The numerical-analytic solution of boundary-value problems for the equation div(? grad ?) = F with piecewise-constant ?

1970 ◽  
Vol 21 (4) ◽  
pp. 477-485 ◽  
Author(s):  
I. I. Lyashko ◽  
I. M. Velikoivanenko ◽  
G. E. Mistetskii
Keyword(s):  
Boundary Value Problems ◽  
Analytic Solution ◽  
Boundary Value ◽  
Piecewise Constant

scholarly journals Numerical computations of the PCD method

2017 ◽  
Vol 37 (1) ◽  
pp. 39-54
Author(s):  
Ahmed Tahiri
Keyword(s):  
Diffusion Equation ◽  
Boundary Value Problems ◽  
Stiffness Matrix ◽  
Boundary Value ◽  
Elasticity Problem ◽  
Numerical Computations ◽  
Piecewise Constant ◽  
Approximate Problem ◽  
Discretization Technique ◽  
Triangular Elements

The PCD (piecewise constant distributions) method is a discretization technique of the boundary value problems in which the unknown distribution and its derivatives are represented by piecewise constant distributions but on distinct meshes. It has the advantage of producing the most sparse stiffness matrix resulting from the approximate problem. In this contribution, we propose a general PCD triangulation by combining rectangular elements and triangular elements. We also apply this discretization technique for the elasticity problem. We end with presentation of numerical results of the proposed method for the 2D diffusion equation.

On One Class of Boundary Value Problems of the Steady-State Filtration Theory in Porous Grounds

2014 ◽  
Vol 1020 ◽  
pp. 367-372
Author(s):  
Suren M. Mkhitaryan ◽  
H.V. Tokmajyan
Keyword(s):  
Steady State ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Filtration Theory ◽  
Steady State Flow ◽  
Fluid Filtration ◽  
Flow Rates ◽  
Piecewise Constant ◽  
Coefficient Of Filtration ◽  
The Ideal

:In the framework of Darcy's law of filtration the investigation results of one class of boundary value problems of the steady-state filtration theory in porous ground base are presented. The plane mixed bounadry value problems on the structural analysis of hydrotechnical con­struction of dam type on filtrating ground base in the form of a layer of finite or infinite thickness are considered. The coefficient of filtration is assumed to be constant, piecewise constant, or changing by the depth of base according to the exponential law, the property of anisotropy of filtration is also taken into account. Axis-symmetric and three-dimentional boundary value problems of the theory of steady-state fluid filtration in a three-dimentional layer of a finite or infinite thickness are discussed. These problems are of the type of Lamb well-known hydrodynamic problems in the theory of steady-state flow of the ideal fluid, when through the circular or rectangular openeing of a rigid screen on the upper bound of the layer the liquid with a definite vertical velocity or with a definite pressure is injected into porous ground base. Here, the fields of velocities and pressures in the layer, as well as flow rates of liquid through the certain sections of the ground base are determined.

Contemporary Methods for the Numerical-Analytic Solution of Boundary-Value Problems in Noncanonical Domains

2020 ◽  
Vol 247 (1) ◽  
pp. 88-107
Author(s):  
P. Shakeri Mobarakeh ◽  
V. T. Grinchenko ◽  
V. V. Popov ◽  
B. Soltannia ◽  
G. M. Zrazhevsky
Keyword(s):  
Boundary Value Problems ◽  
Analytic Solution ◽  
Boundary Value

scholarly journals Galerkin Method for Nonlinear Higher-Order Boundary Value Problems Based on Chebyshev Polynomials

2021 ◽  
Vol 2128 (1) ◽  
pp. 012035
Author(s):  
W. Abbas ◽  
Mohamed Fathy ◽  
M. Mostafa ◽  
A. M. A Hesham
Keyword(s):  
Galerkin Method ◽  
Boundary Value Problems ◽  
Chebyshev Polynomials ◽  
Nonlinear Boundary ◽  
Analytic Solution ◽  
Boundary Value ◽  
Higher Order ◽  
Nonlinear Boundary Value Problems ◽  
Algebraic Set ◽  
The Galerkin Method

Abstract In the current paper, we develop an algorithm to approximate the analytic solution for the nonlinear boundary value problems in higher-order based on the Galerkin method. Chebyshev polynomials are introduced as bases of the solution. Meanwhile, some theorems are deducted to simplify the nonlinear algebraic set resulted from applying the Galerkin method, while Newton’s method is used to solve the resulting nonlinear system. Numerous examples are presented to prove the usefulness and effectiveness of this algorithm in comparison with some other methods.

scholarly journals Analysis of the Sign of the Solution for Certain Second-Order Periodic Boundary Value Problems with Piecewise Constant Arguments

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 1953
Author(s):  
Sebastián Buedo-Fernández ◽  
Daniel Cao Labora ◽  
Rosana Rodríguez-López ◽  
Stepan A. Tersian
Keyword(s):  
Green’S Function ◽  
Boundary Value Problems ◽  
Unique Solution ◽  
Green's Function ◽  
Periodic Boundary ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Second Order ◽  
Piecewise Constant ◽  
Periodic Boundary Value Problems

We find sufficient conditions for the unique solution of certain second-order boundary value problems to have a constant sign. To this purpose, we use the expression in terms of a Green’s function of the unique solution for impulsive linear periodic boundary value problems associated with second-order differential equations with a functional dependence, which is a piecewise constant function. Our analysis lies in the study of the sign of the Green’s function.

scholarly journals The PCD Method on Composite Grid

2015 ◽  
Vol 34 (2) ◽  
pp. 121-145 ◽  
Author(s):  
Ahmed Tahiri
Keyword(s):  
Diffusion Equation ◽  
Boundary Value Problems ◽  
Variational Formulation ◽  
Boundary Value ◽  
Upper Bounds ◽  
Discretization Method ◽  
Two Dimensional ◽  
Piecewise Constant ◽  
Dimensional Diffusion ◽  
Unknown Distribution

We introduce a discretization method of boundary value problems(BVP) in the case of the two dimensional diffusion equation on arectangular mesh with refined zones. The method consists inrepresenting the unknown distribution and its derivatives bypiecewise constant distributions (PCD) on distinct meshes togetherwith an appropriate approximate variational formulation of the exactBVP on this piecewise constant distributions space. This method,named the PCD method, has the advantage of producing the mostcompact possible discrete stencil. Here, we analyze and prove theconvergence of the PCD method and determine upper bounds on itsconvergence rate.

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