Numerical Solutions of Fractional Variable Order Differential Equations via Using Shifted Legendre Polynomials
In this manuscript, an algorithm for the computation of numerical solutions to some variable order fractional differential equations (FDEs) subject to the boundary and initial conditions is developed. We use shifted Legendre polynomials for the required numerical algorithm to develop some operationa...
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| Published in | Computer modeling in engineering & sciences Vol. 134; no. 2; pp. 941 - 955 |
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| Main Author | |
| Format | Journal Article |
| Language | English |
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Tech Science Press
2023
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| Abstract | In this manuscript, an algorithm for the computation of numerical solutions to some variable order fractional differential equations (FDEs) subject to the boundary and initial conditions is developed. We use shifted Legendre polynomials for the required numerical algorithm to develop some operational matrices. Further, operational matrices are constructed using variable order differentiation and integration. We are finding the operational matrices of variable order differentiation and integration by omitting the discretization of data. With the help of aforesaid matrices, considered FDEs are converted to algebraic equations of Sylvester type. Finally, the algebraic equations we get are solved with the help of mathematical software like Matlab or Mathematica to compute numerical solutions. Some examples are given to check the proposed method’s accuracy and graphical representations. Exact and numerical solutions are also compared in the paper for some examples. The efficiency of the method can be enhanced further by increasing the scale level. |
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| AbstractList | In this manuscript, an algorithm for the computation of numerical solutions to some variable order fractional differential equations (FDEs) subject to the boundary and initial conditions is developed. We use shifted Legendre polynomials for the required numerical algorithm to develop some operational matrices. Further, operational matrices are constructed using variable order differentiation and integration. We are finding the operational matrices of variable order differentiation and integration by omitting the discretization of data. With the help of aforesaid matrices, considered FDEs are converted to algebraic equations of Sylvester type. Finally, the algebraic equations we get are solved with the help of mathematical software like Matlab or Mathematica to compute numerical solutions. Some examples are given to check the proposed method’s accuracy and graphical representations. Exact and numerical solutions are also compared in the paper for some examples. The efficiency of the method can be enhanced further by increasing the scale level. |
| Author | Shah, Kamal |
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| Cites_doi | 10.1016/j.camwa.2009.07.006 10.1098/rspa.2019.0498 10.1002/asjc.1687 10.1007/s10915-016-0343-1 10.1017/CBO9780511618352 10.1007/978-3-540-30728-0 10.11121/ijocta.01.2017.00368 10.1016/j.chaos.2019.05.039 10.4249/scholarpedia.3163 10.1112/S146115701700002X 10.1007/978-3-642-84108-8_10 10.1186/1687-1847-2012-8 10.1515/IJNSNS.2001.2.4.365 10.1007/s13398-018-0616-7 10.1016/j.camwa.2010.07.056 10.1007/s00366-020-01227-0 10.1002/num.20504 10.1016/j.cnsns.2007.09.014 10.3390/mca20010093 10.1016/j.mcm.2011.01.037 10.1155/S1110757X04311010 10.1002/mma.5676 10.1515/fca-2019-0003 10.1137/0724020 10.1016/j.aml.2015.02.010 10.1137/16M1097109 10.1016/j.camwa.2014.03.008 10.1155/2013/816803 |
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| SubjectTerms | Algebra Algorithms Approximation Calculus Differential equations Differentiation Fractional calculus Graphical representations Initial conditions Mathematical analysis Methods Numerical analysis Polynomials Researchers |
| Title | Numerical Solutions of Fractional Variable Order Differential Equations via Using Shifted Legendre Polynomials |
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