Higher-Order Singular BVP Emerging in the Semiconductor Industry: Analysis and Computation of Solutions

Document Type : Research Paper

Authors
Pratikshya Mainini 1
Biswajit Pandit 1
B.R. Priyanka 2
Ravi P. Agarwal 3
1 Center of Data Sciences, Siksha ‘O’ Anusandhan, ITER College, Odisha, 751030, India
2 Department of Mathematics, APS College of Engineering, Bengaluru, 560082, India
3 Department of Mathematics and Systems Engineering, Florida Institute of Technology, Melbourne, FL 32901, USA
10.22055/jacm.2024.47592.4747
Abstract
In this work, we consider a class of non-linear, singular, non-self-adjoint, fourth-order, and singular BVP arising in the semiconductor industry. We investigate the theoretical and numerical results of solutions for the governing model corresponding to different boundary conditions by considering the domain as a disk of any radius T > 0. Problems are difficult to analyze the solutions due to their characteristics. Moreover, it has no exact solutions. Additionally, there are two solutions to these problems. So, it becomes difficult to apply any discrete approach to obtain both approximate solutions. We develop an iterative method based on boundary conditions that gives numerical approximations. The convergence analysis of the technique is discussed. Some numerical examples are given to show the efficiency and accuracy of the method.
Keywords
Fourth Order
Singular Boundary Value Problem
Dual Solutions
Non-Self-Adjoint Operators
Iterative Method
Subjects

Publisher’s Note Shahid Chamran University of Ahvaz remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

[1] O’Regan, D., Theory of singular boundary value problems, World Scientific, Galway, 1994.
[2] Swati, Singh, K., Verma, A.K., Singh, M., Higher order Emden–Fowler type equations via uniform Haar wavelet resolution technique, Journal of Computational and Applied Mathematics, 376, 2020, 112836
[3] Rufai, M.A., Ramos, H., Numerical integration of third-order singular boundary-value problems of Emden–Fowler type using hybrid block techniques, Communications in Nonlinear Science and Numerical Simulation, 105, 2022, 106069.
[4] Pandit, B., Rawani, M.K., Verma, A., Cattani, C., Numerical approximation of higher order singular boundary value problem by using Haar functions, Journal of Mathematical Chemistry, 61(3), 2023, 539-568.
[5] Hasan, Y.Q., Zhu, L.M., A note on the use of modified Adomian decomposition method for solving singular boundary value problems of higher-order ordinary differential equations, Communications in Nonlinear Science and Numerical Simulation, 14(8), 2009, 3261-3265.
[6] Verma, A.K., Pandit, B., Agarwal, R.P., On approximate stationary radial solutions for a class of boundary value problems arising in epitaxial growth theory, Journal of Applied and Computational Mechanics, 6(4), 2020, 713-734.
[7] Verma, A.K., Pandit, B., Agarwal, R.P., On multiple solutions for a fourth order nonlinear singular boundary value problems arising in epitaxial growth theory, Mathematical Methods in the Applied Sciences, 44(7), 2021, 5418-5435.
[8] Verma, A.K., Pandit, B., Escudero, C., Numerical solutions for a class of singular boundary value problems arising in the theory of epitaxial growth, Engineering Computations, 37(7), 2020, 2539-2560.
[9] Pandit, B., Verma, A.K., Agarwal, R.P., Numerical approximations for a class of nonlinear higher order singular boundary value problem by using homotopy perturbation and variational iteration method, Computational and Mathematical Methods, 3(6), 2021, e1195.
[10] Escudero, C., Hakl, R., Peral, I., Torres, P.J., On radial stationary solutions to a model of non-equilibrium growth, European Journal of Applied Mathematics, 24(3), 2013, 437-453.
[11] Pandit, B., Verma, A.K., Agarwal, R.P., Existence and nonexistence results for a class of non-self-adjoint fourth-order singular boundary value problems arising in real life, Mathematical Methods in the Applied Sciences, 46(5), 2023, 6077-6110.
[12] Kardar, M., Parisi, G., Zhang, Y.C., Dynamic scaling of growing interfaces, Physical Review Letters, 56(9), 1986, 889.
[13] King, B.B., Stein, O., Winkler, M., A fourth-order parabolic equation modeling epitaxial thin film growth, Journal of Mathematical Analysis and Applications, 286(2), 2003, 459-490.
[14] Singh, R., Kumar, J., An efficient numerical technique for the solution of nonlinear singular boundary value problems, Computer Physics Communications, 185(4), 2014, 1282-1289.
[15] Hosseini, M.M., Nasabzadeh, H., On the convergence of Adomian decomposition method, Applied Mathematics and Computation, 182(1), 2006, 536-543.
[16] Mittal, R.C., Nigam, R., Solution of a class of singular boundary value problems, Numerical Algorithms, 47, 2008, 169-179.
[17] Pandey, R.K., Singh, A.K., On the convergence of a fourth-order method for a class of singular boundary value problems, Journal of Computational and Applied Mathematics, 224(2), 2009, 734-742.
[18] Kanth, A.R., Aruna, K., He’s variational iteration method for treating nonlinear singular boundary value problems, Computers & Mathematics with Applications, 60(3), 2010, 821-829.
[19] Roul, P., Thula, K., A new high-order numerical method for solving singular two-point boundary value problems, Journal of Computational and Applied Mathematics, 343, 2018, 556-574.
[20] Roul, P., Madduri, H., Kassner, K., A sixth-order numerical method for a strongly nonlinear singular boundary value problem governing electrohydrodynamic flow in a circular cylindrical conduit, Applied Mathematics and Computation, 350, 2019, 416-433.
[21] He, J., Verma, L., Pandit, B., Verma, A., Agarwal, R.P., A new Taylor series based numerical method: Simple, reliable, and promising, Journal of Applied and Computational Mechanics, 9(4), 2023, 1122-1134.
[22] Verma, A., Pandit, B., Verma, L., Agarwal, R.P., A review on a class of second order nonlinear singular BVPs, Mathematics, 8(7), 2020, 1045.
[23] Vivek, Kumar, M., Mishra, S.N., Solution of linear and nonlinear singular value problems using operational matrix of integration of Taylor wavelets, Journal of Taibah University for Science, 17(1), 2023, 2241716.
[24] Vivek, Kumar, M., Shahni, J., New algorithm based on Hermite wavelets and Green’s function to solve Thomas-Fermi type differential equations, Numerical Heat Transfer, Part B: Fundamentals, 2024, doi: 10.1080/10407790.2024.2353794.
[25] Ross, K.A., Elementary analysis: the theory of calculus, Second ed., Springer, New York, 2013.
[26] Abbaoui, K., Cherruault, Y., Convergence of Adomian’s method applied to differential equations, Computers & Mathematics with Applications, 28(5), 1994, 103-109.
[27] Adomian, G., A review and of the decomposition method some recent results for nonlinear equation, Mathematical and Computer Modelling, 13(7), 1990, 17-43.
[28] Jyoti, Singh, M., An iterative technique based on HPM for a class of one dimensional, Mathematics and Computers in Simulation, 200(2), 2022, 50-64.
Journal of Applied and Computational Mechanics
Volume 11, Issue 3
Summer 2025
Pages 568-584