Application of the Solution Structure Method in Numerically Solving Poisson’s Equation on the Basis of Atomic Functions

2018 ◽  
Vol 15 (05) ◽  
pp. 1850033 ◽  
Author(s):  
Vedrana Kozulić ◽  
Blaž Gotovac
Keyword(s):  
Differential Equation ◽  
Boundary Value Problem ◽  
Solution Structure ◽  
Analytical Form ◽  
Basis Functions ◽  
Torsion Problem ◽  
Solution Structures ◽  
The Third ◽  
Collocation Technique ◽  
Unknown Component

This paper summarizes the main principles of the solution structure method and presents it in combination with atomic basis functions and a collocation technique. The solution of a boundary value problem is expressed in the form of formulae called solution structures, which depend on three components: the first component describes the geometry of the domain exactly in the analytical form, the second describes all boundary conditions exactly, and the third component, that contains information about the differential equation, is the unknown component represented by a linear combination of atomic basis functions. The proposed method is applied to solve the torsion problem.

scholarly journals On the third order boundary value problems with asymmetric nonlinearity

2011 ◽  
Vol 16 (2) ◽  
pp. 231-241 ◽  
Author(s):  
Sergey Smirnov
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Third Order ◽  
Number Of Solutions ◽  
The Third ◽  
Asymmetric Nonlinearity ◽  
Autonomous Ordinary Differential Equation

The author considers two point third order boundary value problem with asymmetric nonlinearity. The structure and oscillatory properties of solutions of the third order nonlinear autonomous ordinary differential equation are discussed. Results on the estimation of the number of solutions to boundary value problem are provided. An illustrative example is given.

R-Functions in Boundary Value Problems in Mechanics

1995 ◽  
Vol 48 (4) ◽  
pp. 151-188 ◽  
Author(s):  
V. L. Rvachev ◽  
T. I. Sheiko
Keyword(s):  
Boundary Value Problems ◽  
Solution Structure ◽  
Boundary Value ◽  
Analytical Form ◽  
Mathematical Tool ◽  
Analytical Geometry ◽  
Solution Structures ◽  
Application Fields ◽  
Functional Components ◽  
R Functions

Described are the concepts and applications of the R-functions theory in continuum mechanics boundary value problems which model fields of different physical natures. With R-functions there appears the possibility of creating a constructive mathematical tool which incorporates the capabilities of classical continuous analysis and logic algebra. This allows one to overcome the main obstacle which hinders the use of variational methods when solving boundary value problems in domains of complex shape with complex boundary conditions, this obstacle being connected with the construction of so-called coordinate sequences. In contrast to widely used methods of the network type (finite difference, finite and boundary elements), in the R-functions method all the geometric information present in the boundary value problem statement is reduced to analytical form, which allows one to search for a solution in the form of formulae called solution structures containing some indefinite functional components. A method of constructing solution structures satisfying the required conditions of completeness has been developed. The structural formulae include the left-hand sides of the normalized equations of the boundaries of the domains or their regions being considered, thus allowing one to change the solution structure expeditiously when changing the geometric shape. Given in the work is a definition of the basic class of R-functions, solution with their help of the inverse problem of analytical geometry (construction of equations of specified configurations); generalization of the Taylor-Hermite formulae for functional spaces in which points are represented by lines and surfaces; and construction of solution structures of some types of boundary value problems. Shown are the solutions of a number of concrete problems in these application fields with the use of the RL language and POLYE system.

Difference Schemes for Nonlinear BVPS on the Semiaxis

2007 ◽  
Vol 7 (1) ◽  
pp. 25-47 ◽  
Author(s):  
I.P. Gavrilyuk ◽  
M. Hermann ◽  
M.V. Kutniv ◽  
V.L. Makarov
Keyword(s):  
Differential Equation ◽  
Exact Solution ◽  
Boundary Value Problem ◽  
Difference Scheme ◽  
Adaptive Algorithm ◽  
Order Differential Equation ◽  
Constructive Method ◽  
Numerical Examples ◽  
Exact Difference ◽  
The Given

Abstract The scalar boundary value problem (BVP) for a nonlinear second order differential equation on the semiaxis is considered. Under some natural assumptions it is shown that on an arbitrary finite grid there exists a unique three-point exact difference scheme (EDS), i.e., a difference scheme whose solution coincides with the projection of the exact solution of the given differential equation onto the underlying grid. A constructive method is proposed to derive from the EDS a so-called truncated difference scheme (n-TDS) of rank n, where n is a freely selectable natural number. The n-TDS is the basis for a new adaptive algorithm which has all the advantages known from the modern IVP-solvers. Numerical examples are given which illustrate the theorems presented in the paper and demonstrate the reliability of the new algorithm.

Sign in / Sign up

Export Citation Format

Share Document