A Hybrid Local Radial Basis Function Method for the Numerical Modeling of Mixed Diffusion and Wave-Diffusion Equations of Fractional Order Using Caputo’s Derivatives
This article presents an efficient method for the numerical modeling of time fractional mixed diffusion and wave-diffusion equations with two Caputo derivatives of order <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics>...
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| Format: | Article |
| Language: | English |
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MDPI AG
2023-05-01
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| Series: | Fractal and Fractional |
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| Online Access: | https://www.mdpi.com/2504-3110/7/5/381 |
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| author | Raheel Kamal Kamran Saleh M. Alzahrani Talal Alzahrani |
| author_facet | Raheel Kamal Kamran Saleh M. Alzahrani Talal Alzahrani |
| author_sort | Raheel Kamal |
| collection | DOAJ |
| description | This article presents an efficient method for the numerical modeling of time fractional mixed diffusion and wave-diffusion equations with two Caputo derivatives of order <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>0</mn><mo><</mo><mi>α</mi><mo><</mo><mn>1</mn><mo>,</mo></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1</mn><mo><</mo><mi>β</mi><mo><</mo><mn>2</mn></mrow></semantics></math></inline-formula>. The numerical method is based on the Laplace transform technique combined with local radial basis functions. The method consists of three main steps: (i) first, the Laplace transform is used to transform the given time fractional model into an equivalent time-independent inhomogeneous problem in the frequency domain; (ii) in the second step, the local radial basis functions method is utilized to obtain an approximate solution for the reduced problem; (iii) finally, the Stehfest method is employed to convert the obtained solution from the frequency domain back to the time domain. The use of the Laplace transform eliminates the need for classical time-stepping techniques, which often require very small time steps to achieve accuracy. Additionally, the application of local radial basis functions helps overcome issues related to ill-conditioning and sensitivity to shape parameters typically encountered in global radial basis function methods. To validate the efficiency and accuracy of the proposed method, several test problems in regular and irregular domains with uniform and non-uniform nodes are considered. |
| first_indexed | 2024-03-11T03:43:27Z |
| format | Article |
| id | doaj.art-376768249517401ab12f46f6b848af9e |
| institution | Directory Open Access Journal |
| issn | 2504-3110 |
| language | English |
| last_indexed | 2024-03-11T03:43:27Z |
| publishDate | 2023-05-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Fractal and Fractional |
| spelling | doaj.art-376768249517401ab12f46f6b848af9e2023-11-18T01:26:19ZengMDPI AGFractal and Fractional2504-31102023-05-017538110.3390/fractalfract7050381A Hybrid Local Radial Basis Function Method for the Numerical Modeling of Mixed Diffusion and Wave-Diffusion Equations of Fractional Order Using Caputo’s DerivativesRaheel Kamal0Kamran1Saleh M. Alzahrani2Talal Alzahrani3Department of Mathematics, Islamia College Peshawar, Peshawar 25120, Khyber Pakhtunkhwa, PakistanDepartment of Mathematics, Islamia College Peshawar, Peshawar 25120, Khyber Pakhtunkhwa, PakistanMathematics Department, Al-Qunfudhah University College, Umm Al-Qura University, Al-Qunfudhah 28821, Saudi ArabiaMathematics Department, Al-Qunfudhah University College, Umm Al-Qura University, Al-Qunfudhah 28821, Saudi ArabiaThis article presents an efficient method for the numerical modeling of time fractional mixed diffusion and wave-diffusion equations with two Caputo derivatives of order <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>0</mn><mo><</mo><mi>α</mi><mo><</mo><mn>1</mn><mo>,</mo></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1</mn><mo><</mo><mi>β</mi><mo><</mo><mn>2</mn></mrow></semantics></math></inline-formula>. The numerical method is based on the Laplace transform technique combined with local radial basis functions. The method consists of three main steps: (i) first, the Laplace transform is used to transform the given time fractional model into an equivalent time-independent inhomogeneous problem in the frequency domain; (ii) in the second step, the local radial basis functions method is utilized to obtain an approximate solution for the reduced problem; (iii) finally, the Stehfest method is employed to convert the obtained solution from the frequency domain back to the time domain. The use of the Laplace transform eliminates the need for classical time-stepping techniques, which often require very small time steps to achieve accuracy. Additionally, the application of local radial basis functions helps overcome issues related to ill-conditioning and sensitivity to shape parameters typically encountered in global radial basis function methods. To validate the efficiency and accuracy of the proposed method, several test problems in regular and irregular domains with uniform and non-uniform nodes are considered.https://www.mdpi.com/2504-3110/7/5/381mixed diffusion and wave diffusion equationCaputo derivativeLaplace transformlocal radial basis function methodconvergencestability |
| spellingShingle | Raheel Kamal Kamran Saleh M. Alzahrani Talal Alzahrani A Hybrid Local Radial Basis Function Method for the Numerical Modeling of Mixed Diffusion and Wave-Diffusion Equations of Fractional Order Using Caputo’s Derivatives Fractal and Fractional mixed diffusion and wave diffusion equation Caputo derivative Laplace transform local radial basis function method convergence stability |
| title | A Hybrid Local Radial Basis Function Method for the Numerical Modeling of Mixed Diffusion and Wave-Diffusion Equations of Fractional Order Using Caputo’s Derivatives |
| title_full | A Hybrid Local Radial Basis Function Method for the Numerical Modeling of Mixed Diffusion and Wave-Diffusion Equations of Fractional Order Using Caputo’s Derivatives |
| title_fullStr | A Hybrid Local Radial Basis Function Method for the Numerical Modeling of Mixed Diffusion and Wave-Diffusion Equations of Fractional Order Using Caputo’s Derivatives |
| title_full_unstemmed | A Hybrid Local Radial Basis Function Method for the Numerical Modeling of Mixed Diffusion and Wave-Diffusion Equations of Fractional Order Using Caputo’s Derivatives |
| title_short | A Hybrid Local Radial Basis Function Method for the Numerical Modeling of Mixed Diffusion and Wave-Diffusion Equations of Fractional Order Using Caputo’s Derivatives |
| title_sort | hybrid local radial basis function method for the numerical modeling of mixed diffusion and wave diffusion equations of fractional order using caputo s derivatives |
| topic | mixed diffusion and wave diffusion equation Caputo derivative Laplace transform local radial basis function method convergence stability |
| url | https://www.mdpi.com/2504-3110/7/5/381 |
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