Numerical solution of nonlinear diffusion-reaction in porous catalysts using quantum spectral successive linearization method

Document Type : Research Article

Author

• S. Abbasbandy

Department of Applied Mathematics, Faculty of Science, Imam Khomeini International University, Qazvin, Iran.

10.22067/ijnao.2025.94329.1677

Abstract

Significant advances in quantum computing science have been achieved through the development of general-purpose quantum solvers for linear differential equations in several studies. This study employs a hybrid quantum-spectral approach to analyze nonlinear reaction-diffusion dynamics in porous catalysts. Two distinct forms of nonlinearity in the model are examined. The successive linearization method is applied to linearize the governing equations. Within an iterative framework, the final quantum state is constructed by progressively integrating the quantum state from each iteration, enabled by an efficient quantum algorithm. Numerical simulations validate the method’s efficacy, showing favorable agreement with existing literature. Moreover, the approach delivers accurate solutions even for large Thiele modulus values.

Keywords

• Catalyst pellet
• Diffusion and reaction
• Thiele modulus
• Nonlinear reactive transport model

Main Subjects

• Computational Mathematics

---

References

[1] Abbasbandy, S., Approximate solution for the nonlinear model of diffusion and reaction in porous catalysts by means of the homotopy analysis method, Chem. Eng. J., 136 (2008) 144–150.
[2] Abbasbandy, S., Series and rational solutions of the second kind Painlevé equations by using quantum pseudo-spectral method, Int. J. Math. Math. Sci. 2025 (2025) 9705701.
[3] Adomian G., Solving frontier problems of physics: The decomposition method, Kluwer Academic, Dordrecht, 1994.
[4] Ahmad, I., Ilyas, H., Urooj, A., Aslam, M.S., Shoaib, M. and Raja, M.A.Z., Novel applications of intelligent computing paradigms for the analysis of nonlinear reactive transport model of the fluid in soft tissues and microvessels, Neural Comput. Appl., 31 (2019) 9041–9059.

[5] Aris, R., Mathematical theory of diffusion and reaction in permeable catalyst, Oxford University Press, London, 1975.
[6] Arrazola, J.M., Kalajdzievski, T., Weedbrook, C. and Lloyd, S., Quantum algorithm for nonhomogeneous linear partial differential equations, Phys. Rev. A 100 (2019) 032306.
[7] Berry, D.W., High-order quantum algorithm for solving linear differential equations, J. Phys. A 47(10) (2014) 105301.
[8] Berry, D.W., Childs, A.M., Ostrander, A. and Wang, G., Quantum algorithm for linear differential equations with exponentially improved de-pendence on precision, Commun. Math. Phys. 356(3) (2017) 1057–1081.
[9] Biazar, J., Dehghan, M., Houlari, T., Using homotopy analysis method to find the eigenvalues of higher order fractional Sturm–Liouville problems, Iran. J. Numer. Anal. Optim. 10(1) (2020) 49–62.
[10] Childs, A.M., Kothari, R. and Somma, R.D., Quantum linear systems algorithm with exponentially improved dependence on precision, SIAM J. Comput. 46 (2017) 1920–1950.
[11] Childs, A.M. and Liu, J.P., Quantum spectral methods for differential equations, Commun. Math. Phys. 375 (2020) 1427–1457.
[12] Clader, D.B., Jacobs, B.C. and Sprouse, C.R., Preconditioned quantum linear system algorithm, Phys. Rev. Lett. 110 (2013) 250504.
[13] Costa, P., Jordan, S. and Ostrander, A., Quantum algorithm for simulating the wave equation, Phys. Rev. A 99 (2019) 012323.
[14] Costa, P.C.S., Schleich, P., Morales, M.E.S. and Berry, D.W., Further improving quantum algorithms for nonlinear differential equations via higher-order methods and rescaling, Npj Quantum Inf. 11 (1) (2025) 141.
[15] Dana Mazraeh, H. and Parand, K., GEPINN: An innovative hybrid method for a symbolic solution to the Lane–Emden type equation based on grammatical evolution and physics-informed neural networks, Astron. Comput. 48 (2024)100846.

[16] Dana Mazraeh, H. and Parand, K., A three-stage framework combining neural networks and Monte Carlo tree search for approximating analytical solutions to the Thomas–Fermi equation, J. Comput. Sci. 87 (2025)102582.
[17] Dana Mazraeh, H. and Parand, K., An innovative combination of deep Q-networks and context-free grammars for symbolic solutions to differential equations, Eng. Appl. Artif. Intell. 142 (2025) 109733.
[18] Derakhshan, M. and Aminataei, A., Comparison of homotopy perturbation transform method and fractional Adams–Bashforth method for the Caputo–Prabhakar nonlinear fractional differential equations, Iran. J. Numer. Anal. Optim. 10(2) (2020) 63–85.
[19] Ganie, A.H., Rahman, I.U., Sulaiman, M. and Nonlaopon, K., Solution of nonlinear reaction-diffusion model in porous catalysts arising in micro-vessel and soft tissue using a metaheuristic, IEEE Access, 10 (2022) 41813–41827.
[20] Gheorghiu, C.I., Spectral methods for differential problems, Casa Cartii de Stiinta Publishing House, Cluj-Napoca, 2007.
[21] Harrow, A.W., Hassidim, A. and Lloyd, S., Quantum algorithm for linear systems of equations, Phys. Rev. Lett. 103 (2009) 150502.
[22] Hosseini, M.M., A modified pseudospectral method for numerical solution of ordinary differential equations systems, Appl. Math. Comput. 176(2) (2006) 470–475.
[23] Karmishin, A.V., Zhukov, A.I. and Kolosov, V.G., Methods of dynamics calculation and testing for thin-walled structures, Mashinostroyenie, Moscow, 1990.
[24] Kierzenka, J. and Shampine, L.F., A BVP solver based on residual control and the MATLAB PSE, ACM Trans. Math. Software. 27 (2001) 299–316.
[25] Krovi, H., Improved quantum algorithms for linear and nonlinear differential equations, Quantum 7 (2023) 913.

[26] Kyriienko, O., Paine, A.E. and Elfving, V.E., Solving nonlinear differential equations with differentiable quantum circuits, Phys. Rev. A, 103(5) (2021) 052416.
[27] Leyton, S.K. and Osborne, T.J., A quantum algorithm to solve nonlinear differential equations, arXiv:0812.4423 (2008).
[28] Liao, S.J., Beyond perturbation: Introduction to the homotopy analysis method, Chapman and Hall/CRC Press, Boca Raton, 2003.
[29] Liu, J.P., An, D., Fang, D., Wang, J., Low, G.H. and Jordan, S., Efficient quantum algorithm for nonlinear reaction-diffusion equations and energy estimation, Commun. Math. Phys. 404 (2023) 963–1020.
[30] Lyapunov, A.M., General problem on stability of motion (English translation), Taylor and Francis, London, 1992.
[31] Magyari, E., Exact analytical solution of a nonlinear reaction–diffusion model in porous catalysts, Chem. Eng. J. 143 (2008) 167–171.
[32] Mirhosseini-Alizamini, S., Solving linear optimal control problems of the time-delayed systems by Adomian decomposition method, Iran. J. Numer. Anal. Optim. 9(2) (2019) 165–183.
[33] Moitsheki, R.J., Hayat, T., Malik, M.Y. and Mahomed, F.M., Symmetry analysis for the nonlinear model of diffusion and reaction in porous catalysts, Nonlinear Anal. Real World Appl. 11 (2010) 3031–3036.
[34] Nayfeh, A.H., Perturbation methods, John Wiley and Sons, New York, 2000.
[35] Satterfield, C.N., Mass transfer in heterogeneous catalysis, MIT Press, Cambridge, 1970.
[36] Shampine, L.F., Reichelt, M.W. and Kierzenka, J., Solving boundary value problems for ordinary differential equations in MATLAB with bvp4c, MATLAB File Exchange, 2004.
[37] Shen, J., Tang, T., Wang, L.L., Spectral methods: Algorithms, analysis and applications, Springer, Berlin, 2011.

[38] Srinivas, E., Lalu, M. and Phaneendra, K., A numerical approach for singular perturbation problems with an interior layer using an adaptive spline, Iran. J. Numer. Anal. Optim. 12(2) (2022) 355–370.
[39] Thiele, E.W., Relation between catalytic activity and size of particle, Ind. Eng. Chem. 31 (1939) 916–920.
[40] Trefethen, L.N., Spectral methods in MATLAB, Society for Industrial and Applied Mathematics (SIAM), 2000.
[41] Zygelman, B., A first introduction to quantum computing and information, Springer Nature Switzerland AG, 2018.