Modified Variational Iteration Algorithm-II: Convergence and Applications to Diffusion Models
Variational iteration method has been extensively employed to deal with linear and nonlinear differential equations of integer and fractional order. The key property of the technique is its ability and flexibility to investigate linear and nonlinear models conveniently and accurately. The current st...
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| Main Authors: | , , , , |
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| Format: | Article |
| Language: | English |
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Wiley
2020-01-01
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| Series: | Complexity |
| Online Access: | http://dx.doi.org/10.1155/2020/8841718 |
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| author | Hijaz Ahmad Tufail A. Khan Predrag S. Stanimirović Yu-Ming Chu Imtiaz Ahmad |
| author_facet | Hijaz Ahmad Tufail A. Khan Predrag S. Stanimirović Yu-Ming Chu Imtiaz Ahmad |
| author_sort | Hijaz Ahmad |
| collection | DOAJ |
| description | Variational iteration method has been extensively employed to deal with linear and nonlinear differential equations of integer and fractional order. The key property of the technique is its ability and flexibility to investigate linear and nonlinear models conveniently and accurately. The current study presents an improved algorithm to the variational iteration algorithm-II (VIA-II) for the numerical treatment of diffusion as well as convection-diffusion equations. This newly introduced modification is termed as the modified variational iteration algorithm-II (MVIA-II). The convergence of the MVIA-II is studied in the case of solving nonlinear equations. The main advantage of the MVIA-II improvement is an auxiliary parameter which makes sure a fast convergence of the standard VIA-II iteration algorithm. In order to verify the stability, accuracy, and computational speed of the method, the obtained solutions are compared numerically and graphically with the exact ones as well as with the results obtained by the previously proposed compact finite difference method and second kind Chebyshev wavelets. The comparison revealed that the modified version yields accurate results, converges rapidly, and offers better robustness in comparison with other methods used in the literature. Moreover, the basic idea depicted in this study is relied upon the possibility of the MVIA-II being utilized to handle nonlinear differential equations that arise in different fields of physical and biological sciences. A strong motivation for such applications is the fact that any discretization, transformation, or any assumptions are not required for this proposed algorithm in finding appropriate numerical solutions. |
| format | Article |
| id | doaj-art-14dad62befe944ecb6481dc9091c8aaa |
| institution | Kabale University |
| issn | 1076-2787 1099-0526 |
| language | English |
| publishDate | 2020-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Complexity |
| spelling | doaj-art-14dad62befe944ecb6481dc9091c8aaa2025-08-20T03:54:29ZengWileyComplexity1076-27871099-05262020-01-01202010.1155/2020/88417188841718Modified Variational Iteration Algorithm-II: Convergence and Applications to Diffusion ModelsHijaz Ahmad0Tufail A. Khan1Predrag S. Stanimirović2Yu-Ming Chu3Imtiaz Ahmad4Department of Basic Sciences, University of Engineering and Technology, Peshawar, PakistanDepartment of Basic Sciences, University of Engineering and Technology, Peshawar, PakistanFaculty of Science and Mathematics, University of Niš, Višegradska 33, Niš 18000, SerbiaDepartment of Mathematics, Huzhou University, Huzhou 313000, ChinaDepartment of Mathematics, University of Swabi, Swabi, Khyber Pakhtunkhwa, PakistanVariational iteration method has been extensively employed to deal with linear and nonlinear differential equations of integer and fractional order. The key property of the technique is its ability and flexibility to investigate linear and nonlinear models conveniently and accurately. The current study presents an improved algorithm to the variational iteration algorithm-II (VIA-II) for the numerical treatment of diffusion as well as convection-diffusion equations. This newly introduced modification is termed as the modified variational iteration algorithm-II (MVIA-II). The convergence of the MVIA-II is studied in the case of solving nonlinear equations. The main advantage of the MVIA-II improvement is an auxiliary parameter which makes sure a fast convergence of the standard VIA-II iteration algorithm. In order to verify the stability, accuracy, and computational speed of the method, the obtained solutions are compared numerically and graphically with the exact ones as well as with the results obtained by the previously proposed compact finite difference method and second kind Chebyshev wavelets. The comparison revealed that the modified version yields accurate results, converges rapidly, and offers better robustness in comparison with other methods used in the literature. Moreover, the basic idea depicted in this study is relied upon the possibility of the MVIA-II being utilized to handle nonlinear differential equations that arise in different fields of physical and biological sciences. A strong motivation for such applications is the fact that any discretization, transformation, or any assumptions are not required for this proposed algorithm in finding appropriate numerical solutions.http://dx.doi.org/10.1155/2020/8841718 |
| spellingShingle | Hijaz Ahmad Tufail A. Khan Predrag S. Stanimirović Yu-Ming Chu Imtiaz Ahmad Modified Variational Iteration Algorithm-II: Convergence and Applications to Diffusion Models Complexity |
| title | Modified Variational Iteration Algorithm-II: Convergence and Applications to Diffusion Models |
| title_full | Modified Variational Iteration Algorithm-II: Convergence and Applications to Diffusion Models |
| title_fullStr | Modified Variational Iteration Algorithm-II: Convergence and Applications to Diffusion Models |
| title_full_unstemmed | Modified Variational Iteration Algorithm-II: Convergence and Applications to Diffusion Models |
| title_short | Modified Variational Iteration Algorithm-II: Convergence and Applications to Diffusion Models |
| title_sort | modified variational iteration algorithm ii convergence and applications to diffusion models |
| url | http://dx.doi.org/10.1155/2020/8841718 |
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