Analysis of the Parameter-Dependent Multiplicity of Steady-State Profiles of a Strongly Nonlinear Mathematical Model Arising From the Chemical Reactor Theory
محورهای موضوعی : مجله بین المللی ریاضیات صنعتی
M. S. Barikbin
1
,
M. Emamjome
2
,
M. Nabati
3
1 - Department of Mathematics, Takestan Branch, Islamic Azad University, Takestan, Iran.
2 - Department of Mathematics, Golpayegan University of Technology, Golpayegan, Iran.
3 - Department of Basic Sciences, Abadan Faculty of Petroleum
engineering, Petroleum University of Technology, Abadan, Iran
کلید واژه: Iterative technique, Multiple solutions, Convergence, Adiabatic tubular chemical reactor, Reproducing Kernel Hilbert Space, Strongly nonlinear problem,
چکیده مقاله :
In this paper, we study the uniqueness and multiplicity of the solutions of a strongly nonlinear mathematical model arising from chemical reactor theory. The analysis is based on the reproducing kernel Hilbert space method. The main aim of this work is to find how much information can be predicted using numerical computations. The dependence of the number of solutions on the parameters of the model is also studied. Furthermore, the analytical approximations of all branches of solutions can be calculated by the proposed method. The convergence of the proposed method is proved. Some numerical simulations are presented.
در این مقاله ، ما یکتایی و چندگانگی راه حل های یک مدل ریاضی غیرخطی قوی به دست آمده از تئوری راکتور شیمیایی را مطالعه میکنیم. تجزیه و تحلیل بر اساس روش هسته بازتولیدی فضای هیلبرت انجام میگیرد. هدف اصلی این است که بررسی کنیم که چه مقدار از اطلاعات مدل را میتوانیم با استفاده از محاسبات عددی پیش بینی کنیم. همچنین وابستگی تعداد راه حل ها به پارامتر های مسأله نیز بررسی شده است. علاوه بر آن، تقریب های تحلیلی تمام شاخه های راه حل ها را می توان به استفاده از روش پیشنهاد شده به دست آورد. همگرایی روش پیشنهاد شده نیز ثابت شده است و تعدادی شبیه سازی عددی نیز ارائه شده است.
[1] S. Abbasbandy, E. Shivanian, Prediction of multiplicity of solutions of nonlinear boundary value problems: Novel application of homotopy analysis method, Communications in Nonlinear Science and Numerical Simulation 15 (2010) 3830-3846.
[2] S. Abbasbandy, E. Magyari, E. Shivanian, The homotopy analysis method for multiple solutions of nonlinear boundary value problems, Communications in Nonlinear Science and Numerical Simulation 14 (2009) 3530-3536.
[3] S. Abbasbandy, E. Shivanian, Multiple solutions of mixed convection in a porous medium on semi-infinite interval using pseudo-spectral collocation method, Communications in Nonlinear Science and Numerical Simulation 16 (2011) 2745-2752.
[4] S. Abbasbandy, B. Azarnavid, M. S. Alhuthali, A shooting reproducing kernel Hilbert space method for multiple solutions of nonlinear boundary value problems, Journal of Computational and Applied Mathematics 279 (2015) 293-305.
[5] S. Abbasbandy, B. Azarnavid, Some error estimates for the reproducing kernel Hilbert spaces method, Journal of Computational and Applied Mathematics 296 (2016) 789-797.
[6] B. Azarnavid, E. Shivanian, K. Parand, S. Nikmanesh, Multiplicity results by shooting reproducing kernel Hilbert space method for the catalytic reaction in a flat particle, Journal of Theoretical and Computational Chemistry 17 (2018) 185-220.
[7] B. Azarnavid, F. Parvaneh, S. Abbasbandy, Picard-Reproducing Kernel Hilbert Space Method for Solving Generalized Singular Nonlinear Lane-Emden Type Equations, Mathematical Modelling and Analysis 20 (2015) 754-767.
[8] B. Azarnavid, K. Parand, An iterative reproducing kernel method in Hilbert space for the multi{point boundary value problems, Journal of Computational and Applied Mathematics 328 (2018) 151-163.
[9] B. Azarnavid, K. Parand, S. Abbasbandy, An iterative kernel based method for fourth order nonlinear equation with nonlinear boundary condition, Communications in Nonlinear Science and Numerical Simulation 59 (2018) 544-552.
[10] B. Azarnavid, M. Nabati, M. Emamjome, K. Parand, Imposing various boundary conditions on positive definite kernels, Applied Mathematics and Computation 361 (2019) 453-465.
[11] B. Azarnavid, M. Emamjome, M. Nabati, S. Abbasbandy, A reproducing kernel Hilbert space approach in meshless collocation method, Computational and Applied Mathematics 38 (2019) 72-85.
[12] M. Armando, A. M. Salvatore, Boundary value problems for higher order ordinary differential equations, Comment. Math. Univ. Carolin. 3 (1994) 451- 466.
[13] M. G. Cui, Y. Lin, Nonlinear Numerical Analysis in the Reproducing Kernel Space, Nova Science, New York, NY, USA, 2009.
[14] M. Emamjome, B. Azarnavid, H. Roohani Ghehsareh, A reproducing kernel Hilbert space pseudospectral method for numerical investigation of a two-dimensional capillary formation model in tumor angiogenesis problem, Neural Computing and Applications 11 (2017) 1-9.
[15] F. Z. Geng, S. P. Qian, S. Li, A numerical method for singularly perturbed turning point problems with an interior layer, Journal of Computational and Applied Mathematics 255 (2014) 97-105.
[16] M. Heydari, E. Shivanian, B. Azarnavid, S. Abbasbandy, An iterative multistep kernel based method for nonlinear Volterra integral and integro-differential equations of fractional order, Journal of Computational and Applied Mathematics 361 (2019) 97-112.
[17] G. Infante, P. Pietramala, M. Tenuta, Existence and localization of positive solutions for a nonlocal BVP arising in chemical reactor theory, Communications in Nonlinear Science and Numerical Simulation 19 (2014) 2245-2251.
[18] C. Lu, Multiple solutions of a boundary layer problem, Communications in Nonlinear Science and Numerical Simulation 12 (2007) 725-734.
[19] P. Li, H. Chen, Q. Zhang, Multiple positive solutions of n-point boundary value problems on the half-line in Banach spaces, Communications in Nonlinear Science and Numerical Simulation 14 (2009) 2909-2915.
[20] M. Lovo, V. Balakotaiah, Multiplicity features of adiabatic autothermal reactors, AIChE journal 38 (1992) 101-115.
[21] H. R. Marzban, H. R. Tabrizidooz, M. Razzaghi, A composite collocation method for the nonlinear mixed Volterra{Fredholm{Hammerstein integral equations, Communications in Nonlinear Science and Numerical Simulation 16 (2011) 1186-1194.
[22] M. Nabati, M. Emamjome, Mehdi Jalalvand, Reproducing kernel pseudospectral method for the numerical investigation of nonlinear multi-point boundary value problems, Computational and Applied Mathematics 5 (2018) 1-14.
[23] E. Shivanian, F. Abdolrazaghi, On The Existence of Multiple Solutions of a Class of Third-Order Nonlinear Two-Point Boundary Value Problems, Mediterranean Journal of Mathematics 13 (2016) 2339-2351.
