Analysis of Sequential Caputo Fractional Differential Equations versus Non-Sequential Caputo Fractional Differential Equations with Applications
It is known that, from a modeling point of view, fractional dynamic equations are more suitable compared to integer derivative models. In fact, a fractional dynamic equation is referred to as an equation with memory. To demonstrate that the fractional dynamic model is better than the corresponding i...
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| Format: | Article |
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MDPI AG
2022-12-01
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| Series: | Foundations |
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| Online Access: | https://www.mdpi.com/2673-9321/2/4/74 |
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| author | Aghalaya S. Vatsala Govinda Pageni V. Anthony Vijesh |
| author_facet | Aghalaya S. Vatsala Govinda Pageni V. Anthony Vijesh |
| author_sort | Aghalaya S. Vatsala |
| collection | DOAJ |
| description | It is known that, from a modeling point of view, fractional dynamic equations are more suitable compared to integer derivative models. In fact, a fractional dynamic equation is referred to as an equation with memory. To demonstrate that the fractional dynamic model is better than the corresponding integer model, we need to compute the solutions of the fractional differential equations and compare them with an integer model relative to the data available. In this work, we will illustrate that the linear <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mi>q</mi></mrow></semantics></math></inline-formula>-order sequential Caputo fractional differential equations, which are sequential of order <i>q</i> where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>q</mi><mo><</mo><mn>1</mn></mrow></semantics></math></inline-formula> with fractional initial conditions and/or boundary conditions, can be solved. The reason for choosing sequential fractional dynamic equations is that linear non-sequential Caputo fractional dynamic equations with constant coefficients cannot be solved in general. We used the Laplace transform method to solve sequential Caputo fractional initial value problems. We used fractional boundary conditions to compute Green’s function for sequential boundary value problems. In addition, the solution of the sequential dynamic equations yields the solution of the corresponding integer-order differential equations as a special case as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>q</mi><mo>→</mo><mn>1</mn></mrow></semantics></math></inline-formula>. |
| first_indexed | 2024-03-11T09:43:29Z |
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| language | English |
| last_indexed | 2024-03-11T09:43:29Z |
| publishDate | 2022-12-01 |
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| spelling | doaj.art-15460ec08ad343e6be6f24df48f4260a2023-11-16T16:43:06ZengMDPI AGFoundations2673-93212022-12-01241129114210.3390/foundations2040074Analysis of Sequential Caputo Fractional Differential Equations versus Non-Sequential Caputo Fractional Differential Equations with ApplicationsAghalaya S. Vatsala0Govinda Pageni1V. Anthony Vijesh2Department of Mathematics, University of Louisiana at Lafayette, Lafayette, LA 70504, USADepartment of Mathematics, Florida State University Panama City, Panama City, FL 32405, USADepartment of Mathematics, Indian Institute of Technology, Indore 453552, IndiaIt is known that, from a modeling point of view, fractional dynamic equations are more suitable compared to integer derivative models. In fact, a fractional dynamic equation is referred to as an equation with memory. To demonstrate that the fractional dynamic model is better than the corresponding integer model, we need to compute the solutions of the fractional differential equations and compare them with an integer model relative to the data available. In this work, we will illustrate that the linear <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mi>q</mi></mrow></semantics></math></inline-formula>-order sequential Caputo fractional differential equations, which are sequential of order <i>q</i> where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>q</mi><mo><</mo><mn>1</mn></mrow></semantics></math></inline-formula> with fractional initial conditions and/or boundary conditions, can be solved. The reason for choosing sequential fractional dynamic equations is that linear non-sequential Caputo fractional dynamic equations with constant coefficients cannot be solved in general. We used the Laplace transform method to solve sequential Caputo fractional initial value problems. We used fractional boundary conditions to compute Green’s function for sequential boundary value problems. In addition, the solution of the sequential dynamic equations yields the solution of the corresponding integer-order differential equations as a special case as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>q</mi><mo>→</mo><mn>1</mn></mrow></semantics></math></inline-formula>.https://www.mdpi.com/2673-9321/2/4/74sequential Caputo fractional derivativefractional initial and boundary value problemsMittag–Leffler functionsGreen’s function |
| spellingShingle | Aghalaya S. Vatsala Govinda Pageni V. Anthony Vijesh Analysis of Sequential Caputo Fractional Differential Equations versus Non-Sequential Caputo Fractional Differential Equations with Applications Foundations sequential Caputo fractional derivative fractional initial and boundary value problems Mittag–Leffler functions Green’s function |
| title | Analysis of Sequential Caputo Fractional Differential Equations versus Non-Sequential Caputo Fractional Differential Equations with Applications |
| title_full | Analysis of Sequential Caputo Fractional Differential Equations versus Non-Sequential Caputo Fractional Differential Equations with Applications |
| title_fullStr | Analysis of Sequential Caputo Fractional Differential Equations versus Non-Sequential Caputo Fractional Differential Equations with Applications |
| title_full_unstemmed | Analysis of Sequential Caputo Fractional Differential Equations versus Non-Sequential Caputo Fractional Differential Equations with Applications |
| title_short | Analysis of Sequential Caputo Fractional Differential Equations versus Non-Sequential Caputo Fractional Differential Equations with Applications |
| title_sort | analysis of sequential caputo fractional differential equations versus non sequential caputo fractional differential equations with applications |
| topic | sequential Caputo fractional derivative fractional initial and boundary value problems Mittag–Leffler functions Green’s function |
| url | https://www.mdpi.com/2673-9321/2/4/74 |
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