Approximate analytical solution of a class of highly nonlinear time–fractional-order partial differential equations
This article presents a power series technique for obtaining approximate solutions of the time–fractional-order version of a generalised Newell–Whitehead–Segel initial value problem, with the fractional-order derivative described in the Caputo sense. The method assumes a fractional power series in t...
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Elsevier
2025-03-01
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| Series: | Partial Differential Equations in Applied Mathematics |
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| Online Access: | http://www.sciencedirect.com/science/article/pii/S266681812500018X |
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| author | Richard Olu Awonusika |
| author_facet | Richard Olu Awonusika |
| author_sort | Richard Olu Awonusika |
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| description | This article presents a power series technique for obtaining approximate solutions of the time–fractional-order version of a generalised Newell–Whitehead–Segel initial value problem, with the fractional-order derivative described in the Caputo sense. The method assumes a fractional power series in the time variable whose expansion coefficients are functions of the space variable. The proposed approach is based on the generalised Cauchy product of power series and does not require any kind of polynomial or digitisation in the simplification of the nonlinear terms. The application of the generalised Cauchy product enables us to construct explicit recursion formulae for the expansion coefficients of the series solution. The first expansion coefficients are nicely expressed in terms of appropriate integer sequences. Notable special cases of the proposed generalised problem, that include, Newell-Whitehead, Newell–Whitehead–Segel, and Cahn–Allen equations with suitable initial conditions are considered for the purpose of accuracy and reliability of the proposed method. Our numerical results are compared with the exact solutions and other existing results. Comparison of the absolute errors from our method and other published results indicates that the proposed technique is accurate and reliable. Two-dimensional and three-dimensional graphs of results are presented for different fractional-order values 0<μ≤1. It is observed that as the fractional-order μ gets closer to 1, the graphs of the approximate solutions gradually coincide with those of the exact solutions. The convergence rate of the proposed series solutions ranges between 10−15 and 10−18. |
| format | Article |
| id | doaj-art-d8017e767c93424998ec2d6a4ea4df16 |
| institution | Kabale University |
| issn | 2666-8181 |
| language | English |
| publishDate | 2025-03-01 |
| publisher | Elsevier |
| record_format | Article |
| series | Partial Differential Equations in Applied Mathematics |
| spelling | doaj-art-d8017e767c93424998ec2d6a4ea4df162025-01-22T05:44:12ZengElsevierPartial Differential Equations in Applied Mathematics2666-81812025-03-0113101090Approximate analytical solution of a class of highly nonlinear time–fractional-order partial differential equationsRichard Olu Awonusika0Corresponding author.; Department of Mathematical Sciences, Adekunle Ajasin University, P.M.B. 001, Akungba-Akoko, Ondo State, NigeriaThis article presents a power series technique for obtaining approximate solutions of the time–fractional-order version of a generalised Newell–Whitehead–Segel initial value problem, with the fractional-order derivative described in the Caputo sense. The method assumes a fractional power series in the time variable whose expansion coefficients are functions of the space variable. The proposed approach is based on the generalised Cauchy product of power series and does not require any kind of polynomial or digitisation in the simplification of the nonlinear terms. The application of the generalised Cauchy product enables us to construct explicit recursion formulae for the expansion coefficients of the series solution. The first expansion coefficients are nicely expressed in terms of appropriate integer sequences. Notable special cases of the proposed generalised problem, that include, Newell-Whitehead, Newell–Whitehead–Segel, and Cahn–Allen equations with suitable initial conditions are considered for the purpose of accuracy and reliability of the proposed method. Our numerical results are compared with the exact solutions and other existing results. Comparison of the absolute errors from our method and other published results indicates that the proposed technique is accurate and reliable. Two-dimensional and three-dimensional graphs of results are presented for different fractional-order values 0<μ≤1. It is observed that as the fractional-order μ gets closer to 1, the graphs of the approximate solutions gradually coincide with those of the exact solutions. The convergence rate of the proposed series solutions ranges between 10−15 and 10−18.http://www.sciencedirect.com/science/article/pii/S266681812500018XPower seriesGeneralised Cauchy productNewell–Whitehead–Segel equationCahn–Allen equationFractional calculusFractional power series |
| spellingShingle | Richard Olu Awonusika Approximate analytical solution of a class of highly nonlinear time–fractional-order partial differential equations Partial Differential Equations in Applied Mathematics Power series Generalised Cauchy product Newell–Whitehead–Segel equation Cahn–Allen equation Fractional calculus Fractional power series |
| title | Approximate analytical solution of a class of highly nonlinear time–fractional-order partial differential equations |
| title_full | Approximate analytical solution of a class of highly nonlinear time–fractional-order partial differential equations |
| title_fullStr | Approximate analytical solution of a class of highly nonlinear time–fractional-order partial differential equations |
| title_full_unstemmed | Approximate analytical solution of a class of highly nonlinear time–fractional-order partial differential equations |
| title_short | Approximate analytical solution of a class of highly nonlinear time–fractional-order partial differential equations |
| title_sort | approximate analytical solution of a class of highly nonlinear time fractional order partial differential equations |
| topic | Power series Generalised Cauchy product Newell–Whitehead–Segel equation Cahn–Allen equation Fractional calculus Fractional power series |
| url | http://www.sciencedirect.com/science/article/pii/S266681812500018X |
| work_keys_str_mv | AT richardoluawonusika approximateanalyticalsolutionofaclassofhighlynonlineartimefractionalorderpartialdifferentialequations |