scholarly journals Approximations With Polynomial, Trigonometric, Exponential Splines of the Third Order and Boundary Value Problem

2020 ◽  
Vol 14 ◽  
Keyword(s):  
Boundary Value Problem ◽  
Parabolic Problem ◽  
Order Approximation ◽  
Boundary Value ◽  
Numerical Differentiation ◽  
Polynomial Splines ◽  
Third Order ◽  
Numerical Examples ◽  
The Third ◽  
Trigonometric Splines

This paper is devoted to the construction of localapproximations of functions of one and two variables using thepolynomial, the trigonometric, and the exponential splines. Thesesplines are useful for visualizing flows of graphic information.Here, we also discuss the parallelization of computations. Someattention is paid to obtaining two-sided estimates of theapproximations using interval analysis methods. Particularattention is paid to solving the boundary value problem by usingthe polynomial splines and the trigonometric splines of the thirdand fourth order approximation. Using the considered splines,formulas for a numerical differentiation are constructed. Theseformulas are used to construct computational schemes for solvinga parabolic problem. Questions of approximation and stability ofthe obtained schemes are considered. Numerical examples arepresented.

scholarly journals Approximation by the Third-Order Splines on Uniform and Non-uniform Grids and Image Processing

2020 ◽  
Vol 19 ◽  
Keyword(s):  
Order Approximation ◽  
Approximation Error ◽  
Polynomial Splines ◽  
Third Order ◽  
The Third ◽  
Uniform Grids ◽  
Trigonometric Splines ◽  
Image Enlargement ◽  
Third Order Approximation ◽  
Approximation Polynomial

This work is one of a series of papers that is devoted to the further investigation of polynomial splines and trigonometric splines of the third order approximation. Polynomial basis splines are better known and therefore more commonly used. However, the use of trigonometric basis splines often provides a smaller approximation error. In some cases, the use of the trigonometric approximations is preferable to the polynomial approximations. Here we continue to compare these two types of approximation. The Lebesgue functions and constants are discussed for the polynomial splines and the trigonometric splines. The examples of the applications of the splines to image enlargement are given.

ON THE SOLUTIONS OF NONLINEAR THIRD ORDER BOUNDARY VALUE PROBLEMS

2010 ◽  
Vol 15 (1) ◽  
pp. 127-136
Author(s):  
Sergey Smirnov
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Order Equation ◽  
Structure Of Solutions ◽  
Third Order ◽  
Number Of Solutions ◽  
The Third

The author considers a three‐point third order boundary value problem. Properties and the structure of solutions of the third order equation are discussed. Also, a connection between the number of solutions of the boundary value problem and the structure of solutions of the equation is established.

scholarly journals Positive solution for singular third-order BVPs on the half line with first-order derivative dependence

2021 ◽  
Vol 13 (1) ◽  
pp. 105-126
Author(s):  
Abdelhamid Benmezaï ◽  
El-Djouher Sedkaoui
Keyword(s):  
Boundary Value Problem ◽  
Positive Solution ◽  
Positive Constant ◽  
Space Variable ◽  
Boundary Value ◽  
Order Derivative ◽  
Half Line ◽  
Third Order ◽  
First Order ◽  
The Third

Abstract In this paper, we investigate the existence of a positive solution to the third-order boundary value problem { - u ‴ ( t ) + k 2 u ′ ( t ) = φ ( t ) f ( t , u ( t ) , u ′ ( t ) ) ,       t > 0 u ( 0 ) = u ′ ( 0 ) = u ′ ( + ∞ ) = 0 , \left\{ \matrix{- u'''\left( t \right) + {k^2}u'\left( t \right) = \phi \left( t \right)f\left( {t,u\left( t \right),u'\left( t \right)} \right),\,\,\,t > 0 \hfill \cr u\left( 0 \right) = u'\left( 0 \right) = u'\left( { + \infty } \right) = 0, \hfill \cr} \right. where k is a positive constant, ϕ ∈ L1 (0;+ ∞) is nonnegative and does vanish identically on (0;+ ∞) and the function f : ℝ+ × (0;+ ∞) × (0;+ ∞) → ℝ+ is continuous and may be singular at the space variable and at its derivative.

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