A Discrete Grönwall Inequality and Energy Estimates in the Analysis of a Discrete Model for a Nonlinear Time-Fractional Heat Equation
In the present work, we investigate the efficiency of a numerical scheme to solve a nonlinear time-fractional heat equation with sufficiently smooth solutions, which was previously reported in the literature [Fract. Calc. Appl. Anal. <b>16</b>: 892–910 (2013)]. In that article, the autho...
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MDPI AG
2020-09-01
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| Online Access: | https://www.mdpi.com/2227-7390/8/9/1539 |
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| author | Ahmed S. Hendy Jorge E. Macías-Díaz |
| author_facet | Ahmed S. Hendy Jorge E. Macías-Díaz |
| author_sort | Ahmed S. Hendy |
| collection | DOAJ |
| description | In the present work, we investigate the efficiency of a numerical scheme to solve a nonlinear time-fractional heat equation with sufficiently smooth solutions, which was previously reported in the literature [Fract. Calc. Appl. Anal. <b>16</b>: 892–910 (2013)]. In that article, the authors established the stability and consistency of the discrete model using arguments from Fourier analysis. As opposed to that work, in the present work, we use the method of energy inequalities to show that the scheme is stable and converges to the exact solution with order <inline-formula><math display="inline"><semantics><mrow><mi mathvariant="script">O</mi><mo>(</mo><msup><mi>τ</mi><mrow><mn>2</mn><mo>−</mo><mi>α</mi></mrow></msup><mo>+</mo><msup><mi>h</mi><mn>4</mn></msup><mo>)</mo></mrow></semantics></math></inline-formula>, in the case that <inline-formula><math display="inline"><semantics><mrow><mn>0</mn><mo><</mo><mi>α</mi><mo><</mo><mn>1</mn></mrow></semantics></math></inline-formula> satisfies <inline-formula><math display="inline"><semantics><mrow><msup><mn>3</mn><mi>α</mi></msup><mo>≥</mo><mfrac><mn>3</mn><mn>2</mn></mfrac></mrow></semantics></math></inline-formula>, which means that <inline-formula><math display="inline"><semantics><mrow><mn>0.369</mn><mo>⪅</mo><mi>α</mi><mo>≤</mo><mn>1</mn></mrow></semantics></math></inline-formula>. The novelty of the present work lies in the derivation of suitable energy estimates, and a discrete fractional Grönwall inequality, which is consistent with the discrete approximation of the Caputo fractional derivative of order <inline-formula><math display="inline"><semantics><mrow><mn>0</mn><mo><</mo><mi>α</mi><mo><</mo><mn>1</mn></mrow></semantics></math></inline-formula> used for that scheme at <inline-formula><math display="inline"><semantics><msub><mi>t</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msub></semantics></math></inline-formula>. |
| first_indexed | 2024-03-10T16:28:24Z |
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| last_indexed | 2024-03-10T16:28:24Z |
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| spelling | doaj.art-2ae58b3d7271446ba40b227b32129a8f2023-11-20T13:02:45ZengMDPI AGMathematics2227-73902020-09-0189153910.3390/math8091539A Discrete Grönwall Inequality and Energy Estimates in the Analysis of a Discrete Model for a Nonlinear Time-Fractional Heat EquationAhmed S. Hendy0Jorge E. Macías-Díaz1Department of Computational Mathematics and Computer Science, Institute of Natural Sciences and Mathematics, Ural Federal University, 19 Mira St., 620002 Yekaterinburg, RussiaDepartment of Mathematics, School of Digital Technologies, Tallinn University, Narva Rd. 25, 10120 Tallinn, EstoniaIn the present work, we investigate the efficiency of a numerical scheme to solve a nonlinear time-fractional heat equation with sufficiently smooth solutions, which was previously reported in the literature [Fract. Calc. Appl. Anal. <b>16</b>: 892–910 (2013)]. In that article, the authors established the stability and consistency of the discrete model using arguments from Fourier analysis. As opposed to that work, in the present work, we use the method of energy inequalities to show that the scheme is stable and converges to the exact solution with order <inline-formula><math display="inline"><semantics><mrow><mi mathvariant="script">O</mi><mo>(</mo><msup><mi>τ</mi><mrow><mn>2</mn><mo>−</mo><mi>α</mi></mrow></msup><mo>+</mo><msup><mi>h</mi><mn>4</mn></msup><mo>)</mo></mrow></semantics></math></inline-formula>, in the case that <inline-formula><math display="inline"><semantics><mrow><mn>0</mn><mo><</mo><mi>α</mi><mo><</mo><mn>1</mn></mrow></semantics></math></inline-formula> satisfies <inline-formula><math display="inline"><semantics><mrow><msup><mn>3</mn><mi>α</mi></msup><mo>≥</mo><mfrac><mn>3</mn><mn>2</mn></mfrac></mrow></semantics></math></inline-formula>, which means that <inline-formula><math display="inline"><semantics><mrow><mn>0.369</mn><mo>⪅</mo><mi>α</mi><mo>≤</mo><mn>1</mn></mrow></semantics></math></inline-formula>. The novelty of the present work lies in the derivation of suitable energy estimates, and a discrete fractional Grönwall inequality, which is consistent with the discrete approximation of the Caputo fractional derivative of order <inline-formula><math display="inline"><semantics><mrow><mn>0</mn><mo><</mo><mi>α</mi><mo><</mo><mn>1</mn></mrow></semantics></math></inline-formula> used for that scheme at <inline-formula><math display="inline"><semantics><msub><mi>t</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msub></semantics></math></inline-formula>.https://www.mdpi.com/2227-7390/8/9/1539nonlinear fractional heat equationdiscrete energy estimatesdiscrete fractional Grönwall inequalityconvergence and stability analyses |
| spellingShingle | Ahmed S. Hendy Jorge E. Macías-Díaz A Discrete Grönwall Inequality and Energy Estimates in the Analysis of a Discrete Model for a Nonlinear Time-Fractional Heat Equation Mathematics nonlinear fractional heat equation discrete energy estimates discrete fractional Grönwall inequality convergence and stability analyses |
| title | A Discrete Grönwall Inequality and Energy Estimates in the Analysis of a Discrete Model for a Nonlinear Time-Fractional Heat Equation |
| title_full | A Discrete Grönwall Inequality and Energy Estimates in the Analysis of a Discrete Model for a Nonlinear Time-Fractional Heat Equation |
| title_fullStr | A Discrete Grönwall Inequality and Energy Estimates in the Analysis of a Discrete Model for a Nonlinear Time-Fractional Heat Equation |
| title_full_unstemmed | A Discrete Grönwall Inequality and Energy Estimates in the Analysis of a Discrete Model for a Nonlinear Time-Fractional Heat Equation |
| title_short | A Discrete Grönwall Inequality and Energy Estimates in the Analysis of a Discrete Model for a Nonlinear Time-Fractional Heat Equation |
| title_sort | discrete gronwall inequality and energy estimates in the analysis of a discrete model for a nonlinear time fractional heat equation |
| topic | nonlinear fractional heat equation discrete energy estimates discrete fractional Grönwall inequality convergence and stability analyses |
| url | https://www.mdpi.com/2227-7390/8/9/1539 |
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