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Classical and weak solutions of the partial differential equations associated with a class of two-point boundary value problems

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boundary value problems 2018年第1期2018卷 1页

作者： Ma, Ning Wu, Zhen Shandong Univ Sch Math Jinan Shandong Peoples R China

This paper is concerned with a kind of first-order quasilinear parabolic partial differential equations associated with a class of ordinary differential equations with two-point boundary value problems. We prove that the function given by the solution of an ordinary differential equation is the unique solution of a first-order quasilinear parabolic partial differential equation in both classical and weak senses.

关键词： two-point boundary value problems Ordinary differential equations Quasilinear partial differential equations Sobolev weak solutions Viscosity solutions

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two-point boundary value problems associated to functional differential equations of even order solved by iterated splines

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APPLIED NUMERICAL MATHEMATICS 2016年 110卷 128-147页

作者： Bica, Alexandru Mihai Curila, Mircea Curila, Sorin Univ Oradea Fac Elect Engn & Informat Technol Fac Environm Protect Dept Math & Informat Univ St 1 Oradea 410087 Romania

A new iterative numerical method to solve two-point boundary value problems associated to functional differential equations of even order is proposed. The method uses a cubic spline interpolation procedure activated at each iterative step. The convergence of the method is proved and it is tested on some numerical experiments. The notion of numerical stability with respect to the choice of the first iteration is introduced proving that the proposed method is numerically stable in this sense. (C) 2016 IMACS. Published by Elsevier B.V. All rights reserved.

关键词： two-point boundary value problems Functional differential equations of even order Iterative numerical method Natural cubic splines

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Well-posedness of a class of two-point boundary value problems associated with ordinary differential equations

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ADVANCES IN DIFFERENCE EQUATIONS 2018年第1期2018卷 0-0页

作者： Liu, Ruyi Wu, Zhen Shandong Univ Sch Math Jinan Shandong Peoples R China Shandong Univ Zhongtai Secur Inst Financial Studies Jinan Shandong Peoples R China

This paper introduces the regular decoupling field to study the existence and uniqueness of solutions of two-point boundary value problems for a class of ordinary differential equations which can be derived from the maximum principle in optimal control theory. The monotonicity conditions used to guarantee the existence and uniqueness of such equations are initially a special case of the regular decoupling field method. More generally, in case of the homogeneous equations, this paper generalizes the application scope of the monotonicity conditions method by using the linear transformation method. In addition, the linear transformation method can be used to handle the situation where the monotonicity conditions and regular decoupling field method cannot be directly applied. These two methods overall develop the well-posedness theory of two-point boundary value problems which has potential applications in optimal control and partial differential equation theory.

关键词： Ordinary differential equations two-point boundary value problems Regular decoupling field Monotonicity conditions Linear transformation Optimal control theory

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High accuracy variable mesh method for nonlinear two-point boundary value problems in divergence form

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APPLIED MATHEMATICS AND COMPUTATION 2016年 273卷 885-896页

作者： Jain, M. K. Sharma, Sachin Mohanty, R. K. Indian Inst Technol Dept Math New Delhi 110016 India Univ Delhi Fac Math Sci Dept Math Delhi 110007 India South Asian Univ Dept Appl Math New Delhi 110021 India

In this article, using three grid points, we discuss variable mesh method of order three for the numerical solution of nonlinear two-point boundary value problems: (p(x)y ')' = f(x,y), y(0) = A, y(1) = B. We first establish an identity from which general three-point finite difference approximation of various order can be obtained. We obtain a family of thud order discretization using variable mesh for the differential equations. We select the free parameter available in this discretization which leads to the simplest third order method. Numerical results are provided to illustrate the proposed method and their convergence. (C) 2015 Elsevier Inc. All rights reserved.

关键词： Variable mesh Finite difference method Nonlinear equation Divergence form two-point boundary value problems

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A Novel Optimization Algorithm for Solving two-point boundary value problems

A Novel Optimization Algorithm for Solving Two-Point Boundar...

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International Conference on Engineering & MIS (ICEMIS)

作者： Rashaideh, Hasan Al Balqa Appl Univ BAU Prince Abdullah Ben Ghazi Fac IT CS Dept Salt Jordan

ISBN: (纸本)9781509055791

A new algorithm for obtaining the numerical approximation to the solution of two-point boundary value problems based on the use of adaptive Differential Evolution (DE) algorithm is presented. The proposed algorithm might be considered as a variation of the finite difference method in the context that each of the derivatives is replaced by an appropriate difference quotients approximation. This novel approach possesses main advantage as compared to other exiting methods;it can be applied without any limitations on the nature of the problem, type of classification, and the number of mesh points. A test case that include different classes of such equations of different types to demonstrate the efficiency and simplicity of the algorithm is presented.

关键词： Differential Evolution two-point boundary value problems

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EIGENvalueS AND EIGENFUNCTIONS OF DISCONTINUOUS two-point boundary value problems WITH AN EIGENPARAMETER IN THE boundary CONDITION

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ROCKY MOUNTAIN JOURNAL OF MATHEMATICS 2013年第6期43卷 1871-1892页

作者： Caliskan, S. Bayramov, A. Oer, Z. Uslu, S. Yildiz Tech Univ Fac Arts & Sci Dept Math TR-34210 Istanbul Turkey Recep Tayyip Erdogan Univ Fac Educ Dept Math Educ Rize Turkey

In this paper we get asymptotic formulas for eigenvalues and eigenfunctions of discontinuous two-point boundary value problems with the eigenparameter in the boundary conditions with transmission conditions at the two points of discontinuity. When our problem is continuous the obtained results coincide with the corresponding results in [3].

关键词： two-point boundary value problems transmission condition asymptotic of eigenvalues and eigenfunctions discontinuous boundary value problems

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Adaptive Approach to Solving a two-point boundary value Problem under Partial Uncertainty in the Disturbance Field

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AUTOMATION AND REMOTE CONTROL 2021年第1期82卷 93-101页

作者： Vavilov, S. A. St Petersburg State Univ St Petersburg 199034 Russia

We consider the construction of an adaptive algorithm for solving a boundary value problem ensuring that a trajectory issuing from some point hits a finite-size target at a given time under partial uncertainty in the disturbance field. Although the disturbance field has a component that is unknown in explicit form but is still important for hitting a target of given size, we have constructed an iteration procedure for solving the problem in finitely many steps under a number of conditions. The algorithm is based on trial trajectories and uses the measurement of their deviations from the target center as a feedback;this proves sufficient to compensate for the incompleteness of information about the external disturbance field.

关键词： adaptive control two-point boundary value problems uncertainty conditions differential equations

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The Newton-Cotes quadratures for solving a delay differential system

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JOURNAL OF APPLIED MATHEMATICS AND COMPUTING 2022年第6期68卷 4589-4604页

作者： Zhu, Zhi-Qiang Wang, Qi-Ru Guangdong Polytech Normal Univ Dept Informat & Comp Sci Guangzhou 510665 Peoples R China Sun Yat Sen Univ Sch Math Guangzhou 510275 Peoples R China

In this paper, we consider a class of delay differential systems with two-point boundary value conditions. First, we give a criterion to guarantee the existence and uniqueness of solutions for the system. Then, by making use of lower order Newton-Cotes rules, we present a computer-oriented iterative scheme to approximate the exact solution. The efficiency of algorithm is verified by two examples.

关键词： Delay differential systems two-point boundary value problems Existence and uniqueness Numerical solutions The Newton-Cotes Quadratures Green functions

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STATICIZATION AND ITERATED STATICIZATION

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SIAM JOURNAL ON CONTROL AND OPTIMIZATION 2022年第3期60卷 1840-1862页

作者： McEneaney, William M. Zhao, Ruobing Univ Calif San Diego Dept Mech & Aerosp Engn La Jolla CA 92093 USA

Conservative dynamical systems propagate as stationary points of the action functional. Using this representation, it has previously been demonstrated that one may obtain fundamental solutions for two-point boundary value problems for some classes of conservative systems via a solution of an associated dynamic program. It is also known that the gravitational and Coulomb potentials may be represented as stationary points of cubicly parameterized quadratic functionals. Hence, stationary points of the action functional may be represented via iterated ``staticization"" of polynomial functionals, where the staticization operator (introduced and discussed in [J. Differential Equations, 264 (2018), pp. 525--549] and [Automatica J. IFAC, 81 (2017), pp. 56--67]) maps a function to the function value(s) at its stationary (i.e., critical) points. This leads to representations through operations on sets of solutions of differential Riccati equations. A key step in this process is the reordering of staticization operations. Conditions under which this reordering is allowed are obtained, and it is shown that the conditions are satisfied for an astrodynamics problem.

关键词： dynamic programming stationary action staticization two-point boundary value problems conservative dynamical systems

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Fractional-order boundary value problems solutions using advanced numerical technique

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Partial Differential Equations in Applied Mathematics 2025年 13卷

作者： Batool, Asmat Talib, Imran Riaz, Muhammad Bilal Department of Mathematics University of Management and Technology Lahore Pakistan Nonlinear Analysis Group Department of Mathematics Virtual University of Pakistan Lahore Pakistan IT4Innovations VSB – Technical University of Ostrava Ostrava Czech Republic

The main motivation of this study is to extend the use of the operational matrices approach to solve fractional-order two-point boundary value problems (TPBVPs), a method often employed in the literature for solving fractional-order initial value problems. Our proposed approach employs innovative operational matrices, specifically the integral operational matrices based on Chelyshkov polynomials (CPs), a type of orthogonal polynomials. These operational matrices enable us to integrate monomial terms into the algorithm, effectively converting the problem into easily solvable Sylvester-type equations. We provide a comprehensive comparison to demonstrate the accuracy and computational advantages of our proposed approach against existing methods, including the exact solution, the Haar wavelet method (HWM), the Bessel collocation method (BCM), the Pseudo Spectral Method (PSM), the Generalized Adams–Bashforth–Moulton Method (GABMM) and the fractional central difference scheme (FCDS) through numerical examples. Additionally, our proposed approach is well-suited for solving problems with both polynomial and non-polynomial solutions. © 2025 The Authors

关键词： Caputo fractional derivative Chelyshkov polynomials Fractional differential equations Operational matrices approach Orthogonal polynomials two-point boundary value problems

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