Spectral Element Method for Fractional Klein–Gordon Equations Using Interpolating Scaling Functions
The focus of this paper is on utilizing the spectral element method to find the numerical solution of the fractional Klein–Gordon equation. The algorithm employs interpolating scaling functions (ISFs) that meet specific properties and satisfy the multiresolution analysis. Using an orthonormal projec...
| Main Author: | |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Hindawi Limited
2023-01-01
|
| Series: | Advances in Mathematical Physics |
| Online Access: | http://dx.doi.org/10.1155/2023/8453459 |
| _version_ | 1797385479741505536 |
|---|---|
| author | Haifa Bin Jebreen |
| author_facet | Haifa Bin Jebreen |
| author_sort | Haifa Bin Jebreen |
| collection | DOAJ |
| description | The focus of this paper is on utilizing the spectral element method to find the numerical solution of the fractional Klein–Gordon equation. The algorithm employs interpolating scaling functions (ISFs) that meet specific properties and satisfy the multiresolution analysis. Using an orthonormal projection, the equation is mapped to the scaling spaces in this method. A matrix representation of the Caputo fractional derivative of ISFs is presented using matrices representing the fractional integral and derivative operators. Using this matrix, the spectral element method reduces the desired equation to a system of algebraic equations. To find the solution, the generalized minimal residual method (GMRES method) and Newton’s method are used in linear and nonlinear forms of this system, respectively. The method’s convergence is proven, and some illustrative examples confirm it. The method is characterized by its simplicity in implementation, high efficiency, and significant accuracy. |
| first_indexed | 2024-03-08T21:54:49Z |
| format | Article |
| id | doaj.art-2d85deeadf8d432684520e1a5fc07b7c |
| institution | Directory Open Access Journal |
| issn | 1687-9139 |
| language | English |
| last_indexed | 2024-03-08T21:54:49Z |
| publishDate | 2023-01-01 |
| publisher | Hindawi Limited |
| record_format | Article |
| series | Advances in Mathematical Physics |
| spelling | doaj.art-2d85deeadf8d432684520e1a5fc07b7c2023-12-20T05:00:02ZengHindawi LimitedAdvances in Mathematical Physics1687-91392023-01-01202310.1155/2023/8453459Spectral Element Method for Fractional Klein–Gordon Equations Using Interpolating Scaling FunctionsHaifa Bin Jebreen0Department of MathematicsThe focus of this paper is on utilizing the spectral element method to find the numerical solution of the fractional Klein–Gordon equation. The algorithm employs interpolating scaling functions (ISFs) that meet specific properties and satisfy the multiresolution analysis. Using an orthonormal projection, the equation is mapped to the scaling spaces in this method. A matrix representation of the Caputo fractional derivative of ISFs is presented using matrices representing the fractional integral and derivative operators. Using this matrix, the spectral element method reduces the desired equation to a system of algebraic equations. To find the solution, the generalized minimal residual method (GMRES method) and Newton’s method are used in linear and nonlinear forms of this system, respectively. The method’s convergence is proven, and some illustrative examples confirm it. The method is characterized by its simplicity in implementation, high efficiency, and significant accuracy.http://dx.doi.org/10.1155/2023/8453459 |
| spellingShingle | Haifa Bin Jebreen Spectral Element Method for Fractional Klein–Gordon Equations Using Interpolating Scaling Functions Advances in Mathematical Physics |
| title | Spectral Element Method for Fractional Klein–Gordon Equations Using Interpolating Scaling Functions |
| title_full | Spectral Element Method for Fractional Klein–Gordon Equations Using Interpolating Scaling Functions |
| title_fullStr | Spectral Element Method for Fractional Klein–Gordon Equations Using Interpolating Scaling Functions |
| title_full_unstemmed | Spectral Element Method for Fractional Klein–Gordon Equations Using Interpolating Scaling Functions |
| title_short | Spectral Element Method for Fractional Klein–Gordon Equations Using Interpolating Scaling Functions |
| title_sort | spectral element method for fractional klein gordon equations using interpolating scaling functions |
| url | http://dx.doi.org/10.1155/2023/8453459 |
| work_keys_str_mv | AT haifabinjebreen spectralelementmethodforfractionalkleingordonequationsusinginterpolatingscalingfunctions |