A Least Squares Differential Quadrature Method for a Class of Nonlinear Partial Differential Equations of Fractional Order
In this paper a new method called the least squares differential quadrature method (LSDQM) is introduced as a straightforward and efficient method to compute analytical approximate polynomial solutions for nonlinear partial differential equations with fractional time derivatives. LSDQM is a combinat...
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MDPI AG
2020-08-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/8/8/1336 |
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| author | Constantin Bota Bogdan Căruntu Dumitru Ţucu Marioara Lăpădat Mădălina Sofia Paşca |
| author_facet | Constantin Bota Bogdan Căruntu Dumitru Ţucu Marioara Lăpădat Mădălina Sofia Paşca |
| author_sort | Constantin Bota |
| collection | DOAJ |
| description | In this paper a new method called the least squares differential quadrature method (LSDQM) is introduced as a straightforward and efficient method to compute analytical approximate polynomial solutions for nonlinear partial differential equations with fractional time derivatives. LSDQM is a combination of the differential quadrature method and the least squares method and in this paper it is employed to find approximate solutions for a very general class of nonlinear partial differential equations, wherein the fractional derivatives are described in the Caputo sense. The paper contains a clear, step-by-step presentation of the method and a convergence theorem. In order to emphasize the accuracy of LSDQM we included two test problems previously solved by means of other, well-known methods, and observed that our solutions present not only a smaller error but also a much simpler expression. We also included a problem with no known exact solution and the solutions computed by LSDQM are in good agreement with previous ones. |
| first_indexed | 2024-03-10T17:39:34Z |
| format | Article |
| id | doaj.art-3d3e3f45bcda48fbbf5489dd0e0273f5 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T17:39:34Z |
| publishDate | 2020-08-01 |
| publisher | MDPI AG |
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| series | Mathematics |
| spelling | doaj.art-3d3e3f45bcda48fbbf5489dd0e0273f52023-11-20T09:44:59ZengMDPI AGMathematics2227-73902020-08-0188133610.3390/math8081336A Least Squares Differential Quadrature Method for a Class of Nonlinear Partial Differential Equations of Fractional OrderConstantin Bota0Bogdan Căruntu1Dumitru Ţucu2Marioara Lăpădat3Mădălina Sofia Paşca4Department of Mathematics, Politehnica University of Timişoara, 300006 Timişoara, RomaniaDepartment of Mathematics, Politehnica University of Timişoara, 300006 Timişoara, RomaniaDepartment of Mechanical Machinery, Equipment and Transport, Politehnica University of Timişoara, 300222 Timişoara, RomaniaDepartment of Mathematics, Politehnica University of Timişoara, 300006 Timişoara, RomaniaDepartment of Mathematics, Politehnica University of Timişoara, 300006 Timişoara, RomaniaIn this paper a new method called the least squares differential quadrature method (LSDQM) is introduced as a straightforward and efficient method to compute analytical approximate polynomial solutions for nonlinear partial differential equations with fractional time derivatives. LSDQM is a combination of the differential quadrature method and the least squares method and in this paper it is employed to find approximate solutions for a very general class of nonlinear partial differential equations, wherein the fractional derivatives are described in the Caputo sense. The paper contains a clear, step-by-step presentation of the method and a convergence theorem. In order to emphasize the accuracy of LSDQM we included two test problems previously solved by means of other, well-known methods, and observed that our solutions present not only a smaller error but also a much simpler expression. We also included a problem with no known exact solution and the solutions computed by LSDQM are in good agreement with previous ones.https://www.mdpi.com/2227-7390/8/8/1336fractional differential equationsnonlinear partial differential equationsanalytical approximate solutiondifferential quadrature method |
| spellingShingle | Constantin Bota Bogdan Căruntu Dumitru Ţucu Marioara Lăpădat Mădălina Sofia Paşca A Least Squares Differential Quadrature Method for a Class of Nonlinear Partial Differential Equations of Fractional Order Mathematics fractional differential equations nonlinear partial differential equations analytical approximate solution differential quadrature method |
| title | A Least Squares Differential Quadrature Method for a Class of Nonlinear Partial Differential Equations of Fractional Order |
| title_full | A Least Squares Differential Quadrature Method for a Class of Nonlinear Partial Differential Equations of Fractional Order |
| title_fullStr | A Least Squares Differential Quadrature Method for a Class of Nonlinear Partial Differential Equations of Fractional Order |
| title_full_unstemmed | A Least Squares Differential Quadrature Method for a Class of Nonlinear Partial Differential Equations of Fractional Order |
| title_short | A Least Squares Differential Quadrature Method for a Class of Nonlinear Partial Differential Equations of Fractional Order |
| title_sort | least squares differential quadrature method for a class of nonlinear partial differential equations of fractional order |
| topic | fractional differential equations nonlinear partial differential equations analytical approximate solution differential quadrature method |
| url | https://www.mdpi.com/2227-7390/8/8/1336 |
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