Fast Computation of Integrals with Fourier-Type Oscillator Involving Stationary Point
"> Figure 1
<p>(<b>a</b>) Maximum error norm <math display="inline"><semantics> <mrow> <mo>(</mo> <msub> <mi>L</mi> <mo>∞</mo> </msub> <mo>)</mo> </mrow> </semantics></math> for <math display="inline"><semantics> <mrow> <msubsup> <mi>f</mi> <mn>1</mn> <mo>′</mo> </msubsup> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msubsup> <mi>f</mi> <mn>2</mn> <mo>′</mo> </msubsup> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>ω</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>N</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <mn>4</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mn>20</mn> </mrow> </semantics></math>; (<b>b</b>) absolute errors <math display="inline"><semantics> <mrow> <mo>(</mo> <msub> <mi>L</mi> <mrow> <mi>a</mi> <mi>b</mi> <mi>s</mi> </mrow> </msub> <mo>)</mo> </mrow> </semantics></math> of <math display="inline"><semantics> <mrow> <msubsup> <mi>f</mi> <mn>1</mn> <mo>′</mo> </msubsup> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> at different frequencies for fixed nodes <math display="inline"><semantics> <mrow> <mi>N</mi> <mo>=</mo> <mn>20</mn> </mrow> </semantics></math>.</p> "> Figure 2
<p>(<b>a</b>) <math display="inline"><semantics> <msub> <mi>L</mi> <mrow> <mi>a</mi> <mi>b</mi> <mi>s</mi> </mrow> </msub> </semantics></math> scaled by <math display="inline"><semantics> <msup> <mi>ω</mi> <mrow> <mn>3.5</mn> </mrow> </msup> </semantics></math> of the <math display="inline"><semantics> <mrow> <msub> <mi>Q</mi> <mrow> <mi>C</mi> <mo>−</mo> <mi>L</mi> </mrow> </msub> <mrow> <mo>[</mo> <mi>r</mi> <mo>]</mo> </mrow> </mrow> </semantics></math>, (<b>b</b>) <math display="inline"><semantics> <msub> <mi>L</mi> <mrow> <mi>a</mi> <mi>b</mi> <mi>s</mi> </mrow> </msub> </semantics></math> scaled by <math display="inline"><semantics> <msup> <mi>ω</mi> <mn>2</mn> </msup> </semantics></math> of the Filon method (Top) and Levin method (Middle and Bottom) [<a href="#B11-mathematics-07-01160" class="html-bibr">11</a>] for test problem 1.</p> "> Figure 3
<p>(<b>a</b>) <math display="inline"><semantics> <msub> <mi>L</mi> <mrow> <mi>a</mi> <mi>b</mi> <mi>s</mi> </mrow> </msub> </semantics></math> produced by <math display="inline"><semantics> <mrow> <msub> <mi>Q</mi> <mrow> <mi>C</mi> <mo>−</mo> <mi>L</mi> </mrow> </msub> <mrow> <mo>[</mo> <mi>r</mi> <mo>]</mo> </mrow> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msubsup> <mi>Q</mi> <mrow> <mi>h</mi> </mrow> <mn>8</mn> </msubsup> <mrow> <mo>[</mo> <mi>f</mi> <mo>]</mo> </mrow> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msubsup> <mi>Q</mi> <mrow> <mi>H</mi> </mrow> <mi>w</mi> </msubsup> <mrow> <mo>[</mo> <mi>f</mi> <mo>]</mo> </mrow> </mrow> </semantics></math>, (<b>b</b>) <math display="inline"><semantics> <msub> <mi>L</mi> <mrow> <mi>a</mi> <mi>b</mi> <mi>s</mi> </mrow> </msub> </semantics></math> produced by <math display="inline"><semantics> <mrow> <msub> <mi>Q</mi> <mrow> <mi>C</mi> <mo>−</mo> <mi>L</mi> </mrow> </msub> <mrow> <mo>[</mo> <mi>r</mi> <mo>]</mo> </mrow> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msubsup> <mi>Q</mi> <mrow> <mi>h</mi> </mrow> <mn>8</mn> </msubsup> <mrow> <mo>[</mo> <mi>f</mi> <mo>]</mo> </mrow> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msubsup> <mi>Q</mi> <mrow> <mi>H</mi> </mrow> <mi>w</mi> </msubsup> <mrow> <mo>[</mo> <mi>f</mi> <mo>]</mo> </mrow> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mo>(</mo> <mi>N</mi> <mo>,</mo> <mi>n</mi> <mo>=</mo> <mn>10</mn> <mo>)</mo> </mrow> </semantics></math> for test Problem 1.</p> "> Figure 4
<p>(<b>a</b>) Oscillatory behavior of the real part of test Problem 2 for <math display="inline"><semantics> <mrow> <mi>ω</mi> <mo>=</mo> <mn>500</mn> </mrow> </semantics></math>; (<b>b</b>) Oscillation of the real part of test Problem 3 for <math display="inline"><semantics> <mrow> <mi>ω</mi> <mo>=</mo> <mn>100</mn> </mrow> </semantics></math>.</p> "> Figure 5
<p>(<b>a</b>) <math display="inline"><semantics> <msub> <mi>L</mi> <mrow> <mi>a</mi> <mi>b</mi> <mi>s</mi> </mrow> </msub> </semantics></math> produced by Chebyshev–hybrid quadrature (ChQ) scaled by <math display="inline"><semantics> <msup> <mi>ω</mi> <mn>3</mn> </msup> </semantics></math>, (<b>b</b>) <math display="inline"><semantics> <msub> <mi>L</mi> <mrow> <mi>r</mi> <mi>e</mi> <mi>l</mi> </mrow> </msub> </semantics></math> produced by ChQ, Chebyshev–Haar quadrature (CHQ), <math display="inline"><semantics> <mrow> <msubsup> <mi>Q</mi> <mrow> <mi>h</mi> </mrow> <mn>8</mn> </msubsup> <mrow> <mo>[</mo> <mi>f</mi> <mo>]</mo> </mrow> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msubsup> <mi>Q</mi> <mrow> <mi>H</mi> </mrow> <mi>w</mi> </msubsup> <mrow> <mo>[</mo> <mi>f</mi> <mo>]</mo> </mrow> </mrow> </semantics></math> for test Problem 2.</p> "> Figure 6
<p>(<b>a</b>) <math display="inline"><semantics> <msub> <mi>L</mi> <mrow> <mi>r</mi> <mi>e</mi> <mi>l</mi> </mrow> </msub> </semantics></math> produced by ChQ, CHQ, <math display="inline"><semantics> <mrow> <msubsup> <mi>Q</mi> <mrow> <mi>h</mi> </mrow> <mn>8</mn> </msubsup> <mrow> <mo>[</mo> <mi>f</mi> <mo>]</mo> </mrow> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msubsup> <mi>Q</mi> <mrow> <mi>H</mi> </mrow> <mi>w</mi> </msubsup> <mrow> <mo>[</mo> <mi>f</mi> <mo>]</mo> </mrow> </mrow> </semantics></math> for <math display="inline"><semantics> <mrow> <mi>n</mi> <mo>=</mo> <msup> <mn>10</mn> <mn>3</mn> </msup> </mrow> </semantics></math>, (<b>b</b>) <math display="inline"><semantics> <msub> <mi>L</mi> <mrow> <mi>r</mi> <mi>e</mi> <mi>l</mi> </mrow> </msub> </semantics></math> produced by <math display="inline"><semantics> <mrow> <mi>H</mi> <mi>Q</mi> <mn>1</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>H</mi> <mi>Q</mi> <mn>2</mn> </mrow> </semantics></math> reported in [<a href="#B16-mathematics-07-01160" class="html-bibr">16</a>] for <math display="inline"><semantics> <mrow> <mi>n</mi> <mo>=</mo> <msup> <mn>10</mn> <mn>3</mn> </msup> </mrow> </semantics></math>, for test Problem 2.</p> "> Figure 7
<p>(<b>a</b>) <math display="inline"><semantics> <msub> <mi>L</mi> <mrow> <mi>a</mi> <mi>b</mi> <mi>s</mi> </mrow> </msub> </semantics></math> scaled by <math display="inline"><semantics> <msup> <mi>ω</mi> <mn>3</mn> </msup> </semantics></math> of the ChQ, (<b>b</b>) <math display="inline"><semantics> <msub> <mi>L</mi> <mrow> <mi>a</mi> <mi>b</mi> <mi>s</mi> </mrow> </msub> </semantics></math> produced by ChQ, CHQ, <math display="inline"><semantics> <mrow> <msubsup> <mi>Q</mi> <mrow> <mi>h</mi> </mrow> <mn>8</mn> </msubsup> <mrow> <mo>[</mo> <mi>f</mi> <mo>]</mo> </mrow> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msubsup> <mi>Q</mi> <mrow> <mi>H</mi> </mrow> <mi>w</mi> </msubsup> <mrow> <mo>[</mo> <mi>f</mi> <mo>]</mo> </mrow> </mrow> </semantics></math> on increasing <span class="html-italic">N</span> for test Problem 3.</p> "> Figure 8
<p>(<b>a</b>) <math display="inline"><semantics> <msub> <mi>L</mi> <mrow> <mi>a</mi> <mi>b</mi> <mi>s</mi> </mrow> </msub> </semantics></math> produce by Filon-type method scaled by <math display="inline"><semantics> <msup> <mi>ω</mi> <mn>2</mn> </msup> </semantics></math> in [<a href="#B11-mathematics-07-01160" class="html-bibr">11</a>], (<b>b</b>) <math display="inline"><semantics> <msub> <mi>L</mi> <mrow> <mi>a</mi> <mi>b</mi> <mi>s</mi> </mrow> </msub> </semantics></math> scaled by <math display="inline"><semantics> <msup> <mi>ω</mi> <mn>3</mn> </msup> </semantics></math> of ChQ for test Problem 4.</p> "> Figure 9
<p>(<b>a</b>) <math display="inline"><semantics> <msub> <mi>L</mi> <mrow> <mi>a</mi> <mi>b</mi> <mi>s</mi> </mrow> </msub> </semantics></math> produced by ChQ, CHQ, <math display="inline"><semantics> <mrow> <msubsup> <mi>Q</mi> <mrow> <mi>h</mi> </mrow> <mn>8</mn> </msubsup> <mrow> <mo>[</mo> <mi>f</mi> <mo>]</mo> </mrow> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msubsup> <mi>Q</mi> <mrow> <mi>H</mi> </mrow> <mi>w</mi> </msubsup> <mrow> <mo>[</mo> <mi>f</mi> <mo>]</mo> </mrow> </mrow> </semantics></math> for fixed <math display="inline"><semantics> <mi>ω</mi> </semantics></math>, (<b>b</b>) <math display="inline"><semantics> <msub> <mi>L</mi> <mrow> <mi>a</mi> <mi>b</mi> <mi>s</mi> </mrow> </msub> </semantics></math> produced by ChQ, CHQ, <math display="inline"><semantics> <mrow> <msubsup> <mi>Q</mi> <mrow> <mi>h</mi> </mrow> <mn>8</mn> </msubsup> <mrow> <mo>[</mo> <mi>f</mi> <mo>]</mo> </mrow> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msubsup> <mi>Q</mi> <mrow> <mi>H</mi> </mrow> <mi>w</mi> </msubsup> <mrow> <mo>[</mo> <mi>f</mi> <mo>]</mo> </mrow> </mrow> </semantics></math> for fixed nodal points of test Problem 4.</p> ">
Abstract
:1. Introduction
2. Evaluation Procedure
2.1. Levin Quadrature
2.2. Chebyshev Differentiation Matrix and Its Approximation
2.3. Chebyshev–Levin Quadrature
2.4. Adaptive Splitting
- If is used to compute the integral having a stationary point, then Chebyshev–hybrid quadrature is given by
- If is used for computing the integral having stationary point, then Chebyshev–Haar quadrature can be written as
- ;
- ; (Slitting parameter.)
- ;
- , for ; (where r(x) is the amplitude function of (1).)
- ; (The coefficient matrix of the ODE (8) and I is the identity matrix.)
- ; (The approximate solution of the ODE (8).)
- ; (Approximate Chebshev solution of integral having no stationary point.)
- ; (Approximate hybrid solution of the integral having stationary point.)
- ChQ = App1 + App2; (Solution of integral (1)).
3. Error Bounds
- i.
- or
- ii.
- and is monotonic.
4. Numerical Examples and Discussion
5. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
Symbols | Discription |
Chebyshev–Levin quadrature | |
Quadrature based on hybrid functions | |
Quadrature baed on Haar wavelet | |
ChQ | Splitting procedure with Chebyshev–hybrid quadrature |
CHQ | Splitting procedure with Chebyshev–Haar quadrature |
Splitting parameter | |
k | Order of the stationary point |
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mp | mp |
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Zaman, S.; Hussain, I.; Singh, D. Fast Computation of Integrals with Fourier-Type Oscillator Involving Stationary Point. Mathematics 2019, 7, 1160. https://doi.org/10.3390/math7121160
Zaman S, Hussain I, Singh D. Fast Computation of Integrals with Fourier-Type Oscillator Involving Stationary Point. Mathematics. 2019; 7(12):1160. https://doi.org/10.3390/math7121160
Chicago/Turabian StyleZaman, Sakhi, Irshad Hussain, and Dhananjay Singh. 2019. "Fast Computation of Integrals with Fourier-Type Oscillator Involving Stationary Point" Mathematics 7, no. 12: 1160. https://doi.org/10.3390/math7121160