A Fourier Series Technique for Approximate Solutions of Modified Anomalous Time-Fractional Sub-Diffusion Equations
This article presents an innovative approximating technique for addressing modified anomalous time-fractional sub-diffusion equations (MAFSDEs) of the Caputo type. These equations generalize classical diffusion equations, which involve fractional derivatives with respect to time, capturing the non-l...
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MDPI AG
2024-12-01
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| Series: | Fractal and Fractional |
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| Online Access: | https://www.mdpi.com/2504-3110/8/12/718 |
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| author | Samad Kheybari Farzaneh Alizadeh Mohammad Taghi Darvishi Kamyar Hosseini |
| author_facet | Samad Kheybari Farzaneh Alizadeh Mohammad Taghi Darvishi Kamyar Hosseini |
| author_sort | Samad Kheybari |
| collection | DOAJ |
| description | This article presents an innovative approximating technique for addressing modified anomalous time-fractional sub-diffusion equations (MAFSDEs) of the Caputo type. These equations generalize classical diffusion equations, which involve fractional derivatives with respect to time, capturing the non-local and history-dependent behavior typical in sub-diffusion processes. In such a model, the particle transports slower than in a standard diffusion, often due to obstacles or memory effects in the medium. The core of the proposed technique involves transforming the original problem into a family of independent fractional-order ordinary differential equations (FODEs). This transformation is achieved using the Fourier expansion method. Each of these resulting FODEs is defined under initial value conditions which are derived from the initial condition of the original problem. To solve them, for each resulting FODE, some secondary initial value problems are introduced. By solving these secondary initial value problems, some particular solutions are obtained and then we combine them linearly in an optimal manner. This combination is essential to estimate the solution of the original problem. To evaluate the accuracy and effectiveness of the proposed scheme, we conduct a various test problem. For each problem, we analyze the observed convergence order indicators and compare them with those from other methods. Our comparison demonstrates that the proposed technique provides enhanced precision and reliability in respect with the current numerical approaches in the literature. |
| format | Article |
| id | doaj-art-c321df35d0c4487e8d3f9d33b21f1586 |
| institution | DOAJ |
| issn | 2504-3110 |
| language | English |
| publishDate | 2024-12-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Fractal and Fractional |
| spelling | doaj-art-c321df35d0c4487e8d3f9d33b21f15862025-08-20T02:53:35ZengMDPI AGFractal and Fractional2504-31102024-12-0181271810.3390/fractalfract8120718A Fourier Series Technique for Approximate Solutions of Modified Anomalous Time-Fractional Sub-Diffusion EquationsSamad Kheybari0Farzaneh Alizadeh1Mohammad Taghi Darvishi2Kamyar Hosseini3Faculty of Art and Science, University of Kyrenia, TRNC, Mersin 10, Kyrenia 99320, TurkeyFaculty of Art and Science, University of Kyrenia, TRNC, Mersin 10, Kyrenia 99320, TurkeyDepartment of Mathematics, Razi University, Kermanshah 67149, IranMathematics Research Center, Near East University, TRNC, Mersin 10, Nicosia 99138, TurkeyThis article presents an innovative approximating technique for addressing modified anomalous time-fractional sub-diffusion equations (MAFSDEs) of the Caputo type. These equations generalize classical diffusion equations, which involve fractional derivatives with respect to time, capturing the non-local and history-dependent behavior typical in sub-diffusion processes. In such a model, the particle transports slower than in a standard diffusion, often due to obstacles or memory effects in the medium. The core of the proposed technique involves transforming the original problem into a family of independent fractional-order ordinary differential equations (FODEs). This transformation is achieved using the Fourier expansion method. Each of these resulting FODEs is defined under initial value conditions which are derived from the initial condition of the original problem. To solve them, for each resulting FODE, some secondary initial value problems are introduced. By solving these secondary initial value problems, some particular solutions are obtained and then we combine them linearly in an optimal manner. This combination is essential to estimate the solution of the original problem. To evaluate the accuracy and effectiveness of the proposed scheme, we conduct a various test problem. For each problem, we analyze the observed convergence order indicators and compare them with those from other methods. Our comparison demonstrates that the proposed technique provides enhanced precision and reliability in respect with the current numerical approaches in the literature.https://www.mdpi.com/2504-3110/8/12/718Caputo-type fractional derivativemodified anomalous time-fractional sub-diffusion problemFourier expansion methodsecondary initial value problem |
| spellingShingle | Samad Kheybari Farzaneh Alizadeh Mohammad Taghi Darvishi Kamyar Hosseini A Fourier Series Technique for Approximate Solutions of Modified Anomalous Time-Fractional Sub-Diffusion Equations Fractal and Fractional Caputo-type fractional derivative modified anomalous time-fractional sub-diffusion problem Fourier expansion method secondary initial value problem |
| title | A Fourier Series Technique for Approximate Solutions of Modified Anomalous Time-Fractional Sub-Diffusion Equations |
| title_full | A Fourier Series Technique for Approximate Solutions of Modified Anomalous Time-Fractional Sub-Diffusion Equations |
| title_fullStr | A Fourier Series Technique for Approximate Solutions of Modified Anomalous Time-Fractional Sub-Diffusion Equations |
| title_full_unstemmed | A Fourier Series Technique for Approximate Solutions of Modified Anomalous Time-Fractional Sub-Diffusion Equations |
| title_short | A Fourier Series Technique for Approximate Solutions of Modified Anomalous Time-Fractional Sub-Diffusion Equations |
| title_sort | fourier series technique for approximate solutions of modified anomalous time fractional sub diffusion equations |
| topic | Caputo-type fractional derivative modified anomalous time-fractional sub-diffusion problem Fourier expansion method secondary initial value problem |
| url | https://www.mdpi.com/2504-3110/8/12/718 |
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