A Second-Order Time Discretization for Second Kind Volterra Integral Equations with Non-Smooth Solutions
In this paper, a novel second-order method based on a change of variable and the symmetrical and repeated quadrature formula is presented for numerical solving second kind Volterra integral equations with non-smooth solutions. Applying the discrete Grönwall inequality with weak singularity, the conv...
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MDPI AG
2023-06-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/11/12/2594 |
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| author | Boya Zhou Xiujun Cheng |
| author_facet | Boya Zhou Xiujun Cheng |
| author_sort | Boya Zhou |
| collection | DOAJ |
| description | In this paper, a novel second-order method based on a change of variable and the symmetrical and repeated quadrature formula is presented for numerical solving second kind Volterra integral equations with non-smooth solutions. Applying the discrete Grönwall inequality with weak singularity, the convergence order <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">O</mi><mo>(</mo><msup><mi>N</mi><mrow><mo>−</mo><mn>2</mn></mrow></msup><mo>)</mo></mrow></semantics></math></inline-formula> in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>L</mi><mo>∞</mo></msup></semantics></math></inline-formula> norm is proved, where <i>N</i> refers to the number of time steps. Numerical results are conducted to verify the efficiency and accuracy of the method. |
| first_indexed | 2024-03-11T02:12:05Z |
| format | Article |
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| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-11T02:12:05Z |
| publishDate | 2023-06-01 |
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| spelling | doaj.art-1627ee0247b34862b033cfafca0c4bfa2023-11-18T11:27:05ZengMDPI AGMathematics2227-73902023-06-011112259410.3390/math11122594A Second-Order Time Discretization for Second Kind Volterra Integral Equations with Non-Smooth SolutionsBoya Zhou0Xiujun Cheng1School of Mathematics and Big Data, Foshan University, Foshan 528225, ChinaCollege of Science, Zhejiang Sci-Tech University, Hangzhou 310018, ChinaIn this paper, a novel second-order method based on a change of variable and the symmetrical and repeated quadrature formula is presented for numerical solving second kind Volterra integral equations with non-smooth solutions. Applying the discrete Grönwall inequality with weak singularity, the convergence order <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">O</mi><mo>(</mo><msup><mi>N</mi><mrow><mo>−</mo><mn>2</mn></mrow></msup><mo>)</mo></mrow></semantics></math></inline-formula> in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>L</mi><mo>∞</mo></msup></semantics></math></inline-formula> norm is proved, where <i>N</i> refers to the number of time steps. Numerical results are conducted to verify the efficiency and accuracy of the method.https://www.mdpi.com/2227-7390/11/12/2594volterra equationsweakly singular kernelsa discrete Grönwall inequalitychange of variableerror analysis |
| spellingShingle | Boya Zhou Xiujun Cheng A Second-Order Time Discretization for Second Kind Volterra Integral Equations with Non-Smooth Solutions Mathematics volterra equations weakly singular kernels a discrete Grönwall inequality change of variable error analysis |
| title | A Second-Order Time Discretization for Second Kind Volterra Integral Equations with Non-Smooth Solutions |
| title_full | A Second-Order Time Discretization for Second Kind Volterra Integral Equations with Non-Smooth Solutions |
| title_fullStr | A Second-Order Time Discretization for Second Kind Volterra Integral Equations with Non-Smooth Solutions |
| title_full_unstemmed | A Second-Order Time Discretization for Second Kind Volterra Integral Equations with Non-Smooth Solutions |
| title_short | A Second-Order Time Discretization for Second Kind Volterra Integral Equations with Non-Smooth Solutions |
| title_sort | second order time discretization for second kind volterra integral equations with non smooth solutions |
| topic | volterra equations weakly singular kernels a discrete Grönwall inequality change of variable error analysis |
| url | https://www.mdpi.com/2227-7390/11/12/2594 |
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