Finite Difference–Collocation Method for the Generalized Fractional Diffusion Equation
In this paper, an approximate method combining the finite difference and collocation methods is studied to solve the generalized fractional diffusion equation (GFDE). The convergence and stability analysis of the presented method are also established in detail. To ensure the effectiveness and the ac...
| Main Authors: | , , , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2022-07-01
|
| Series: | Fractal and Fractional |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2504-3110/6/7/387 |
| _version_ | 1797406306172141568 |
|---|---|
| author | Sandeep Kumar Rajesh K. Pandey Kamlesh Kumar Shyam Kamal Thach Ngoc Dinh |
| author_facet | Sandeep Kumar Rajesh K. Pandey Kamlesh Kumar Shyam Kamal Thach Ngoc Dinh |
| author_sort | Sandeep Kumar |
| collection | DOAJ |
| description | In this paper, an approximate method combining the finite difference and collocation methods is studied to solve the generalized fractional diffusion equation (GFDE). The convergence and stability analysis of the presented method are also established in detail. To ensure the effectiveness and the accuracy of the proposed method, test examples with different scale and weight functions are considered, and the obtained numerical results are compared with the existing methods in the literature. It is observed that the proposed approach works very well with the generalized fractional derivatives (GFDs), as the presence of scale and weight functions in a generalized fractional derivative (GFD) cause difficulty for its discretization and further analysis. |
| first_indexed | 2024-03-09T03:23:32Z |
| format | Article |
| id | doaj.art-710f5fd22d6c42259c48c01a04c45c44 |
| institution | Directory Open Access Journal |
| issn | 2504-3110 |
| language | English |
| last_indexed | 2024-03-09T03:23:32Z |
| publishDate | 2022-07-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Fractal and Fractional |
| spelling | doaj.art-710f5fd22d6c42259c48c01a04c45c442023-12-03T15:04:54ZengMDPI AGFractal and Fractional2504-31102022-07-016738710.3390/fractalfract6070387Finite Difference–Collocation Method for the Generalized Fractional Diffusion EquationSandeep Kumar0Rajesh K. Pandey1Kamlesh Kumar2Shyam Kamal3Thach Ngoc Dinh4Department of Mathematical Sciences, Indian Institute of Technology (BHU) Varanasi, Varanasi 221005, Uttar Pradesh, IndiaDepartment of Mathematical Sciences, Indian Institute of Technology (BHU) Varanasi, Varanasi 221005, Uttar Pradesh, IndiaDepartment of Mathematics, Manav Rachana University, Faridabad 121004, Haryana, IndiaDepartment of Electrical Engineering, Indian Institute of Technology (BHU) Varanasi, Varanasi 221005, Uttar Pradesh, IndiaConservatoire National des Arts et Métiers (CNAM), Cedric-Laetitia, 292 Rue St-Martin, CEDEX 03, 75141 Paris, FranceIn this paper, an approximate method combining the finite difference and collocation methods is studied to solve the generalized fractional diffusion equation (GFDE). The convergence and stability analysis of the presented method are also established in detail. To ensure the effectiveness and the accuracy of the proposed method, test examples with different scale and weight functions are considered, and the obtained numerical results are compared with the existing methods in the literature. It is observed that the proposed approach works very well with the generalized fractional derivatives (GFDs), as the presence of scale and weight functions in a generalized fractional derivative (GFD) cause difficulty for its discretization and further analysis.https://www.mdpi.com/2504-3110/6/7/387generalized Caputo derivatefractional diffusion equationfinite difference methodcollocation methoderrorstability and convergence analysis |
| spellingShingle | Sandeep Kumar Rajesh K. Pandey Kamlesh Kumar Shyam Kamal Thach Ngoc Dinh Finite Difference–Collocation Method for the Generalized Fractional Diffusion Equation Fractal and Fractional generalized Caputo derivate fractional diffusion equation finite difference method collocation method error stability and convergence analysis |
| title | Finite Difference–Collocation Method for the Generalized Fractional Diffusion Equation |
| title_full | Finite Difference–Collocation Method for the Generalized Fractional Diffusion Equation |
| title_fullStr | Finite Difference–Collocation Method for the Generalized Fractional Diffusion Equation |
| title_full_unstemmed | Finite Difference–Collocation Method for the Generalized Fractional Diffusion Equation |
| title_short | Finite Difference–Collocation Method for the Generalized Fractional Diffusion Equation |
| title_sort | finite difference collocation method for the generalized fractional diffusion equation |
| topic | generalized Caputo derivate fractional diffusion equation finite difference method collocation method error stability and convergence analysis |
| url | https://www.mdpi.com/2504-3110/6/7/387 |
| work_keys_str_mv | AT sandeepkumar finitedifferencecollocationmethodforthegeneralizedfractionaldiffusionequation AT rajeshkpandey finitedifferencecollocationmethodforthegeneralizedfractionaldiffusionequation AT kamleshkumar finitedifferencecollocationmethodforthegeneralizedfractionaldiffusionequation AT shyamkamal finitedifferencecollocationmethodforthegeneralizedfractionaldiffusionequation AT thachngocdinh finitedifferencecollocationmethodforthegeneralizedfractionaldiffusionequation |