Article
Mathematics, Applied
Yong Ma, Ying Wang, Cesar T. Ledesma
Summary: This paper studies the positive solutions of the Lane-Emden equation perturbed by a non-homogeneous potential. The behavior of the solutions depends on the exponent p, with stable solutions obtained when p satisfies certain conditions and unstable solutions obtained otherwise. The existence of extremal solutions provides additional solutions.
ADVANCES IN NONLINEAR ANALYSIS
(2022)
Article
Mathematics, Applied
Fatih Idiz, Gamze Tanoglu, Nasser Aghazadeh
Summary: In this research, we study the numerical solution of fractional Lane-Emden type equations using Legendre wavelets and the quasilinearization technique. The proposed method effectively overcomes the singularity in the equations.
NUMERICAL ALGORITHMS
(2023)
Article
Mathematics, Applied
M. Abdelhakem, Y. H. Youssri
Summary: This research combines two methodical spectral Legendre's derivative algorithms to numerically solve various types of equations, with a focus on stability and convergence analyses. The proposed algorithms are found to be effective and accurate in solving the problems.
APPLIED NUMERICAL MATHEMATICS
(2021)
Article
Mathematics
Huyuan Chen, Laurent Veron
Summary: We study the isolated singularities of functions satisfying (E) (-4)sv +/-|v|p-1v = 0 in S2\{0}, v=0 inRN\S2, where 0 < s < 1, p > 1 and S2 is a bounded domain containing the origin. We use the Caffarelli-Silvestre extension to R+ x RN. We emphasize the obtention of a priori estimates and analyse the set of self-similar solutions via energy methods to characterize the singularities.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2023)
Article
Mathematics, Applied
Julee Shahni, Randhir Singh, Carlo Cattani
Summary: Two efficient numerical algorithms, Bernoulli uniform collocation method and Bernoulli Chebyshev collocation method, are proposed for solving 3rd-order Lane-Emden-Fowler boundary value problems. The singularity at x = 0 is avoided by transforming the problem into its integral form. By using the Bernoulli collocation method, the resulting integral equation is converted into a system of nonlinear equations to be solved numerically. The high accuracy and efficiency of the proposed method are demonstrated by comparing the results with other known techniques.
APPLIED NUMERICAL MATHEMATICS
(2023)
Article
Mathematics, Applied
Arup Kumar Sahoo, S. Chakraverty
Summary: This paper investigates the impact of the curriculum learning process in a multilayer neural network for solving the Lane-Emden and Emden-Fowler models. Gradually expanding the domain, the training process starts in a small domain. The symplectic neural network trial solution is used in solving the titled models. Feedforward neural network and error back-propagation algorithms are employed to minimize the error function and adjust the parameters. The algorithm's consistency is demonstrated by solving multiple problems. By calculating different types of errors (MSE up to 1E-10), the excellent agreement between the current simulations and existing results is shown.
EAST ASIAN JOURNAL ON APPLIED MATHEMATICS
(2023)
Article
Mathematics, Interdisciplinary Applications
Kholoud Saad Albalawi, Ashish Kumar, Badr Saad Alkahtani, Pranay Goswami
Summary: This paper aims to solve general fractional Lane-Emden-Fowler differential equations using the Haar wavelet collocation method. The method transforms the equation into a nonlinear system, which is solved for Haar coefficients using Newton's method. The paper presents the construction of higher-order Lane-Emden-Fowler equations, discusses convergence rate and stability analysis, and demonstrates the accuracy and efficiency of the approach through numerical simulations.
FRACTAL AND FRACTIONAL
(2023)
Article
Mathematics, Applied
Arup Kumar Sahoo, S. Chakraverty
Summary: This paper investigates the impact of curriculum learning on a multilayer neural network (NN) for solving the Lane-Emden and Emden-Fowler models. The authors start with training the NN in a small domain and gradually expand the domain. They use the symplectic NN trial solution for solving the models and employ feedforward NN and error back-propagation algorithms for parameter optimization. The consistency of the algorithm is demonstrated by solving multiple problems and comparing the results with existing ones.
EAST ASIAN JOURNAL ON APPLIED MATHEMATICS
(2023)
Article
Computer Science, Interdisciplinary Applications
Higinio Ramos, Mufutau Ajani Rufai
Summary: An optimized pair of hybrid block techniques is presented and applied to integrate second-order singular initial value problems of Lane-Emden-Fowler type. The proposed one-step hybrid block technique is obtained by using three intermediate points. The obtained block formulas are paired with suitable formulas to avoid singularity issues. Numerical simulations of real-world application problems confirm the superiority and robust performance of the proposed scheme.
MATHEMATICS AND COMPUTERS IN SIMULATION
(2022)
Article
Chemistry, Multidisciplinary
Diksha Tiwari, Amit K. Verma, Carlo Cattani
Summary: This article discusses the construction of a general method based on orthogonal polynomials for capturing solutions near singular points of nonlinear singular boundary value problems. It also explores the convergence of this method and uses orthogonal polynomial wavelets to effectively deal with singularities in solving such problems. The importance and accuracy of the proposed methods are demonstrated by solving Lane-Emden type problems and achieving convergence to exact or known solutions with increased resolution.
JOURNAL OF MATHEMATICAL CHEMISTRY
(2022)
Article
Materials Science, Multidisciplinary
Jian-Jie Wan, Jie Gu, Zhao-Yang Wu, Fan Wu, Jiao Li, Hao-Xue Qiao
Summary: In this paper, the pseudospectral-fitting method is proposed to obtain the radial wave function in position space numerically, and the analytical method is used to calculate the radial wave function in momentum space to avoid numerical difficulties. Finally, the high-accuracy Compton profile of the ns (n = 1-10) orbitals of hydrogenic ions has been calculated at arbitrary momentum using Gauss-Legendre quadrature.
RESULTS IN PHYSICS
(2023)
Article
Computer Science, Interdisciplinary Applications
K. Parand, A. A. Aghaei, S. Kiani, T. Ilkhas Zadeh, Z. Khosravi
Summary: In this paper, an Artificial Neural Network (ANN) method is proposed for solving Lane-Emden type equations, which are important nonlinear singular second order differential equations. Legendre and Chebyshev blocks are introduced as a new efficient neural network architecture based on mathematical properties of Jacobi polynomials to approximate the solution. The use of these blocks significantly increases accuracy, as demonstrated by comparisons with other models and well-known approximate results.
ENGINEERING WITH COMPUTERS
(2023)
Article
Mathematics, Applied
Craig Cowan, Abdolrahman Razani
Summary: This work discusses the existence of positive singular solutions under certain conditions, as well as the case where one or both of the solutions are Holder continuous.
DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS
(2021)
Article
Mathematics, Applied
Fahima Hebhoub, Khaled Zennir, Tosiya Miyasita, Mohamed Biomy
Summary: This study considers a new coupled system for nonlinear wave equations of Emden-Fowler type involving boundary value and initial values. Under certain conditions, the viscoelastic terms are shown to make the problem dissipative, leading to global solutions not existing in L-2 norm beyond a finite time.
Article
Mathematics, Applied
Haochuan Huang, Rui Huang, Jingxue Yin
Summary: This paper focuses on singular curve and critical curve for the Lane-Emden type equation, filling the gap in previous results regarding exponents. By constructing a special blow-up sequence, the convex assumption on the domain Omega is removed, extending the applicability to various cases.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2021)
Article
Engineering, Environmental
Amir Mosayebi, Mohammad Ali Mehrpouya, Reza Abedini
CHEMICAL ENGINEERING JOURNAL
(2016)
Article
Automation & Control Systems
M. Shahini, M. A. Mehrpouya
IMA JOURNAL OF MATHEMATICAL CONTROL AND INFORMATION
(2018)
Article
Mathematics, Applied
M. A. Mehrpouya, M. Shamsi, M. Razzaghi
Article
Automation & Control Systems
M. A. Mehrpouya, M. Shamsi, V. Azhmyakov
JOURNAL OF THE FRANKLIN INSTITUTE-ENGINEERING AND APPLIED MATHEMATICS
(2014)
Article
Acoustics
Mohammad Ali Mehrpouya, Shirin Fallahi
JOURNAL OF VIBRATION AND CONTROL
(2015)
Article
Engineering, Civil
Hossein Heidary, Mohammad Ali Mehrpouya
THIN-WALLED STRUCTURES
(2019)
Article
Engineering, Aerospace
Mohammad A. Mehrpouya
PROCEEDINGS OF THE INSTITUTION OF MECHANICAL ENGINEERS PART G-JOURNAL OF AEROSPACE ENGINEERING
(2020)
Article
Mathematics, Applied
Mohammad Ali Mehrpouya, Haijun Peng
Summary: This paper introduces a robust pseudospectral method for solving nonlinear optimal control problems efficiently. The method first derives the first-order necessary conditions of optimality based on Pontryagin's minimum principle, and then converts the nonlinear optimal control problem into a system of nonlinear algebraic equations using the pseudospectral method. An optimization approach is introduced to simplify the need for a good initial guess, addressing the challenge of solving the system of nonlinear equations. Numerical results of benchmark examples are presented, along with the computational features of the proposed method.
INTERNATIONAL JOURNAL OF COMPUTER MATHEMATICS
(2021)
Article
Automation & Control Systems
M. Shahini, M. A. Mehrpouya
Summary: This paper investigates an efficient numerical method for solving infinite horizon fractional optimal control problems using transformed orthogonal functions to approximate state and control functions, converting the original problem into a nonlinear programming problem. The results show that the method can solve the problem on the original time interval with high convergence rate.
EUROPEAN JOURNAL OF CONTROL
(2021)
Article
Physics, Multidisciplinary
H. Heidary, M. A. Mehrpouya, A. Ghalee, A. R. Oskouei
Summary: This paper introduces a magnetic yoke design using magnetic flux leakage to detect notches on plate edges and demonstrates how the leaked magnetic field is influenced by model parameters. The leaked magnetic field is shown through numerical methods, and an experimental setup is proposed to validate the numerical analysis.
EUROPEAN PHYSICAL JOURNAL PLUS
(2021)
Article
Physics, Multidisciplinary
M. A. Mehrpouya, R. Salehi
Summary: This paper proposes a less sensitive robust numerical scheme for accurate solution of sensitive boundary value problems, utilizing an orthogonal collocation approach for discretization and converting the problem to the solution of nonlinear algebraic equations. The method transforms the system of equations into an optimization problem by considering the values of the solution at collocation points as decision parameters, achieving good results even with poor initial guesses and a small number of discretization points.
EUROPEAN PHYSICAL JOURNAL PLUS
(2021)
Article
Mathematics, Applied
Mohammad Ali Mehrpouya, Mahmood Khaksar-E Oshagh
Summary: This paper presents an efficient computational algorithm for solving Hamiltonian boundary value problems related to bang-bang optimal control problems. The algorithm utilizes an indirect shooting method with control parameterization to accurately capture the values of the control function and switching points. By combining a homotopic approach, the method addresses the challenge of initial guess in solving the shooting equation, leading to accurate solutions without prior assumptions on optimal control structure and sensitivity to unknown parameters.
COMPUTATIONAL METHODS FOR DIFFERENTIAL EQUATIONS
(2021)
