Document Type : Research Paper

**Authors**

Faculty of Mathematical Sciences, Shahrood University of Technology, Shahrood, Iran.

**Abstract**

In this article, the Chebyshev pseudo-spectral (CPS) method is presented for solving Troesch’s problem, which is a singular, highly sensitive, and nonlinear boundary problem and occurs in the consideration of the confinement of a plasma column by radiation pressure. Here, a continuous time optimization (CTO) problem corresponding to Troesch’s problem is first proposed. Then, the Chebyshev pseudo-spectral method is used to convert the CTO problem to a discrete-time optimization problem its optimal solution can be found by nonlinear programming methods. The feasibility and convergence of the generated approximate solutions are analyzed. The proposed method is used to solve various kinds of Troesch’s equations. The obtained results have been compared with approximate solutions resulting from well known numerical methods. It can be confirmed that the numerical solutions resulting from this method are completely acceptable and accurate, compared with other techniques.

**Keywords**

- Troesch’s problem
- Nonlinear programming
- Chebyshev pseudo-spectral method
- Continuous and discrete time optimization

**Main Subjects**

- [1] M. Abramowitz and I. Stegun, Handbook of mathematical functions, Dover, New York, 1972.
- [2] L. Bougoffa and M.A. Al-khadhi, New explicit solutions for Troesch’s boundary value problem, Appl. Math. Inform. Sci., 3(1) (2009), 89–96.
- [3] C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang, Spectral Methods in Fluid Dynamics, Springer, New York, 1988.
- [4] S. Chang and I. Chang, A new algorithm for calculating one-dimensional differential transform of non-linear functions, Applied Mathematics and Computation, 195(2) (2008), 799-808.
- [5] S. H. Chang, Numerical solution of Troesch’s problem by simple shooting method, Appl. Math. Comput., 216 (2010), 3303-3306.
- [6] S. H. Chang, A variational iteration method for solving Troesch’s problem, J. Comp. Appl. Math., 234 (2011), 3043-3047.
- [7] E. Deeba, S.A. Khuri, and S. Xie, An algorithm for solving boundary value problems, J. Comput. Phys., 159 (2000), 125-138.
- [8] M. Delkhosh and K. Parand, Generalized pseudospectral method: Theory and applications, Journal of Computational Science, 34 (2019), 11-32.
- [9] M. Delkhosh and K. Parand, A new computational method based on fractional Lagrange functions to solve multiterm fractional differential equations, Numer Algor 88 (2021), 729766.
- [10] M. Delkhosh, K. Parand, and D.D. Ganji, An efficient numerical method to solve the boundary layer flow of an eyring-powell non-newtonian fluid, Journal of Applied and Computational Mechanics, 5(2) (2019), 454-467.
- [11] G. Elnagar, M. A. Kazemi, and M. Razzaghi, The pseudospectral legendre method for discretizing optimal control problem, IEEE Transactions on Automatic Control, 40(10) (1995) 1793-1796.
- [12] G. Elnagar and M. A. Kazemi, Pseudospectral Chebyshev optimal control of constrained nonlinear dynamical systems, Comput. Optim. Appl., 11 (1998) 195-217.
- [13] A. Erdelyi, W. Magnus, F. Oberhettinger, and F. Tricomi, Higher rtanscendental functions, McGraw-Hill, New York, 1953.
- [14] U. Erdogan and T. Ozis, A smart nonstandard finite difference scheme for second order nonlinear boundary value problems, J. Comput. Phys., 230 (2011), 6464-6474.
- [15] H. R. Erfanian, M. H. Noori Skandari, and A. V. Kamyad, Control of a class of nonsmooth dynamical systems, Journal of Vibration and Control, 21(11) (2015), 2212-2222.
- [16] F. Fahroo and I. M. Ross, Costate estimation by a legendre pseudospectral method, Journal of Guidance, Control, and Dynamics, 24(2) (2001), 270-277.
- [17] F. Fahroo and I. M. Ross, Direct trajectory optimization by a Chebyshev pseudospectral method, Journal of Guidance, Control and Dynamics, 25(1) (2002), 160-166.
- [18] X. Feng, L. Mei, and G. He, An efficient algorithm for solving Troesch’s problem, Appl. Math. Comput., 189 (2007), 500-507.
- [19] F. M. Villanueva, Maneuverable reentry vehicle trajectory optimization using pseudospectral method, 2022 IEEE Engineering International Research Conference (EIRCON), (2022), 1-4.
- [20] G. Freud, Orthogonal polynomials, Pregamon Press, Elmsford, 1971.
- [21] M. Ghaznavi and M. H. Noori Skandari, An efficient pseudo-spectral method for nonsmooth dynamical systems, Iran. J. Sci. Technol. Trans. Sci, 42(2) (2018), 635-646.
- [22] D. Gidaspow and B. Baker, A model for discharge of storage batteries, Journal of the Electrochemical Society, 120(8) (1973), 1005-1010.
- [23] Q. Gong, W. Kang, and I. M. Ross, A pseudospectral method for the optimal control of constrained feedback linearizable systems, IEEE Trans. Autom. Control, 51(7) (2006), 1115-1129.
- [24] Q. Gong, I. Michael Ross, W. Kang, and F. Fahroo, Connections between the covector mapping theorem and convergence of pseudospectral methods for optimal control, Computational Optimization and Applications, 41(3) (2008), 307 - 335.
- [25] S. A. Khuri, A numerical algorithm for solving Troesch’s problem, Int. J. Computer Math., 80 (2003), 493-498.
- [26] S. A. Khuri and A. Sayfy, Troesch’s problem: B-spline collocation approach, Math. Comput. Modelling, 54 (2011), 1907-1918.
- [27] H. V. Leal, Y. Khan, G. F. Anaya, A. H. May, A. S. Reyes, U. F. Nino, V. J. Fernandez, and D. P. Diaz, A general solution for Troesch’s problem, Mathematical Problems in Engineering, 2012, Article ID 208375.
- [28] Y. Li, W. Chen, and L. Yang, Multistage linear Gauss pseudospectral method for piecewise continuous nonlinear optimal control problems, IEEE Transactions on Aerospace and Electronic Systems, 57(4), (2021) 2298-2310.
- [29] V. L. Makarov and D. V. Dragunov, An efficient approach for solving stiff nonlinear boundary value problems, Journal of Computational and Applied Mathematics, 345 (2019), 452-470.
- [30] S. H. Mirmoradia, I. Hosseinpoura, S. Ghanbarpourb, and A. Barari,Application of an approximate analytical method to nonlinear Troesch’s problem, Applied Mathematical Sciences, 3 (2009), 1579-1585.
- [31] F. Mohammadizadeh, H. A. Tehrani, and M. H. Noori Skandari, Chebyshev pseudo-spectral method for optimal control problem of Burgers equation, Iranian Journal of Numerical Analysis and Optimization, 9(2) (2019), 77-102.
- [32] S. Momani, S. Abuasad, and Z. Odibat, Variational iteration method for solving nonlinear boundary value problems, Applied Mathematics and Computation, 183(2) (2006), 1351-1358.
- [33] M. Nabati and M. Jalalvand, Solution of Troesch’s problem through double exponential Sinc-Galerkin method, Computational Methods for Differential Equations, 5(2) (2017), 141-157.
- [34] M. H. Noori Skandari and M. Ghaznavi, Chebyshev pseudo-spectral method for Bratu’s problem, Iran. J. Sci. Technol. Trans. Sci, 41(4) (2017), 913-921.
- [35] M. H. Noori Skandari and M. Ghaznavi, A numerical method for solving shortest path problems, Calcolo, 14(1) (2018), 1-14.
- [36] M. H. Noori Skandari and M. Ghaznavi, A novel technique for a class of singular boundary value problems, Computational Methods for Differential Equations, 6(1) (2018), 40-52.
- [37] M. H. Noori Skandari, M. Mahmoudi, J. Vahidi, and M. Ghovatmand, Legendre pseudo-spectral method for solving multi-pantograph delay differential equations, Journal of New Researches in Mathematics, (2022), In press.
- [38] M. H. Noori Skandari, A. V. Kamyad and S. Effati, Generalized Euler-Lagrange equation for nonsmooth calculus of variations, Nonlinear Dynamics, 75(1-2) (2014), 85-100.
- [39] E. Polak, Optimization: algorithms and consistent approximations, Springer, Heidelberg, 1997.
- [40] K. Parand, S. Latifi, M. Delkhosh, and M. M. Moayeri, Generalized Lagrangian Jacobi Gauss collocation method for solving unsteady isothermal gas through a micro-nano porous medium, The European Physical Journal Plus, 133(28) (2018).
- [41] M. A. Z. Raja, Stochastic numerical techniques for solving Troesch’s Problem, Information Sciences, 279 (2014), 860-873.
- [42] S. M. Roberts and J. S. Shipman, On the closed-form solution of Troesch’s problem, J. Comput. Phys., 21 (1976), 291-304.
- [43] A. Saadatmandi and T. Abdolahi-Niasar, Numerical solution of Troeschs problem using Christov rational Functions, Computational Methods for Differential Equations, 3 (2015), 123-133.
- [44] M. Scott, On the conversion of boundary-value problems into stable initial-value problems via several invariant imbedding algorithms, in: A.K. Aziz (Ed.), Numerical Solutions of Boundary-Value Problems for Ordinary Differential Equations, 1975.
- [45] H. Temimi and H. Kurkcu, An accurate asymptotic approximation and precise numerical solution of highly sensitive Troeschs problem, Applied Mathematics and Computation, 235 (2014), 253 260.
- [46] L. N. Trefethen, Spectral methods in MATLAB, Society for industrial and applied mathematics, Philadelphia, 2000.
- [47] B. A. Troesch, A simple approach to a sensitive two-point boundary value problem, J. Comput. Phys., 21 (1976), 279-290.
- [48] E. Weibel, On the confinement of a plasma by magne-tostatic fields, Physics of Fluids. 2(1) (1959), 52-56.
- [49] M. Youssef and G. Baumann, Troeschs problem solved by Sinc methods, Mathematics and Computers in Simulation, 162 (2019), 31-44.
- [50] M. Zarebnia and M. Sajjadian, The sinc-Galerkin method for solving Troeschs problem, Mathematical and Computer Modelling, 56 (2012), 218-228.
- [51] A. E. Zuniga, L. M. l Palacios-Pineda, I. H. Jimenez-Cedeno, O. M. Romero, and D. O. Trejo, A fractal model for current generation in porous electrodes, Journal of Electroanalytical Chemistry, 880 (2021), 114883.

October 2024

Pages 827-841

**Receive Date:**24 April 2023**Revise Date:**26 July 2023**Accept Date:**16 April 2024