technique for fractional nonlinear differential equations, Appl. Math. Comput., 220 (2013), 630-648.

45. İ. Çelik, Haar wavelet approximation for magnetohydrodynamic flow equations, Appl. Math. Model., 37 (2013), 3894-3902.    

46. İ. Çelik, Haar wavelet method for solving generalized burgers-huxley equation, Arab Journal of Mathematical Sciences, 18 (2012), 25-37.    

47. G. Hariharan, K. Kannan, K. Sharma, Haar wavelet method for solving fisher's equation, Appl. Math. Comput., 211 (2009), 284-292.

48. L. Wang, Y. Ma, Z. Meng, Haar wavelet method for solving fractional partial differential equations numerically, Appl. Math. Comput., 227 (2014), 66-76.

49. H. Kaur, R. Mittal, V. Mishra, Haar wavelet solutions of nonlinear oscillator equations, Appl. Math. Model., 38 (2014), 4958-4971.    

50. Z. Shi, Y. Y. Cao, Q. j. Chen, Solving 2d and 3d poisson equations and biharmonic equations by the haar wavelet method, Appl. Math. Model., 36 (2012), 5143-5161.    

51. G. Jin, X. Xie, Z. Liu, The haar wavelet method for free vibration analysis of functionally graded cylindrical shells based on the shear deformation theory, Composite Structures, 108 (2014), 435-448.    

52. S. Kumar, A. Kumar, S. Abbas, et al. A modified analytical approach with existence and uniqueness for fractional cauchy reaction-diffusion equations, Advances in Difference Equations, 2020 (2020), 1-18.    

53. S. Kumar, A. Kumar, S. Momani, et al. Numerical solutions of nonlinear fractional model arising in the appearance of the stripe patterns in two-dimensional systems, Advances in Difference Equations, 2019 (2019), 413.

54. M. Jleli, S. Kumar, R. Kumar, et al. Analytical approach for time fractional wave equations in the sense of yang-abdel-aty-cattani via the homotopy perturbation transform method, Alexan. Eng. J., 2019.

55. S. Kumar, K. S. Nisar, R. Kumar, et al. A new rabotnov fractional-exponential function based fractional derivative for diffusion equation under external force, Mathematical Methods in Applied Science, 2020.

56. B. Ghanbari, S. Kumar, R. Kumar, A study of behaviour for immune and tumor cells in immunogenetic tumour model with non-singular fractional derivative, Chaos, Soliton and Fractals, 133 (2020), 109619.

57. A. El-Ajou, M. N. Oqielat, Z. Al-Zhour, et al. Solitary solutions for time-fractional nonlinear dispersive pdes in the sense of conformable fractional derivative, Chaos: An Interdisciplinary Journal of Nonlinear Science, 29 (2019), 093102.

58. E. F. D. Goufo, S. Kumar, S. B. Mugisha, Similarities in a fifth-order evolution equation with and with no singular kernel, Chaos, Soliton and Fractals, 130 (2020), 109467.

59. S. Bhatter, A. Mathur, D. Kumar, et al. A new analysis of fractional drinfeld-sokolov-wilson model with exponential memory, Physica A: Statistical Mechanics and its Applications, 537 (2020), 122578.

60. P. Veeresha, D. G. Prakasha, D. Kumar, An efficient technique for nonlinear time-fractional klein- fock-gordon equation, Appl. Math. Comput., 364 (2020), 124637.

61. D. Kumar, J. Singh, M. A. Al-Qurashi, et al. A new fractional sirs-si malaria disease model with application of vaccines, antimalarial drugs, and spraying, Advances in Difference Equations, 2019 (2019), 278.

62. T. Hull, W. Enright, B. Fellen, et al. Comparing numerical methods for ordinary differential equations, SIAM J. Numer. Anal., 9 (1972), 603-637.    

63. O. S. Board, Ocean acidification: a national strategy to meet the challenges of a changing ocean, National Academies Press, 2010.

64. S. Muthukaruppan, I. Krishnaperumal, R. Lakshmanan, Theoretical analysis of mass transfer with chemical reaction using absorption of carbon dioxide into phenyl glycidyl ether solution, Appl. Math. Ser. B, 3 (2012), 1179-1186.    

65. S. Kumar, M. M. Rashidi, New analytical method for gas dynamics equation arising in shock fronts, Comput. Phys. Commun., 185 (2014), 1947-1954.    

66. Y. Li, W. Zhao, Haar wavelet operational matrix of fractional order integration and its applications in solving the fractional order differential equations, Appl. Math. Comput., 216 (2010), 2276-2285.

67. M. Srivastava, S. K. Agrawal, S. Das, Synchronization of chaotic fractional order lotka-volterra system, Int. J. Nonlinear Sci., 13 (2012), 482-494.

68. K. Diethelm, N. J. Ford, Multi-order fractional differential equations and their numerical solution, Appl. Math. Comput., 154 (2004), 621-640.

69. H. Aminikhah, An analytical approximation to the solution of chemical kinetics system, J. King Saud Univ. Sci., 23 (2011), 167-170.    

70. J. S. Duan, R. Rach, A. M. Wazwaz, Steady-state concentrations of carbon dioxide absorbed into phenyl glycidyl ether solutions by the adomian decomposition method, J. Math. Chem., 53 (2015), 1054-1067.    

71. M. A. AL-Jawary, G. H. Radhi, The variational iteration method for calculating carbon dioxide absorbed into phenyl glycidyl ether, Iosr Journal of Mathematics, 11 (2015), 99-105.

© 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (

Download full text in PDF

Export Citation

Article outline

Show full outline
Copyright © AIMS Press All Rights Reserved