Numerical approximation of time-fractional nonlinear partial integro-differential equation using fractional Euler and cubic trigonometric B-Spline methods
Nonlinear mathematical problems arise due to existence of important complex nonlinear phenomena in engineering and science. In this article, a class of time-fractional nonlinear parabolic partial integro-differential equations is solved numerically by combination of fractional Euler and cubic trigon...
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| Format: | Article |
| Language: | English |
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Elsevier
2025-09-01
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| Series: | Partial Differential Equations in Applied Mathematics |
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| Online Access: | http://www.sciencedirect.com/science/article/pii/S2666818125001500 |
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| author | Mehwish Saleem Arshed Ali Fazal-i-Haq Hassan Khan |
| author_facet | Mehwish Saleem Arshed Ali Fazal-i-Haq Hassan Khan |
| author_sort | Mehwish Saleem |
| collection | DOAJ |
| description | Nonlinear mathematical problems arise due to existence of important complex nonlinear phenomena in engineering and science. In this article, a class of time-fractional nonlinear parabolic partial integro-differential equations is solved numerically by combination of fractional Euler and cubic trigonometric B-spline collocation methods. Backward finite difference formula is employed for time-fractional Caputo derivative to get an unconditional stable scheme. The memory(integral) term is evaluated using a second order quadrature rule. Fractional Euler method for Caputo derivative is used in computing the nonlinear memory term. At each time level, cubic trigonometric B-spline functions are applied to obtain the solution in spatial dimension which reduces the problem to a system of algebraic equations. This method has the ability to handle any kind of nonlinearity without using iterative processes. Efficiency and reliability of the current method is analyzed for the fractional-order via three highly nonlinear test problems with variable coefficients. The rate of convergence of the proposed method is also computed in temporal and spatial dimensions. |
| format | Article |
| id | doaj-art-653704bf5e66485792dad41ca556ded8 |
| institution | Kabale University |
| issn | 2666-8181 |
| language | English |
| publishDate | 2025-09-01 |
| publisher | Elsevier |
| record_format | Article |
| series | Partial Differential Equations in Applied Mathematics |
| spelling | doaj-art-653704bf5e66485792dad41ca556ded82025-08-20T03:51:08ZengElsevierPartial Differential Equations in Applied Mathematics2666-81812025-09-011510122310.1016/j.padiff.2025.101223Numerical approximation of time-fractional nonlinear partial integro-differential equation using fractional Euler and cubic trigonometric B-Spline methodsMehwish Saleem0Arshed Ali1 Fazal-i-Haq2Hassan Khan3Department of Mathematics, Islamia College Peshawar, Khyber Pakhtunkhwa, PakistanDepartment of Mathematics, Islamia College Peshawar, Khyber Pakhtunkhwa, PakistanDepartment of Mathematics, Statistics and Computer Science, The University of Agricultural, Peshawar, Khyber Pakhtunkhwa, PakistanDepartment of Mathematics, Abdul Wali Khan University, Mardan, Khyber Pakhtunkhwa, Pakistan; Corresponding author.Nonlinear mathematical problems arise due to existence of important complex nonlinear phenomena in engineering and science. In this article, a class of time-fractional nonlinear parabolic partial integro-differential equations is solved numerically by combination of fractional Euler and cubic trigonometric B-spline collocation methods. Backward finite difference formula is employed for time-fractional Caputo derivative to get an unconditional stable scheme. The memory(integral) term is evaluated using a second order quadrature rule. Fractional Euler method for Caputo derivative is used in computing the nonlinear memory term. At each time level, cubic trigonometric B-spline functions are applied to obtain the solution in spatial dimension which reduces the problem to a system of algebraic equations. This method has the ability to handle any kind of nonlinearity without using iterative processes. Efficiency and reliability of the current method is analyzed for the fractional-order via three highly nonlinear test problems with variable coefficients. The rate of convergence of the proposed method is also computed in temporal and spatial dimensions.http://www.sciencedirect.com/science/article/pii/S2666818125001500Fractional Euler methodCaputo fractional derivativeCubic trigonometric B-spline functionsFractional partial integro-differential equation |
| spellingShingle | Mehwish Saleem Arshed Ali Fazal-i-Haq Hassan Khan Numerical approximation of time-fractional nonlinear partial integro-differential equation using fractional Euler and cubic trigonometric B-Spline methods Partial Differential Equations in Applied Mathematics Fractional Euler method Caputo fractional derivative Cubic trigonometric B-spline functions Fractional partial integro-differential equation |
| title | Numerical approximation of time-fractional nonlinear partial integro-differential equation using fractional Euler and cubic trigonometric B-Spline methods |
| title_full | Numerical approximation of time-fractional nonlinear partial integro-differential equation using fractional Euler and cubic trigonometric B-Spline methods |
| title_fullStr | Numerical approximation of time-fractional nonlinear partial integro-differential equation using fractional Euler and cubic trigonometric B-Spline methods |
| title_full_unstemmed | Numerical approximation of time-fractional nonlinear partial integro-differential equation using fractional Euler and cubic trigonometric B-Spline methods |
| title_short | Numerical approximation of time-fractional nonlinear partial integro-differential equation using fractional Euler and cubic trigonometric B-Spline methods |
| title_sort | numerical approximation of time fractional nonlinear partial integro differential equation using fractional euler and cubic trigonometric b spline methods |
| topic | Fractional Euler method Caputo fractional derivative Cubic trigonometric B-spline functions Fractional partial integro-differential equation |
| url | http://www.sciencedirect.com/science/article/pii/S2666818125001500 |
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