Matrix Method by Genocchi Polynomials for Solving Nonlinear Volterra Integral Equations with Weakly Singular Kernels
<p>The plots of the Genocchi polynomials.</p> "> Figure 2
<p>Plot of comparison between the exact and approximate solutions of Example 1 for <math display="inline"><semantics> <mrow> <mi>N</mi> <mo>=</mo> <mn>5</mn> </mrow> </semantics></math>.</p> "> Figure 3
<p>Plot of the absolute error with <math display="inline"><semantics> <mrow> <mi>N</mi> <mo>=</mo> <mn>5</mn> <mo></mo> </mrow> </semantics></math> for Example 1.</p> "> Figure 4
<p>Plot of the absolute error with <math display="inline"><semantics> <mrow> <mi>N</mi> <mo>=</mo> <mn>10</mn> </mrow> </semantics></math> for Example 1.</p> "> Figure 5
<p>Plot of the absolute error with <math display="inline"><semantics> <mrow> <mi>N</mi> <mo>=</mo> <mn>15</mn> </mrow> </semantics></math> for Example 1.</p> "> Figure 6
<p>Plot of the absolute error with <math display="inline"><semantics> <mrow> <mi>N</mi> <mo>=</mo> <mn>20</mn> </mrow> </semantics></math> for Example 1.</p> "> Figure 7
<p>Plot of approximate solutions by our method (Genocchi polynomials) with different values of <math display="inline"><semantics> <mrow> <mi>N</mi> </mrow> </semantics></math> on the interval <math display="inline"><semantics> <mrow> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mi>ε</mi> <mo stretchy="false">]</mo> </mrow> </semantics></math> with <math display="inline"><semantics> <mrow> <mi>ε</mi> <mo>=</mo> <mn>0.002</mn> </mrow> </semantics></math> for Example 1.</p> ">
Abstract
:1. Introduction
2. Genocchi Polynomials and Their Properties
2.1. Definition of the Genocchi Polynomials
0 | 1 | 2 | 4 | 6 | |
0 | 1 | −1 | 1 | −3 |
2.2. Approximation of Arbitrary Function by Applying Genocchi Polynomials
2.3. Using the Matrix Approach to Compute the Genocchi Approximation Coefficients
3. Implementation of the Genocchi Polynomial Method for Solving Nonlinear Volterra Integral Equations with Weakly Singular Kernels
4. Error Analysis
5. Illustrative Examples
6. Conclusions and Future Work
Author Contributions
Funding
Conflicts of Interest
References
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0.0 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000 |
0.2 | 0.585076 | 0.584768 | 0.584793 | 0.584797 | 0.584804 |
0.4 | 0.736620 | 0.736795 | 0.736802 | 0.736804 | 0.736806 |
0.6 | 0.843508 | 0.843427 | 0.843431 | 0.843434 | 0.843433 |
0.8 | 0.928164 | 0.928313 | 0.928317 | 0.928319 | 0.928318 |
1.0 | 1.00041 | 0.99996 | 1.000001 | 1.000001 | 1.000000 |
Computing Time (s) | ||
---|---|---|
5 | 5.98532 × 10−4 | 0.321 |
10 | 1.14944 × 10−4 | 0.357 |
15 | 4.48214 × 10−5 | 0.420 |
20 | 2.85973 × 10−5 | 0.451 |
0.0 | 0.000000000 | 0.0000000000 | 0.00000000 | 0.00000000 |
0.2 | 0.000272294 | 0.0000359627 | 0.00001089 | 6.67632 × 10−6 |
0.4 | 0.000185829 | 0.0000111075 | 3.8548 × 10−6 | 2.25757 × 10−6 |
0.6 | 0.000075540 | 5.5081 × 10−6 | 1.83763 × 10−6 | 9.72768 × 10−6 |
0.8 | 0.000153622 | 4.62581 × 10−6 | 1.11576 × 10−6 | 2.18163 × 10−6 |
1.0 | 0.000406512 | 0.0000397521 | 8.73408× 10−6 | 9.0017 × 10−6 |
Euler’s Method [10] | Our Method (Genocchi Polynomials) | ||||||||
---|---|---|---|---|---|---|---|---|---|
N | N | ||||||||
80 | 0.67 × 10−2 | 6.60 × 10−3 | 6.50 × 10−3 | 6.30 × 10−3 | 5 | 0.000 | 1.851 × 10−3 | 2.343 × 10−3 | 2.427 × 10−3 |
160 | 3.21 × 10−3 | 3.10 × 10−3 | 3,10 × 10−3 | 3.03 × 10−3 | 10 | 0.000 | 6.595 × 10−4 | 6.704 × 10−4 | 6.704 × 10−4 |
320 | 1.55 × 10−3 | 1.50 × 10−3 | 1.50 × 10−3 | 1.50 × 10−3 | 15 | 0.000 | 3.193 × 10−4 | 3.193 × 10−4 | 3.178 × 10−4 |
640 | 753 × 10−4 | 7.40 × 10−4 | 7.20 × 10−4 | 7.20 × 10−4 | 20 | 0.000 | 2.305 × 10−4 | 2.305 × 10−4 | 2.305 × 10−4 |
Computing Time (s) | ||
---|---|---|
5 | 1.912914 × 10−4 | 0.351 |
10 | 1.087754 × 10−4 | 0.402 |
15 | 9.106063 × 10−5 | 0.457 |
20 | 7.200394 × 10−5 | 0.530 |
0.0000 | 1.11022 × 10−16 | 1.12022 × 10−16 | 0.000000000 | 0.000000000 |
0.0004 | 0.000507344 | 0.000477524 | 0.000214749 | 0.0000875408 |
0.0008 | 0.000800212 | 0.000750437 | 0.000336102 | 0.0001366621 |
0.0012 | 0.001041861 | 0.000973492 | 0.000434214 | 0.0001761062 |
0.0016 | 0.001254042 | 0.001167461 | 0.000518585 | 0.0002097873 |
0.002 | 0.001445845 | 0.001341082 | 0.000593241 | 0.000239374 |
Euler’s Method [10] | Our Method (Genocchi Polynomials) | ||||||
---|---|---|---|---|---|---|---|
N | N | ||||||
40 | 3.60 × 10−2 | 3.00 × 10−2 | 1.70 × 10−2 | 5 | 6.165 × 10−3 | 7.441 × 10−3 | 9.878 × 10−3 |
80 | 2.10 × 10−2 | 1.7 × 10−2 | 9.10 × 10−3 | 10 | 3.378 × 10−3 | 3.386 × 10−3 | 4.162 × 10−3 |
160 | 1.01 × 10−2 | 8.4 × 10−3 | 4.00 × 10−3 | 12 | 2.785 × 10−3 | 3.113 × 10−3 | 3.222 × 10−3 |
320 | 4.00 × 10−3 | 3.00 × 10−3 | 1.30 × 10−3 | 15 | 2.151 × 10−3 | 2.311 × 10−3 | 2.232 × 10−3 |
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Hashemizadeh, E.; Ebadi, M.A.; Noeiaghdam, S. Matrix Method by Genocchi Polynomials for Solving Nonlinear Volterra Integral Equations with Weakly Singular Kernels. Symmetry 2020, 12, 2105. https://doi.org/10.3390/sym12122105
Hashemizadeh E, Ebadi MA, Noeiaghdam S. Matrix Method by Genocchi Polynomials for Solving Nonlinear Volterra Integral Equations with Weakly Singular Kernels. Symmetry. 2020; 12(12):2105. https://doi.org/10.3390/sym12122105
Chicago/Turabian StyleHashemizadeh, Elham, Mohammad Ali Ebadi, and Samad Noeiaghdam. 2020. "Matrix Method by Genocchi Polynomials for Solving Nonlinear Volterra Integral Equations with Weakly Singular Kernels" Symmetry 12, no. 12: 2105. https://doi.org/10.3390/sym12122105