Boundary-Value Problems for a Class of Operator Differential Equations of High Order with Variable Coefficients

2003 ◽  
Vol 74 (5/6) ◽  
pp. 761-771 ◽  
Author(s):  
A. R. Aliev
Keyword(s):  
Differential Equations ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Variable Coefficients ◽  
High Order

CONDITIONS OF EXISTENCE THE TRACE OF FUNCTIONS FROM SPACE WITH MULTIWEIGHTED DERIVATIVES IN A SPECIAL POINT

2020 ◽  
pp. 123-128
Author(s):  
А. Kalybay ◽  
◽  
Zh. Keulimzhaeva ◽  
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Functional Space ◽  
Variable Coefficients ◽  
Weight Functions ◽  
Weight Sobolev Space ◽  
Strongly Degenerate

When solving differential equations with variable coefficients, especially when the coefficients degenerate at the boundary of a given domain, problems arise in the formulation of boundary value problems. Usually, differential equations with variable coefficients are investigated in a suitable weight functional space. Often in the role of such spaces the weight Sobolev space or various generalizations are considered, which are currently sufficiently studied. However, in some cases, when the coefficients of the considered differential equation are strongly degenerate, the formulation of boundary value problems becomes problematic. In this work, we consider the so-called space with multiweighted derivatives, where after each derivative, the function is multiplied by the weight function and then the next derivative is taken. By controlling the behavior of the weight functions on the boundary, strongly degenerate equations can be investigated. Here we investigate the existence of traces on the boundary of a function from such spaces.

Continuous Solution for Boundary Value Problems on Non Rectangular Geometry

2013 ◽  
Author(s):  
P. Venkataraman
Keyword(s):  
Differential Equations ◽  
Boundary Value Problems ◽  
Continuous Solution ◽  
Boundary Value ◽  
Finite Difference Methods ◽  
High Order ◽  
Coupled Systems ◽  
Discontinuous Solutions ◽  
Finite Elements Analysis ◽  
Order Continuous

Boundary value problems on a rectangular domain can also be solved through non domain discretization techniques. These methods yield high order continuous solutions. One such method, based on Bézier functions and developed by the author, can solve linear, nonlinear, ordinary, partial, single, or coupled systems of differential equations, using the same consistent approach. In this paper the technique is extended to non-rectangular domains. This provides a mesh free alternate to the family of finite element or finite difference methods that are currently used to solve these problems. The problem requires no transformation and the setup is direct and simple. The solution is established by minimizing the error in the residuals of the differential equations and the error in the boundary conditions over the domain. In addition, the high order continuous solution is available in polynomial form. The solution is obtained through the application of standard unconstrained optimization. Three examples are discussed. The first example is Poisson’s equation in a circular region. The second is the solution to the Laplace equation over a five sided domain with one of the sides circular. The third example is a simple nonlinear extension of the differential equation in the second example. Directly accommodating a nonlinear boundary value problem, without any change, is the strength of this approach. Solutions are compared with ones from finite elements analysis where available. This approach can provide continuous solution to problems where previously only discontinuous solutions were available.

scholarly journals Numerical solution for initial and boundary value problems of high-order ordinary differential equations using Hermite neural network algorithm with improved extreme learning machine

2021 ◽  
Author(s):  
Yanfei Lu ◽  
Futian Weng ◽  
Hongli Sun
Keyword(s):  
Neural Network ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Ordinary Differential Equations ◽  
Boundary Value Problems ◽  
Extreme Learning Machine ◽  
Boundary Value ◽  
Initial Boundary ◽  
High Order ◽  
Learning Machine

Abstract In this paper we put forth Hermite neural network (HNN) algorithm with improved extreme learning machine (IELM) to solve initial/boundary value problems of high-order ordinary differential equation(ODEs) and high-order system of ordinary differential equations (SODEs). The model function was expressed as a sum of two terms, where the first term contains no adjustable parameters but satisfies the initial/boundary conditions, the second term involved a single-layered neural network structure with IELM and Hermite basis functions to be trained. The approximate solution is presented in closed form by means of HNN algorithm, whose parameters are obtained by solving a system of linear equations utilizing IELM, which reduces the complexity of the problem. Numerical results demonstrate that the method is effective and reliable for solving high-order ODEs and high-order SODEs with initial and boundary conditions.Mathematics Subject Classification (2020) 34A30 ; 65D15

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