On the Modified Numerical Methods for Partial Differential Equations Involving Fractional Derivatives
This paper provides both analytical and numerical solutions of (PDEs) involving time-fractional derivatives. We implemented three powerful techniques, including the modified variational iteration technique, the modified Adomian decomposition technique, and the modified homotopy analysis technique, t...
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| Format: | Article |
| Language: | English |
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MDPI AG
2023-09-01
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| Series: | Axioms |
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| Online Access: | https://www.mdpi.com/2075-1680/12/9/901 |
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| author | Fahad Alsidrani Adem Kılıçman Norazak Senu |
| author_facet | Fahad Alsidrani Adem Kılıçman Norazak Senu |
| author_sort | Fahad Alsidrani |
| collection | DOAJ |
| description | This paper provides both analytical and numerical solutions of (PDEs) involving time-fractional derivatives. We implemented three powerful techniques, including the modified variational iteration technique, the modified Adomian decomposition technique, and the modified homotopy analysis technique, to obtain an approximate solution for the bounded space variable <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ν</mi></semantics></math></inline-formula>. The Laplace transformation is used in the time-fractional derivative operator to enhance the proposed numerical methods’ performance and accuracy and find an approximate solution to time-fractional Fornberg–Whitham equations. To confirm the accuracy of the proposed methods, we evaluate homogeneous time-fractional Fornberg–Whitham equations in terms of non-integer order and variable coefficients. The obtained results of the modified methods are shown through tables and graphs. |
| first_indexed | 2024-03-10T23:02:05Z |
| format | Article |
| id | doaj.art-12f81af73e3b463a974847bfc6893c8c |
| institution | Directory Open Access Journal |
| issn | 2075-1680 |
| language | English |
| last_indexed | 2024-03-10T23:02:05Z |
| publishDate | 2023-09-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Axioms |
| spelling | doaj.art-12f81af73e3b463a974847bfc6893c8c2023-11-19T09:33:17ZengMDPI AGAxioms2075-16802023-09-0112990110.3390/axioms12090901On the Modified Numerical Methods for Partial Differential Equations Involving Fractional DerivativesFahad Alsidrani0Adem Kılıçman1Norazak Senu2Department of Mathematics, College of Science and Arts, Qassim University, Al Methnab 51931, Qassim, Saudi ArabiaInstitute for Mathematical Research, Universiti Putra Malaysia, Serdang 43400, Selangor, MalaysiaInstitute for Mathematical Research, Universiti Putra Malaysia, Serdang 43400, Selangor, MalaysiaThis paper provides both analytical and numerical solutions of (PDEs) involving time-fractional derivatives. We implemented three powerful techniques, including the modified variational iteration technique, the modified Adomian decomposition technique, and the modified homotopy analysis technique, to obtain an approximate solution for the bounded space variable <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ν</mi></semantics></math></inline-formula>. The Laplace transformation is used in the time-fractional derivative operator to enhance the proposed numerical methods’ performance and accuracy and find an approximate solution to time-fractional Fornberg–Whitham equations. To confirm the accuracy of the proposed methods, we evaluate homogeneous time-fractional Fornberg–Whitham equations in terms of non-integer order and variable coefficients. The obtained results of the modified methods are shown through tables and graphs.https://www.mdpi.com/2075-1680/12/9/901fractional derivativespartial differential equationnumerical methodsnonlinear equationintegral transformsLaplace transformation |
| spellingShingle | Fahad Alsidrani Adem Kılıçman Norazak Senu On the Modified Numerical Methods for Partial Differential Equations Involving Fractional Derivatives Axioms fractional derivatives partial differential equation numerical methods nonlinear equation integral transforms Laplace transformation |
| title | On the Modified Numerical Methods for Partial Differential Equations Involving Fractional Derivatives |
| title_full | On the Modified Numerical Methods for Partial Differential Equations Involving Fractional Derivatives |
| title_fullStr | On the Modified Numerical Methods for Partial Differential Equations Involving Fractional Derivatives |
| title_full_unstemmed | On the Modified Numerical Methods for Partial Differential Equations Involving Fractional Derivatives |
| title_short | On the Modified Numerical Methods for Partial Differential Equations Involving Fractional Derivatives |
| title_sort | on the modified numerical methods for partial differential equations involving fractional derivatives |
| topic | fractional derivatives partial differential equation numerical methods nonlinear equation integral transforms Laplace transformation |
| url | https://www.mdpi.com/2075-1680/12/9/901 |
| work_keys_str_mv | AT fahadalsidrani onthemodifiednumericalmethodsforpartialdifferentialequationsinvolvingfractionalderivatives AT ademkılıcman onthemodifiednumericalmethodsforpartialdifferentialequationsinvolvingfractionalderivatives AT norazaksenu onthemodifiednumericalmethodsforpartialdifferentialequationsinvolvingfractionalderivatives |