Numerical Methods for Caputo–Hadamard Fractional Differential Equations with Graded and Non-Uniform Meshes

Numerical Methods for Caputo–Hadamard Fractional Differential Equations with Graded and Non-Uniform Meshes

We consider the predictor-corrector numerical methods for solving Caputo–Hadamard fractional differential equations with the graded meshes <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo form="...

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Autores principales: Charles Wing Ho Green, Yanzhi Liu, Yubin Yan
Formato: article
Lenguaje:EN
Publicado: MDPI AG 2021
Materias:
predictor-corrector method
Caputo–Hadamard fractional derivative
graded meshes
error estimates
Mathematics
QA1-939
Acceso en línea:https://doaj.org/article/7f323a305db14369a814159008966cac
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institution DOAJ
collection DOAJ
language EN
topic predictor-corrector method
Caputo–Hadamard fractional derivative
graded meshes
error estimates
Mathematics
QA1-939
spellingShingle predictor-corrector method
Caputo–Hadamard fractional derivative
graded meshes
error estimates
Mathematics
QA1-939
Charles Wing Ho Green
Yanzhi Liu
Yubin Yan
Numerical Methods for Caputo–Hadamard Fractional Differential Equations with Graded and Non-Uniform Meshes
description We consider the predictor-corrector numerical methods for solving Caputo–Hadamard fractional differential equations with the graded meshes <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo form="prefix">log</mo><msub><mi>t</mi><mi>j</mi></msub><mo>=</mo><mo form="prefix">log</mo><mi>a</mi><mo>+</mo><mfenced separators="" open="(" close=")"><mo form="prefix">log</mo><mfrac><msub><mi>t</mi><mi>N</mi></msub><mi>a</mi></mfrac></mfenced><msup><mfenced open="(" close=")"><mfrac><mi>j</mi><mi>N</mi></mfrac></mfenced><mi>r</mi></msup><mo>,</mo><mspace width="0.166667em"></mspace><mi>j</mi><mo>=</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>N</mi></mrow></semantics></math></inline-formula> with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>a</mi><mo>≥</mo><mn>1</mn></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>r</mi><mo>≥</mo><mn>1</mn></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo form="prefix">log</mo><mi>a</mi><mo>=</mo><mo form="prefix">log</mo><msub><mi>t</mi><mn>0</mn></msub><mo><</mo><mo form="prefix">log</mo><msub><mi>t</mi><mn>1</mn></msub><mo><</mo><mo>⋯</mo><mo><</mo><mo form="prefix">log</mo><msub><mi>t</mi><mi>N</mi></msub><mo>=</mo><mo form="prefix">log</mo><mi>T</mi></mrow></semantics></math></inline-formula> is a partition of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>[</mo><mo form="prefix">log</mo><msub><mi>t</mi><mn>0</mn></msub><mo>,</mo><mo form="prefix">log</mo><mi>T</mi><mo>]</mo></mrow></semantics></math></inline-formula>. We also consider the rectangular and trapezoidal methods for solving Caputo–Hadamard fractional differential equations with the non-uniform meshes <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo form="prefix">log</mo><msub><mi>t</mi><mi>j</mi></msub><mo>=</mo><mo form="prefix">log</mo><mi>a</mi><mo>+</mo><mfenced separators="" open="(" close=")"><mo form="prefix">log</mo><mfrac><msub><mi>t</mi><mi>N</mi></msub><mi>a</mi></mfrac></mfenced><mfrac><mrow><mi>j</mi><mo>(</mo><mi>j</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mrow><mi>N</mi><mo>(</mo><mi>N</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow></mfrac><mo>,</mo></mrow></semantics></math></inline-formula><inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>j</mi><mo>=</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>N</mi></mrow></semantics></math></inline-formula>. Under the weak smoothness assumptions of the Caputo–Hadamard fractional derivative, e.g., <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mmultiscripts><mi>D</mi><mrow><mi>C</mi><mi>H</mi></mrow><mrow></mrow></mmultiscripts><mrow><mi>a</mi><mo>,</mo><mi>t</mi></mrow><mi>α</mi></msubsup><mi>y</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>∉</mo><msup><mi>C</mi><mn>1</mn></msup><mrow><mo>[</mo><mi>a</mi><mo>,</mo><mi>T</mi><mo>]</mo></mrow></mrow></semantics></math></inline-formula> with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>2</mn><mo>)</mo></mrow></semantics></math></inline-formula>, the optimal convergence orders of the proposed numerical methods are obtained by choosing the suitable graded mesh ratio <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>r</mi><mo>≥</mo><mn>1</mn></mrow></semantics></math></inline-formula>. The numerical examples are given to show that the numerical results are consistent with the theoretical findings.
format article
author Charles Wing Ho Green
Yanzhi Liu
Yubin Yan
author_facet Charles Wing Ho Green
Yanzhi Liu
Yubin Yan
author_sort Charles Wing Ho Green
title Numerical Methods for Caputo–Hadamard Fractional Differential Equations with Graded and Non-Uniform Meshes
title_short Numerical Methods for Caputo–Hadamard Fractional Differential Equations with Graded and Non-Uniform Meshes
title_full Numerical Methods for Caputo–Hadamard Fractional Differential Equations with Graded and Non-Uniform Meshes
title_fullStr Numerical Methods for Caputo–Hadamard Fractional Differential Equations with Graded and Non-Uniform Meshes
title_full_unstemmed Numerical Methods for Caputo–Hadamard Fractional Differential Equations with Graded and Non-Uniform Meshes
title_sort numerical methods for caputo–hadamard fractional differential equations with graded and non-uniform meshes
publisher MDPI AG
publishDate 2021
url https://doaj.org/article/7f323a305db14369a814159008966cac
work_keys_str_mv AT charleswinghogreen numericalmethodsforcaputohadamardfractionaldifferentialequationswithgradedandnonuniformmeshes
AT yanzhiliu numericalmethodsforcaputohadamardfractionaldifferentialequationswithgradedandnonuniformmeshes
AT yubinyan numericalmethodsforcaputohadamardfractionaldifferentialequationswithgradedandnonuniformmeshes
_version_ 1718431894357409792
spelling oai:doaj.org-article:7f323a305db14369a814159008966cac2021-11-11T18:16:50ZNumerical Methods for Caputo–Hadamard Fractional Differential Equations with Graded and Non-Uniform Meshes10.3390/math92127282227-7390https://doaj.org/article/7f323a305db14369a814159008966cac2021-10-01T00:00:00Zhttps://www.mdpi.com/2227-7390/9/21/2728https://doaj.org/toc/2227-7390We consider the predictor-corrector numerical methods for solving Caputo–Hadamard fractional differential equations with the graded meshes <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo form="prefix">log</mo><msub><mi>t</mi><mi>j</mi></msub><mo>=</mo><mo form="prefix">log</mo><mi>a</mi><mo>+</mo><mfenced separators="" open="(" close=")"><mo form="prefix">log</mo><mfrac><msub><mi>t</mi><mi>N</mi></msub><mi>a</mi></mfrac></mfenced><msup><mfenced open="(" close=")"><mfrac><mi>j</mi><mi>N</mi></mfrac></mfenced><mi>r</mi></msup><mo>,</mo><mspace width="0.166667em"></mspace><mi>j</mi><mo>=</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>N</mi></mrow></semantics></math></inline-formula> with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>a</mi><mo>≥</mo><mn>1</mn></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>r</mi><mo>≥</mo><mn>1</mn></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo form="prefix">log</mo><mi>a</mi><mo>=</mo><mo form="prefix">log</mo><msub><mi>t</mi><mn>0</mn></msub><mo><</mo><mo form="prefix">log</mo><msub><mi>t</mi><mn>1</mn></msub><mo><</mo><mo>⋯</mo><mo><</mo><mo form="prefix">log</mo><msub><mi>t</mi><mi>N</mi></msub><mo>=</mo><mo form="prefix">log</mo><mi>T</mi></mrow></semantics></math></inline-formula> is a partition of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>[</mo><mo form="prefix">log</mo><msub><mi>t</mi><mn>0</mn></msub><mo>,</mo><mo form="prefix">log</mo><mi>T</mi><mo>]</mo></mrow></semantics></math></inline-formula>. We also consider the rectangular and trapezoidal methods for solving Caputo–Hadamard fractional differential equations with the non-uniform meshes <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo form="prefix">log</mo><msub><mi>t</mi><mi>j</mi></msub><mo>=</mo><mo form="prefix">log</mo><mi>a</mi><mo>+</mo><mfenced separators="" open="(" close=")"><mo form="prefix">log</mo><mfrac><msub><mi>t</mi><mi>N</mi></msub><mi>a</mi></mfrac></mfenced><mfrac><mrow><mi>j</mi><mo>(</mo><mi>j</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mrow><mi>N</mi><mo>(</mo><mi>N</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow></mfrac><mo>,</mo></mrow></semantics></math></inline-formula><inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>j</mi><mo>=</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>N</mi></mrow></semantics></math></inline-formula>. Under the weak smoothness assumptions of the Caputo–Hadamard fractional derivative, e.g., <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mmultiscripts><mi>D</mi><mrow><mi>C</mi><mi>H</mi></mrow><mrow></mrow></mmultiscripts><mrow><mi>a</mi><mo>,</mo><mi>t</mi></mrow><mi>α</mi></msubsup><mi>y</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>∉</mo><msup><mi>C</mi><mn>1</mn></msup><mrow><mo>[</mo><mi>a</mi><mo>,</mo><mi>T</mi><mo>]</mo></mrow></mrow></semantics></math></inline-formula> with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>2</mn><mo>)</mo></mrow></semantics></math></inline-formula>, the optimal convergence orders of the proposed numerical methods are obtained by choosing the suitable graded mesh ratio <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>r</mi><mo>≥</mo><mn>1</mn></mrow></semantics></math></inline-formula>. The numerical examples are given to show that the numerical results are consistent with the theoretical findings.Charles Wing Ho GreenYanzhi LiuYubin YanMDPI AGarticlepredictor-corrector methodCaputo–Hadamard fractional derivativegraded mesheserror estimatesMathematicsQA1-939ENMathematics, Vol 9, Iss 2728, p 2728 (2021)

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