ELZAKI TRANSFORM BASED ACCELERATED HOMOTOPY PERTURBATION METHOD FOR MULTI-DIMENSIONAL SMOLUCHOWSKI'S COAGULATION AND COUPLED COAGULATION-FRAGMENTATION EQUATIONS

Gourav Arora; Rajesh Kumar; Youcef Mammeri

doi:10.11948/20240004

2024 Volume 14 Issue 5

Article Contents

Article navigation > Journal of Applied Analysis & Computation > 2024 > 14(5): 2922-2953

Next Article Previous Article

Gourav Arora, Rajesh Kumar, Youcef Mammeri. ELZAKI TRANSFORM BASED ACCELERATED HOMOTOPY PERTURBATION METHOD FOR MULTI-DIMENSIONAL SMOLUCHOWSKI'S COAGULATION AND COUPLED COAGULATION-FRAGMENTATION EQUATIONS[J]. Journal of Applied Analysis & Computation, 2024, 14(5): 2922-2953. doi: 10.11948/20240004

Citation:

Gourav Arora, Rajesh Kumar, Youcef Mammeri. ELZAKI TRANSFORM BASED ACCELERATED HOMOTOPY PERTURBATION METHOD FOR MULTI-DIMENSIONAL SMOLUCHOWSKI'S COAGULATION AND COUPLED COAGULATION-FRAGMENTATION EQUATIONS[J]. Journal of Applied Analysis & Computation, 2024, 14(5): 2922-2953. doi: 10.11948/20240004

ELZAKI TRANSFORM BASED ACCELERATED HOMOTOPY PERTURBATION METHOD FOR MULTI-DIMENSIONAL SMOLUCHOWSKI'S COAGULATION AND COUPLED COAGULATION-FRAGMENTATION EQUATIONS

1.
Department of Mathematics, Birla Institute of Technology and Science Pilani, Pilani Campus, Rajasthan-333031, India
2.
Institut Camille Jordan CNRS UMR 5208, Université Jean Monnet, 42100 Saint-Etienne, France

Author Bio: Email: rajesh.kumar@pilani.bits-pilani.ac.in(R. Kumar); Email: youcef.mammeri@math.cnrs.fr(Y. Mammeri)
Corresponding author: Email: P20190421@pilani.bits-pilani.ac.in(G. Arora)

Abstract

Abstract

This article aims to establish a semi-analytical approach based on the homotopy perturbation method (HPM) to find the closed form or approximated solutions for the population balance equations such as Smoluchowski's coagulation, fragmentation, coupled coagulation-fragmentation and bivariate coagulation equations. An accelerated form of the HPM is combined with the Elzaki transformation to improve the accuracy and efficiency of the method. One of the significant advantages of the technique lies over the classic numerical methods as it allows solving the linear and non-linear differential equations without discretization. Further, it has benefits over the existing semi-analytical techniques such as Adomian decomposition method (ADM), optimized decomposition method (ODM), and homotopy analysis method (HAM) in the sense that computation of Adomian polynomials and convergence parameters are not required. The novelty of the scheme is shown by comparing the numerical findings with the existing results obtained via ADM, HPM, HAM and ODM for non-linear coagulation equation. This motivates us to extend the scheme for solving the other models mentioned above. The supremacy of the proposed scheme is demonstrated by taking several numerical examples for each problem. The error between exact and series solutions provided in graphs and tables show the accuracy and applicability of the method. In addition to this, convergence of the series solution is also the key attraction of the work.
- Population balance equation /
- aggregation equation /
- semi-analytical technique /
- Elzaki transformation /
- accelerated homotopy perturbation method /
- series solution /
- convergence analysis
MSC: 45K05, 34A34, 65R20, 35C10

FullText(HTML)

References

[1]	R. Ahrens and S. Le Borne, Fft-based evaluation of multivariate aggregation integrals in population balance equations on uniform tensor grids, Journal of Computational and Applied Mathematics, 2018, 338, 280–297. doi: 10.1016/j.cam.2018.02.013 CrossRef Google Scholar
[2]	G. Arora, S. Hussain and R. Kumar, Comparison of variational iteration and adomian decomposition methods to solve growth, aggregation and aggregation-breakage equations, Journal of Computational Science, 2023, 67, 101973. doi: 10.1016/j.jocs.2023.101973 CrossRef Google Scholar
[3]	G. Arora, R. Kumar and Y. Mammeri, Homotopy perturbation and adomian decomposition methods for condensing coagulation and Lifshitz-Slyzov models, GEM-International Journal on Geomathematics, 2023, 14(1), 4. doi: 10.1007/s13137-023-00215-y CrossRef Google Scholar
[4]	M. Attarakih, M. Jaradat, C. Drumm, et al., A multivariate sectional quadrature method of moments for the solution of the population balance equation, Computer Aided Chemical Engineering, 2010, 28, 1551–1556. Google Scholar
[5]	R. P. Batycky, J. Hanes, R. Langer and D. A. Edwards, A theoretical model of erosion and macromolecular drug release from biodegrading microspheres, Journal of Pharmaceutical Sciences, 1997, 86(12), 1464–1477. doi: 10.1021/js9604117 CrossRef Google Scholar
[6]	M. Ben-Romdhane and H. Temimi, An iterative numerical method for solving the lane–emden initial and boundary value problems, International Journal of Computational Methods, 2018, 15(4), 1850020. doi: 10.1142/S0219876218500202 CrossRef Google Scholar
[7]	Y. Bie, X. Cui and Z. Li, A coupling approach of state-based peridynamics with node-based smoothed finite element method, Computer Methods in Applied Mechanics and Engineering, 2018, 331, 675–700. doi: 10.1016/j.cma.2017.11.022 CrossRef Google Scholar
[8]	H. Briesen, Simulation of crystal size and shape by means of a reduced two-dimensional population balance model, Chemical Engineering Science, 2006, 61(1), 104–112. doi: 10.1016/j.ces.2004.11.062 CrossRef Google Scholar
[9]	M. Dehghan, Y. Rahmani, D. D. Ganji, et al., Convection–radiation heat transfer in solar heat exchangers filled with a porous medium: Homotopy perturbation method versus numerical analysis, Renewable Energy, 2015, 74, 448–455. doi: 10.1016/j.renene.2014.08.044 CrossRef Google Scholar
[10]	A. Dutta, Z. Pınar, D. Constales and T. Öziş, Population balances involving aggregation and breakage through homotopy approaches, International Journal of Chemical Reactor Engineering, 2018, 16(6). Google Scholar
[11]	T. M. Elzaki, Application of new transform "elzaki transform" to partial differential equations, Global Journal of Pure and Applied Mathematics, 2011, 7(1), 65–70. Google Scholar
[12]	T. M. Elzaki, The new integral transform elzaki transform, Global Journal of Pure and Applied Mathematics, 2011, 7(1), 57–64. Google Scholar
[13]	T. M. Elzaki, et al., On the new integral transform "elzaki transform" fundamental properties investigations and applications, Global Journal of Mathematical Sciences: Theory and Practical, 2012, 4(1), 1–13. Google Scholar
[14]	J. Favero and P. Lage, The dual-quadrature method of generalized moments using automatic integration packages, Computers & Chemical Engineering, 2012, 38, 1–10. Google Scholar
[15]	J. Fernandez-Diaz and G. Gomez-Garcia, Exact solution of smoluchowski's continuous multi-component equation with an additive kernel, EPL (Europhysics Letters), 2007, 78(5), 56002. doi: 10.1209/0295-5075/78/56002 CrossRef Google Scholar
[16]	D. Ganji and A. Sadighi, Application of he's Homotopy-Perturbation method to nonlinear coupled systems of reaction-diffusion equations, International Journal of Nonlinear Sciences and Numerical Simulation, 2006, 7(4), 411–418. Google Scholar
[17]	A. K. Giri and E. Hausenblas, Convergence analysis of sectional methods for solving aggregation population balance equations: The fixed pivot technique, Nonlinear Analysis: Real World Applications, 2013, 14(6), 2068–2090. doi: 10.1016/j.nonrwa.2013.03.002 CrossRef Google Scholar
[18]	Z. Hammouch and T. Mekkaoui, A Laplace-Variational Iteration Method for Solving the Homogeneous Smoluchowski Coagulation Equation, 2010. Google Scholar
[19]	A. Hasseine, S. Senouci, M. Attarakih and H. -J. Bart, Two analytical approaches for solution of population balance equations: Particle breakage process, Chemical Engineering & Technology, 2015, 38(9), 1574–1584. Google Scholar
[20]	J. -H. He, Application of homotopy perturbation method to nonlinear wave equations, Chaos, Solitons & Fractals, 2005, 26(3), 695–700. Google Scholar
[21]	S. Hussain, G. Arora and R. Kumar, Semi-analytical methods for solving non-linear differential equations: A review, Journal of Mathematical Analysis and Applications, 2023, 127821. Google Scholar
[22]	S. Hussain, G. Arora and R. Kumar, An efficient semi-analytical technique to solve multi-dimensional Burgers' equation, Computational and Applied Mathematics, 2024, 43(1), 11. doi: 10.1007/s40314-023-02512-6 CrossRef Google Scholar
[23]	S. Hussain and R. Kumar, Elzaki projected differential transform method for multi-dimensional aggregation and combined aggregation-breakage equations, Journal of Computational Science, 2024, 75, 102211. doi: 10.1016/j.jocs.2024.102211 CrossRef Google Scholar
[24]	S. Jasrotia and P. Singh, Accelerated homotopy perturbation elzaki transformation method for solving nonlinear partial differential equations, in Journal of Physics: Conference Series, 2267, IOP Publishing, 2022, 012106. Google Scholar
[25]	G. Kaur, R. Singh and H. Briesen, Approximate solutions of aggregation and breakage population balance equations, Journal of Mathematical Analysis and Applications, 2022, 512(2), 126166. doi: 10.1016/j.jmaa.2022.126166 CrossRef Google Scholar
[26]	G. Kaur, R. Singh, M. Singh, et al., Analytical approach for solving population balances: A homotopy perturbation method, Journal of Physics A: Mathematical and Theoretical, 2019, 52(38), 385201. doi: 10.1088/1751-8121/ab2cf5 CrossRef Google Scholar
[27]	S. Kaushik, S. Hussain and R. Kumar, Laplace transform-based approximation methods for solving pure aggregation and breakage equations, Mathematical Methods in the Applied Sciences, 2023, 46(16), 17402–17421. doi: 10.1002/mma.9507 CrossRef Google Scholar
[28]	S. Kaushik and R. Kumar, A novel optimized decomposition method for Smoluchowski's aggregation equation, Journal of Computational and Applied Mathematics, 2023, 419, 114710. doi: 10.1016/j.cam.2022.114710 CrossRef Google Scholar
[29]	A. Khidir, A note on the solution of general Falkner-Skan problem by two novel semi-analytical techniques, Propulsion and Power Research, 2015, 4(4), 212–220. doi: 10.1016/j.jppr.2015.11.001 CrossRef Google Scholar
[30]	Y. P. Kim and J. H. Seinfeld, Simulation of multicomponent aerosol condensation by the moving sectional method, Journal of Colloid and Interface Science, 1990, 135(1), 185–199. doi: 10.1016/0021-9797(90)90299-4 CrossRef Google Scholar
[31]	P. Lage, Comments on the "an analytical solution to the population balance equation with coalescence and breakage-the special case with constant number of particles" by dp patil and jrg andrews[chemical engineering science, 53(3) 599–601], Chemical Engineering Science, 2002, 19(57), 4253–4254. Google Scholar
[32]	K. Lee and T. Matsoukas, Simultaneous coagulation and break-up using constant-n monte carlo, Powder Technology, 2000, 110(1–2), 82–89. doi: 10.1016/S0032-5910(99)00270-3 CrossRef Google Scholar
[33]	G. Madras and B. J. McCoy, Reversible crystal growth–dissolution and aggregation–breakage: Numerical and moment solutions for population balance equations, Powder Technology, 2004, 143, 297–307. Google Scholar
[34]	A. W. Mahoney and D. Ramkrishna, Efficient solution of population balance equations with discontinuities by finite elements, Chemical Engineering Science, 2002, 57(7), 1107–1119. doi: 10.1016/S0009-2509(01)00427-4 CrossRef Google Scholar
[35]	A. Majumder, V. Kariwala, S. Ansumali and A. Rajendran, Lattice Boltzmann method for population balance equations with simultaneous growth, nucleation, aggregation and breakage, Chemical Engineering Science, 2012, 69(1), 316–328. doi: 10.1016/j.ces.2011.10.051 CrossRef Google Scholar
[36]	N. V. Mantzaris, P. Daoutidis and F. Srienc, Numerical solution of multi-variable cell population balance models: I. finite difference methods, Computers & Chemical Engineering, 2001, 25(11–12), 1411–1440. Google Scholar
[37]	T. Matsoukas, T. Kim and K. Lee, Bicomponent aggregation with composition-dependent rates and the approach to well-mixed state, Chemical Engineering Science, 2009, 64(4), 787–799. doi: 10.1016/j.ces.2008.04.060 CrossRef Google Scholar
[38]	Z. Odibat, An optimized decomposition method for nonlinear ordinary and partial differential equations, Physica A: Statistical Mechanics and its Applications, 2020, 541, 123323. doi: 10.1016/j.physa.2019.123323 CrossRef Google Scholar
[39]	D. Ramkrishna, Population balances: Theory and Pplications to Particulate Systems in Engineering, Elsevier, 2000. Google Scholar
[40]	M. Ranjbar, H. Adibi and M. Lakestani, Numerical solution of homogeneous Smoluchowski's coagulation equation, International Journal of Computer Mathematics, 2010, 87(9), 2113–2122. doi: 10.1080/00207160802617012 CrossRef Google Scholar
[41]	M. J. Rhodes, Introduction to Particle Technology, John Wiley & Sons, 2008. Google Scholar
[42]	W. T. Scott, Analytic studies of cloud droplet coalescence i, Journal of Atmospheric Sciences, 1968, 25(1), 54–65. doi: 10.1175/1520-0469(1968)025<0054:ASOCDC>2.0.CO;2 CrossRef Google Scholar
[43]	M. Singh, Accurate and efficient approximations for generalized population balances incorporating coagulation and fragmentation, Journal of Computational Physics, 2021, 435, 110215. doi: 10.1016/j.jcp.2021.110215 CrossRef Google Scholar
[44]	M. Singh, New finite volume approach for multidimensional smoluchowski equation on nonuniform grids, Studies in Applied Mathematics, 2021, 147(3), 955–977. doi: 10.1111/sapm.12415 CrossRef Google Scholar
[45]	M. Singh, J. Kumar, A. Bück and E. Tsotsas, An improved and efficient finite volume scheme for bivariate aggregation population balance equation, Journal of Computational and Applied Mathematics, 2016, 308, 83–97. doi: 10.1016/j.cam.2016.04.037 CrossRef Google Scholar
[46]	M. Singh, T. Matsoukas, A. B. Albadarin and G. Walker, New volume consistent approximation for binary breakage population balance equation and its convergence analysis, ESAIM: Mathematical Modelling and Numerical Analysis, 2019, 53(5), 1695–1713. doi: 10.1051/m2an/2019036 CrossRef Google Scholar
[47]	M. Singh, V. Ranade, O. Shardt and T. Matsoukas, Challenges and opportunities concerning numerical solutions for population balances: A critical review, Journal of Physics A: Mathematical and Theoretical, 2022, 55(38), 383002. doi: 10.1088/1751-8121/ac8a42 CrossRef Google Scholar
[48]	R. Singh, J. Saha and J. Kumar, Adomian decomposition method for solving fragmentation and aggregation population balance equations, Journal of Applied Mathematics and Computing, 2015, 48(1), 265–292. Google Scholar
[49]	M. Singh, R. Singh, S. Singh, et al., Discrete finite volume approach for multidimensional agglomeration population balance equation on unstructured grid, Powder Technology, 2020, 376, 229–240. doi: 10.1016/j.powtec.2020.08.022 CrossRef Google Scholar
[50]	M. Singh and G. Walker, Finite volume approach for fragmentation equation and its mathematical analysis, Numerical Algorithms, 2022, 89(2), 465–486. doi: 10.1007/s11075-021-01122-9 CrossRef Google Scholar
[51]	J. Su, Z. Gu, Y. Li, et al., Solution of population balance equation using quadrature method of moments with an adjustable factor, Chemical Engineering Science, 2007, 62(21), 5897–5911. doi: 10.1016/j.ces.2007.06.016 CrossRef Google Scholar
[52]	H. Temimi, A. R. Ansari and A. M. Siddiqui, An approximate solution for the static beam problem and nonlinear integro-differential equations, Computers & Mathematics with Applications, 2011, 62(8), 3132–3139. Google Scholar
[53]	H. Temimi and M. Ben-Romdhane, Numerical solution of Falkner-Skan equation by iterative transformation method, Mathematical Modelling and Analysis, 2018, 23(1), 139–151. doi: 10.3846/mma.2018.009 CrossRef Google Scholar
[54]	H. Temimi, M. Ben-Romdhane, S. El-Borgi and Y. -J. Cha, Time-delay effects on controlled seismically excited linear and nonlinear structures, International Journal of Structural Stability and Dynamics, 2016, 16(7), 1550031. doi: 10.1142/S0219455415500315 CrossRef Google Scholar
[55]	H. Temimi and H. Kurkcu, An accurate asymptotic approximation and precise numerical solution of highly sensitive Troesch's problem, Applied Mathematics and Computation, 2014, 235, 253–260. doi: 10.1016/j.amc.2014.03.022 CrossRef Google Scholar
[56]	N. Yadav, M. Singh, S. Singh, et al., A note on homotopy perturbation approach for nonlinear coagulation equation to improve series solutions for longer times, Chaos, Solitons & Fractals, 2023, 173, 113628. Google Scholar
[57]	Z. Yin and H. Liu, Numerical Simulation of Nanoparticles Diffusion and Coagulation in a Twin-Jet Via a Temom Method, International Journal of Numerical Methods for Heat & Fluid Flow, 2014. Google Scholar