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A Pseudo-Spectral Scheme for Systems of two-point boundary value problems with Left and Right Sided Fractional Derivatives and Related Integral Equations

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Computer Modeling in Engineering & Sciences 2021年第7期128卷 21-41页

作者： I.G.Ameen N.A.Elkot M.A.Zaky A.S.Hendy E.H.Doha Department of Mathematics Faculty of ScienceAl-Azhar UniversityCairoEgypt Department of Mathematics Faculty of ScienceCairo UniversityGiza12613Egypt Department of Applied Mathematics Physics DivisionNational Research CentreDokkiCairo12622Egypt Department of Computational Mathematics and Computer Science Institute of Natural Sciences and MathematicsUral Federal UniversityYekaterinburg620002Russia Department of Mathematics Faculty of ScienceBenha UniversityBenha13511Egypt

We target here to solve numerically a class of nonlinear fractional two-point boundary value problems involving left-and right-sided fractional *** main ingredient of the proposed method is to recast the problem into an equivalent system of weakly singular integral ***,a Legendre-based spectral collocation method is developed for solving the transformed ***,we can make good use of the advantages of the Gauss quadrature *** present the construction and analysis of the collocation *** results can be indirectly applied to solve fractional optimal control problems by considering the corresponding Euler–Lagrange *** numerical examples are given to confirm the convergence analysis and robustness of the scheme.

关键词： Spectral collocation method weakly singular integral equations two-point boundary value problems convergence analysis

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A reliable algorithm for a class of singular nonlinear two-point boundary value problems arising in physiology

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INTERNATIONAL JOURNAL OF MODELING SIMULATION AND SCIENTIFIC COMPUTING 2023年第5期14卷

作者： Gupta, Sumit Kumar, Devendra Singh, Jagdev Swami Keshvan & Inst Technol Management & Gramotha Dept Math Jaipur 302017 Rajasthan India Univ Rajasthan Dept Math Jaipur 302004 Rajasthan India JECRC Univ Dept Math Jaipur 303905 Rajasthan India

In this paper, we present a reliable numerical algorithm to determine approximate solutions of the two-point boundary value problems having Robin boundary conditions that naturally occur in the investigation of distinct tumor growth issues, the dispersal of heat sources in the person head and steady state oxygen diffusion in spherical cell possessing Michaelis-Menten uptake kinetics. This approach is based on a modified concept of Adomian polynomials (AP), and the two-step Adomian decomposition method (TSADM) merged with Pade approximants. Furthermore a Maple package ADMP is applied to solve various problems, which is very easy to use and efficient and needed to input the system of equations with initial or boundary conditions and diverse essential parameters to deliver the analytic approximate solutions within a few seconds. The suggested scheme does not require linearization, perturbations, guessing the initial terms, a set of basis function or other limiting presumptions, which yields the solutions in closed form. Many examples are examined to make clear the scope and validity of the package ADMP.

关键词： TSADM AP two-point boundary value problems ADMP

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High-order orthogonal spline collocation methods for two-point boundary value problems with interfaces

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MATHEMATICS AND COMPUTERS IN SIMULATION 2020年 174卷 102-122页

作者： Bhal, Santosh Kumar Danumjaya, P. Fairweather, G. BITS Goa Campus Dept Math Sancoale 403726 Goa India Amer Math Soc Math Reviews Ann Arbor MI 48103 USA

Orthogonal spline collocation methods (OSC) are used to solve two-point boundary value problems (BVPs) with interfaces. We first consider the one-dimensional Helmholtz equation with piecewise wave numbers solved using the standard OSC approach. For the solution of self-adjoint two-point BVPs with interfaces, we employ OSC with monomial bases of degree r, where r = 3, 4. In each case, the results of numerical experiments involving numerous examples from the literature exhibit optimal accuracy in the L-infinity and L-2 norms of order r + 1, and order r accuracy in the H-1 norm. Moreover, superconvergence of order 2r - 2 in the nodal error in the OSC approximation and also in its derivative when r = 4 is observed. Each OSC approach gives rise to almost block diagonal linear systems which are solved using standard software. (C) 2020 International Association for Mathematics and Computers in Simulation (IMACS). Published by Elsevier B.V. All rights reserved.

关键词： two-point boundary value problems Helmholtz equation Interfaces Orthogonal spline collocation methods Monomial bases Almost block diagonal linear systems Optimal global error estimates Superconvergence

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An efficient collocation method with convergence rates based on Muntz spaces for solving nonlinear fractional two-point boundary value problems

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COMPUTATIONAL & APPLIED MATHEMATICS 2020年第4期39卷 1-23页

作者： Erfani, S. Javadi, S. Babolian, E. Kharazmi Univ Fac Math Sci & Comp 50 Taleghani Ave Tehran Iran

The main purpose of this work is to develop effective spectral collocation methods for solving numerically general form of non-linear fractional two-point boundary value problems with two end-point singularities. Specifically, we define two classes of Muntz-Legendre polynomials to deal with the singular behaviors of the solutions at both end-points, which enhance greatly the accuracy of the numerical solutions. Important and practical formulas for ordinary derivatives of Muntz-Legendre polynomials are derived. Then using a three-term recurrence formula and Jacobi-Gauss quadrature rules, applicable and stable numerical approaches for computing left and right Caputo fractional derivatives of these basis functions are provided. One of the important features of the present method is to show how to recover spectral accuracy from the end-point singularities for the coupled systems of fractional differential equations with left and right fractional derivatives using the mentioned approximation basis. Moreover, we present a detailed analysis with accurate convergence rates of the singular behavior of the solution to a rather general system of nonlinear fractional differential equations including left and right fractional derivatives. Numerical results show the expected convergence rates. Finally, some numerical examples are given to demonstrate the performance of the proposed schemes.

关键词： Muntz-Legendre polynomials Caputo fractional derivative Collocation method Convergence rate two-point boundary value problems

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Numerical solution of two-point nonlinear boundary value problems via Legendre-Picard iteration method

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MATHEMATICS AND COMPUTERS IN SIMULATION 2022年 199卷 133-159页

作者： Tafakkori-Bafghi, M. Loghmani, G. B. Heydari, M. Yazd Univ Dept Math Yazd Iran

The aim of the present work is to introduce an effective numerical method for solving two-point nonlinear boundary value problems. The proposed iterative scheme, called the Legendre-Picard iteration method is based on the Picard iteration technique, shifted Legendre polynomials and Legendre-Gauss quadrature formula. In the Legendre-Picard iteration method, the boundary value problem is reduced to an iterative formula for updating the coefficients of the approximate solution in each step and with a straightforward manner, the integrals of the shifted Legendre polynomials are calculated. In addition, to reduce the CPU time, a vector-matrix scheme of the Legendre-Picard iteration method is constructed. The convergence analysis of the method is studied. Five nonlinear boundary value problems are given to illustrate the validity of the Legendre-Picard iteration method. Numerical results indicate the good performance and the precision of the proposed procedure.(c) 2022 International Association for Mathematics and Computers in Simulation (IMACS). Published by Elsevier B.V. All rights reserved.

关键词： two-point boundary value problems Picard iteration method Shifted Legendre polynomials Legendre-Gauss quadrature formula vector-matrix structure

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Corrigendum to “two-point boundary value problems associated to functional differential equations of even order solved by iterated splines” [Appl. Numer. Math. 110 (2016) 128–147]

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Applied Numerical Mathematics 2021年 165卷 620-621页

作者： Alexandru Mihai Bica Department of Mathematics and Informatics University of Oradea Universitatii Street no. 1 410087 Oradea Romania

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A Spectral Quasilinearization Parametric Method for Nonlinear two-point boundary value problems

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BULLETIN OF THE MALAYSIAN MATHEMATICAL SCIENCES SOCIETY 2019年第1期42卷 1-13页

作者： Ghorbani, Asghar Gachpazan, Morteza Ferdowsi Univ Mashhad Fac Math Sci Dept Appl Math Mashhad Iran

In this paper, we develop an efficient explicit method based on the spectral parametric iteration method and quasilinearization scheme, which can be used for the efficient numerical solution of nonlinear stiff/nonstiff two-point boundary value problems. The method derived here has the advantage that it does not require the solution of nonlinear systems of equations. We derive the method, which requires one evaluation of the Jacobian and one LU decomposition per step. Some numerical experiments on nonlinear stiff/nonstiff problems show the efficiency and accuracy of the method. Moreover, the method provides us a simple way to control and modify the convergence rate of the solution.

关键词： Spectral parametric iteration method Quasilinearization method two-point boundary value problems Nonlinear boundary conditions Van Der Pol equation

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Numerical study of self-adjoint singularly perturbed two-point boundary value problems using collocation method with error estimation

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Journal of Ocean Engineering and Science 2018年第3期3卷 237-243页

作者： Khalid K.Ali A.R.Hadhoud M.A.Shaalan Faculty of Science Al-Azhar UniversityCairoEgypt Faculty of Science Menoufia UniversityShebeen El-KoomEgypt Higher Technological Institute 10th of Ramadan CityEgypt

A numerical treatment for self-adjoint singularly perturbed second-order two-point boundary value problems using trigonometric quintic B-splines is presented,which depend on different engineering *** method is found to have a truncation error of O(h 6)and converges to the exact solution at O(h 4).The numerical examples show that our method is very effective and the maximum absolute error is acceptable.

关键词： Self-adjoint singularly-perturbation problems two-point boundary value problems Trigonometric quintic B-spline collocation method

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Biologically inspired computing framework for solving two-point boundary value problems using differential evolution

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NEURAL COMPUTING & APPLICATIONS 2017年第8期28卷 2165-2179页

作者： Fateh, Muhammad Faisal Zameer, Aneela Mirza, Nasir M. Mirza, Sikander M. Raja, Muhammad Asif Zahoor Pakistan Inst Engn & Appl Sci Dept Elect Engn Islamabad 45650 Pakistan Pakistan Inst Engn & Appl Sci Dept Comp & Informat Sci Islamabad 45650 Pakistan Pakistan Inst Engn & Appl Sci Dept Phys & Appl Math Islamabad 45650 Pakistan COMSATs Inst Informat Technol Dept Elect Engn Attock Campus Attock Pakistan

In the present study, a design of biologically inspired computing framework is presented for solving second-order two-point boundary value problems (BVPs) by differential evolution (DE) algorithm employing finite difference-based cost function. The DE has been implemented to minimize the combined residue from all nodes in a least square sense. The proposed methodology has been evaluated using five numerical examples in linear and nonlinear regime of BVPs in order to demonstrate the process and check the efficacy of the implementation. The assessment and validation of the DE algorithm have been carried out by comparing the DE-computed results with exact solution as well as with the corresponding data obtained using continuous genetic algorithms. These benchmark comparisons clearly establish DE as a competitive solver in this domain in terms of computational competence and precision.

关键词： two-point boundary value problems Differential evolution Evolutionary algorithms

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SOLVING two-point boundary value problems FOR A WAVE EQUATION VIA THE PRINCIPLE OF STATIONARY ACTION AND OPTIMAL CONTROL

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SIAM JOURNAL ON CONTROL AND OPTIMIZATION 2017年第4期55卷 2151-2205页

作者： Dower, Peter M. McEneaney, William M. Univ Melbourne Dept Elect & Elect Engn Parkville Vic 3010 Australia Univ Calif San Diego Dept Mech & Aerosp Engn La Jolla CA 92093 USA

A new approach to solving two-point boundary value problems for a wave equation is developed. This new approach exploits the principle of stationary action to reformulate and solve such problems in the framework of optimal control. In particular, an in finite dimensional optimal control problem is posed so that the wave equation dynamics and temporal boundary data of interest are captured via the characteristics of the associated Hamiltonian and choice of terminal payoff, respectively. In order to solve this optimal control problem for any such terminal payoff, and hence solve any two-point boundary value problem corresponding to the boundary data encapsulated by that terminal payoff, a fundamental solution to the optimal control problem is constructed. Specifically, the optimal control problem corresponding to any given terminal payoff can be solved via a max-plus convolution of this fundamental solution with the specified terminal payoff. Crucially, the fundamental solution is shown to be a quadratic functional that is de fined with respect to the unique solution of a set of operator differential equations, and computable using spectral methods. An example is presented in which this fundamental solution is computed and applied to solve a two-point boundary value problem for the wave equation of interest.

关键词： two-point boundary value problems wave equation stationary action optimal control fundamental solution

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