Applications of (<i>h</i>,<i>q</i>)-Time Scale Calculus to the Solution of Partial Differential Equations <xref rid="fn1-csmf-2463464" ref-type="fn">†</xref>
In this article, we developed the idea of <i>q</i>-time scale calculus in quantum geometry. It includes the <i><i>q</i>-</i>time scale integral operators and<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline&qu...
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| Format: | Article |
| Language: | English |
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MDPI AG
2023-04-01
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| Series: | Computer Sciences & Mathematics Forum |
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| Online Access: | https://www.mdpi.com/2813-0324/7/1/56 |
| _version_ | 1827575152780509184 |
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| author | Hussain Ali Ghulam Muhammad Munawwar Ali Abbas |
| author_facet | Hussain Ali Ghulam Muhammad Munawwar Ali Abbas |
| author_sort | Hussain Ali |
| collection | DOAJ |
| description | In this article, we developed the idea of <i>q</i>-time scale calculus in quantum geometry. It includes the <i><i>q</i>-</i>time scale integral operators and<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo> </mo><msub><mo>∆</mo><mi>q</mi></msub></mrow></semantics></math></inline-formula>-differentials. It analyzes the fundamental principles which follow the calculus of <i>q</i>-time scales compared with the Leibnitz–Newton usual calculus and have few crucial consequences. The <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mo>∆</mo><mi>q</mi></msub></mrow></semantics></math></inline-formula>-differential reduced method of transformations was proposed to work out on partial <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="sans-serif">Δ</mi><mi>q</mi></msub></mrow></semantics></math></inline-formula>-differential equations in time scale. With easily computable coefficients, the result is calculated in the version of a power series which is convergent. The performance and effectiveness of the proposed procedure are also illustrated, and Matlab software is applied for calculation with the support of some fascinating examples. It changes when <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>σ</mi><mfenced><mi>t</mi></mfenced><mo>=</mo><mi>t</mi></mrow></semantics></math></inline-formula> and <i>q</i> = 1; then, the solution merges with usual calculus for the mentioned initial value problem. The finding of the present work is that the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="sans-serif">Δ</mi><mi>q</mi></msub></mrow></semantics></math></inline-formula>-differential transformation reduced method is convenient and efficient. |
| first_indexed | 2024-03-08T20:52:10Z |
| format | Article |
| id | doaj.art-d15f91afe59e4a98a82115a7d34fe62e |
| institution | Directory Open Access Journal |
| issn | 2813-0324 |
| language | English |
| last_indexed | 2024-03-08T20:52:10Z |
| publishDate | 2023-04-01 |
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| series | Computer Sciences & Mathematics Forum |
| spelling | doaj.art-d15f91afe59e4a98a82115a7d34fe62e2023-12-22T14:02:12ZengMDPI AGComputer Sciences & Mathematics Forum2813-03242023-04-01715610.3390/IOCMA2023-14388Applications of (<i>h</i>,<i>q</i>)-Time Scale Calculus to the Solution of Partial Differential Equations <xref rid="fn1-csmf-2463464" ref-type="fn">†</xref>Hussain Ali0Ghulam Muhammad1Munawwar Ali Abbas2Department of Mathematics, University of Baltistan, Skardu 16200, PakistanDepartment of Mathematics, University of Baltistan, Skardu 16200, PakistanDepartment of Mathematics, University of Baltistan, Skardu 16200, PakistanIn this article, we developed the idea of <i>q</i>-time scale calculus in quantum geometry. It includes the <i><i>q</i>-</i>time scale integral operators and<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo> </mo><msub><mo>∆</mo><mi>q</mi></msub></mrow></semantics></math></inline-formula>-differentials. It analyzes the fundamental principles which follow the calculus of <i>q</i>-time scales compared with the Leibnitz–Newton usual calculus and have few crucial consequences. The <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mo>∆</mo><mi>q</mi></msub></mrow></semantics></math></inline-formula>-differential reduced method of transformations was proposed to work out on partial <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="sans-serif">Δ</mi><mi>q</mi></msub></mrow></semantics></math></inline-formula>-differential equations in time scale. With easily computable coefficients, the result is calculated in the version of a power series which is convergent. The performance and effectiveness of the proposed procedure are also illustrated, and Matlab software is applied for calculation with the support of some fascinating examples. It changes when <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>σ</mi><mfenced><mi>t</mi></mfenced><mo>=</mo><mi>t</mi></mrow></semantics></math></inline-formula> and <i>q</i> = 1; then, the solution merges with usual calculus for the mentioned initial value problem. The finding of the present work is that the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="sans-serif">Δ</mi><mi>q</mi></msub></mrow></semantics></math></inline-formula>-differential transformation reduced method is convenient and efficient.https://www.mdpi.com/2813-0324/7/1/56Δ<i>q</i>-differential<i><i>q</i>-</i>time scale<i>q</i>-Integral operatorsΔ<i>q</i>-differential reduced transform methodpartial differential equations |
| spellingShingle | Hussain Ali Ghulam Muhammad Munawwar Ali Abbas Applications of (<i>h</i>,<i>q</i>)-Time Scale Calculus to the Solution of Partial Differential Equations <xref rid="fn1-csmf-2463464" ref-type="fn">†</xref> Computer Sciences & Mathematics Forum Δ<i>q</i>-differential <i><i>q</i>-</i>time scale <i>q</i>-Integral operators Δ<i>q</i>-differential reduced transform method partial differential equations |
| title | Applications of (<i>h</i>,<i>q</i>)-Time Scale Calculus to the Solution of Partial Differential Equations <xref rid="fn1-csmf-2463464" ref-type="fn">†</xref> |
| title_full | Applications of (<i>h</i>,<i>q</i>)-Time Scale Calculus to the Solution of Partial Differential Equations <xref rid="fn1-csmf-2463464" ref-type="fn">†</xref> |
| title_fullStr | Applications of (<i>h</i>,<i>q</i>)-Time Scale Calculus to the Solution of Partial Differential Equations <xref rid="fn1-csmf-2463464" ref-type="fn">†</xref> |
| title_full_unstemmed | Applications of (<i>h</i>,<i>q</i>)-Time Scale Calculus to the Solution of Partial Differential Equations <xref rid="fn1-csmf-2463464" ref-type="fn">†</xref> |
| title_short | Applications of (<i>h</i>,<i>q</i>)-Time Scale Calculus to the Solution of Partial Differential Equations <xref rid="fn1-csmf-2463464" ref-type="fn">†</xref> |
| title_sort | applications of i h i i q i time scale calculus to the solution of partial differential equations xref rid fn1 csmf 2463464 ref type fn † xref |
| topic | Δ<i>q</i>-differential <i><i>q</i>-</i>time scale <i>q</i>-Integral operators Δ<i>q</i>-differential reduced transform method partial differential equations |
| url | https://www.mdpi.com/2813-0324/7/1/56 |
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