Optimization of the Approximate Integration Formula Using the Discrete Analogue of a High-Order Differential Operator
It is known that discrete analogs of differential operators play an important role in constructing optimal quadrature, cubature, and difference formulas. Using discrete analogs of differential operators, optimal interpolation, quadrature, and difference formulas exact for algebraic polynomials, trig...
| Main Authors: | , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2023-07-01
|
| Series: | Mathematics |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2227-7390/11/14/3114 |
| _version_ | 1797588409069338624 |
|---|---|
| author | Kholmat Shadimetov Aziz Boltaev Roman Parovik |
| author_facet | Kholmat Shadimetov Aziz Boltaev Roman Parovik |
| author_sort | Kholmat Shadimetov |
| collection | DOAJ |
| description | It is known that discrete analogs of differential operators play an important role in constructing optimal quadrature, cubature, and difference formulas. Using discrete analogs of differential operators, optimal interpolation, quadrature, and difference formulas exact for algebraic polynomials, trigonometric and exponential functions can be constructed. In this paper, we construct a discrete analogue <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>D</mi><mi>m</mi></msub><mrow><mo>(</mo><mi>h</mi><mi>β</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> of the differential operator <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mfrac><msup><mi>d</mi><mrow><mn>2</mn><mi>m</mi></mrow></msup><mrow><mi>d</mi><msup><mi>x</mi><mrow><mn>2</mn><mi>m</mi></mrow></msup></mrow></mfrac><mo>+</mo><mn>2</mn><mfrac><msup><mi>d</mi><mi>m</mi></msup><mrow><mi>d</mi><msup><mi>x</mi><mi>m</mi></msup></mrow></mfrac><mo>+</mo><mn>1</mn></mrow></semantics></math></inline-formula> in the Hilbert space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi>W</mi><mn>2</mn><mrow><mo>(</mo><mi>m</mi><mo>,</mo><mn>0</mn><mo>)</mo></mrow></msubsup></semantics></math></inline-formula>. We develop an algorithm for constructing optimal quadrature formulas exact on exponential-trigonometric functions using a discrete operator. Based on this algorithm, in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>m</mi><mo>=</mo><mn>2</mn></mrow></semantics></math></inline-formula>, we give an optimal quadrature formula exact for trigonometric functions. Finally, we present the rate of convergence of the optimal quadrature formula in the Hilbert space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi>W</mi><mn>2</mn><mrow><mo>(</mo><mn>2</mn><mo>,</mo><mn>0</mn><mo>)</mo></mrow></msubsup></semantics></math></inline-formula> for the case <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>m</mi><mo>=</mo><mn>2</mn></mrow></semantics></math></inline-formula>. |
| first_indexed | 2024-03-11T00:51:33Z |
| format | Article |
| id | doaj.art-2e9902bb50314018b61b1b402c61773f |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-11T00:51:33Z |
| publishDate | 2023-07-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-2e9902bb50314018b61b1b402c61773f2023-11-18T20:20:45ZengMDPI AGMathematics2227-73902023-07-011114311410.3390/math11143114Optimization of the Approximate Integration Formula Using the Discrete Analogue of a High-Order Differential OperatorKholmat Shadimetov0Aziz Boltaev1Roman Parovik2Department of Informatics and Computer Graphics, Tashkent State Transport University, 1 Odilkhodjayev Str., Tashkent 100167, UzbekistanDepartment of Informatics and Computer Graphics, Tashkent State Transport University, 1 Odilkhodjayev Str., Tashkent 100167, UzbekistanInternational Integrative Research Laboratory of Extreme Phenomena of Kamchatka, Vitus Bering Kamchatka State University, 4 Pogranichnaya St., Petropavlovsk-Kamchatskiy 683032, RussiaIt is known that discrete analogs of differential operators play an important role in constructing optimal quadrature, cubature, and difference formulas. Using discrete analogs of differential operators, optimal interpolation, quadrature, and difference formulas exact for algebraic polynomials, trigonometric and exponential functions can be constructed. In this paper, we construct a discrete analogue <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>D</mi><mi>m</mi></msub><mrow><mo>(</mo><mi>h</mi><mi>β</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> of the differential operator <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mfrac><msup><mi>d</mi><mrow><mn>2</mn><mi>m</mi></mrow></msup><mrow><mi>d</mi><msup><mi>x</mi><mrow><mn>2</mn><mi>m</mi></mrow></msup></mrow></mfrac><mo>+</mo><mn>2</mn><mfrac><msup><mi>d</mi><mi>m</mi></msup><mrow><mi>d</mi><msup><mi>x</mi><mi>m</mi></msup></mrow></mfrac><mo>+</mo><mn>1</mn></mrow></semantics></math></inline-formula> in the Hilbert space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi>W</mi><mn>2</mn><mrow><mo>(</mo><mi>m</mi><mo>,</mo><mn>0</mn><mo>)</mo></mrow></msubsup></semantics></math></inline-formula>. We develop an algorithm for constructing optimal quadrature formulas exact on exponential-trigonometric functions using a discrete operator. Based on this algorithm, in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>m</mi><mo>=</mo><mn>2</mn></mrow></semantics></math></inline-formula>, we give an optimal quadrature formula exact for trigonometric functions. Finally, we present the rate of convergence of the optimal quadrature formula in the Hilbert space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi>W</mi><mn>2</mn><mrow><mo>(</mo><mn>2</mn><mo>,</mo><mn>0</mn><mo>)</mo></mrow></msubsup></semantics></math></inline-formula> for the case <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>m</mi><mo>=</mo><mn>2</mn></mrow></semantics></math></inline-formula>.https://www.mdpi.com/2227-7390/11/14/3114differential operatordiscrete analogueHilbert spacediscrete argument functionsoptimal quadrature formula |
| spellingShingle | Kholmat Shadimetov Aziz Boltaev Roman Parovik Optimization of the Approximate Integration Formula Using the Discrete Analogue of a High-Order Differential Operator Mathematics differential operator discrete analogue Hilbert space discrete argument functions optimal quadrature formula |
| title | Optimization of the Approximate Integration Formula Using the Discrete Analogue of a High-Order Differential Operator |
| title_full | Optimization of the Approximate Integration Formula Using the Discrete Analogue of a High-Order Differential Operator |
| title_fullStr | Optimization of the Approximate Integration Formula Using the Discrete Analogue of a High-Order Differential Operator |
| title_full_unstemmed | Optimization of the Approximate Integration Formula Using the Discrete Analogue of a High-Order Differential Operator |
| title_short | Optimization of the Approximate Integration Formula Using the Discrete Analogue of a High-Order Differential Operator |
| title_sort | optimization of the approximate integration formula using the discrete analogue of a high order differential operator |
| topic | differential operator discrete analogue Hilbert space discrete argument functions optimal quadrature formula |
| url | https://www.mdpi.com/2227-7390/11/14/3114 |
| work_keys_str_mv | AT kholmatshadimetov optimizationoftheapproximateintegrationformulausingthediscreteanalogueofahighorderdifferentialoperator AT azizboltaev optimizationoftheapproximateintegrationformulausingthediscreteanalogueofahighorderdifferentialoperator AT romanparovik optimizationoftheapproximateintegrationformulausingthediscreteanalogueofahighorderdifferentialoperator |