Nonlinear self-adjointness, conserved vectors, and traveling wave structures for the kinetics of phase separation dependent on ternary alloys in iron (Fe-Cr-Y (Y = Mo, Cu))
The present exploration is concerned with fundamental elements corresponding to the phase decomposition in (Fe-Cr-Mo) and (Fe-Cr-Cu) ternary composites. For the ternary composites of iron, we examine the dynamical behavior of the phase separation. The dynamic of this separation is depicted by a mode...
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| Format: | Article |
| Language: | English |
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Elsevier
2021-06-01
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| Series: | Results in Physics |
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| Online Access: | http://www.sciencedirect.com/science/article/pii/S221137972100303X |
| _version_ | 1819142011507179520 |
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| author | Muhammad Bilal Riaz Dumitru Baleanu Adil Jhangeer Naseem Abbas |
| author_facet | Muhammad Bilal Riaz Dumitru Baleanu Adil Jhangeer Naseem Abbas |
| author_sort | Muhammad Bilal Riaz |
| collection | DOAJ |
| description | The present exploration is concerned with fundamental elements corresponding to the phase decomposition in (Fe-Cr-Mo) and (Fe-Cr-Cu) ternary composites. For the ternary composites of iron, we examine the dynamical behavior of the phase separation. The dynamic of this separation is depicted by a model known as the Cahn-Hilliard equation. The nonlinear self-adjointness for the model under consideration is taken into account. The conserved quantities are calculated with the help of the direct method. For each symmetry generator, we have reduced the considered equation into non-linear ordinary differential equations (ODEs). Also, we have computed the optimal system of the equation under study to find the similarity reduction. Also, the traveling wave structures of the Cahn-Hilliard equation are obtained with the modified simple equation (MSE) technique. Moreover, solitary wave structures is exhibited graphically in the form of 3D, 2D and contour plots. |
| first_indexed | 2024-12-22T12:03:34Z |
| format | Article |
| id | doaj.art-ce49d0647bf8402bab4378f9d2c6ea89 |
| institution | Directory Open Access Journal |
| issn | 2211-3797 |
| language | English |
| last_indexed | 2024-12-22T12:03:34Z |
| publishDate | 2021-06-01 |
| publisher | Elsevier |
| record_format | Article |
| series | Results in Physics |
| spelling | doaj.art-ce49d0647bf8402bab4378f9d2c6ea892022-12-21T18:26:30ZengElsevierResults in Physics2211-37972021-06-0125104151Nonlinear self-adjointness, conserved vectors, and traveling wave structures for the kinetics of phase separation dependent on ternary alloys in iron (Fe-Cr-Y (Y = Mo, Cu))Muhammad Bilal Riaz0Dumitru Baleanu1Adil Jhangeer2Naseem Abbas3Department of Mathematics, University of Management and Technology, PakistanDepartment of Mathematics, Cankaya University, 06530 Ankara, Turkey; Institute of Space Sciences, Magurele, 077125 Bucharest, Romania; Department of Medical Research, China Medical University Hospital, China Medical University, Taichung, Taiwan; Corresponding author.Department of Mathematics, Namal Institute, Talagang Road, Mianwali 42250, PakistanDepartment of Mathematics, Quaid-I-Azam University 45320, Islamabad 44000, PakistanThe present exploration is concerned with fundamental elements corresponding to the phase decomposition in (Fe-Cr-Mo) and (Fe-Cr-Cu) ternary composites. For the ternary composites of iron, we examine the dynamical behavior of the phase separation. The dynamic of this separation is depicted by a model known as the Cahn-Hilliard equation. The nonlinear self-adjointness for the model under consideration is taken into account. The conserved quantities are calculated with the help of the direct method. For each symmetry generator, we have reduced the considered equation into non-linear ordinary differential equations (ODEs). Also, we have computed the optimal system of the equation under study to find the similarity reduction. Also, the traveling wave structures of the Cahn-Hilliard equation are obtained with the modified simple equation (MSE) technique. Moreover, solitary wave structures is exhibited graphically in the form of 3D, 2D and contour plots.http://www.sciencedirect.com/science/article/pii/S221137972100303XConvective-diffusive Cahn-Hilliard equationNonlinear self-adjointnessDirect methodSimilarity reductionSolitary wave solutions |
| spellingShingle | Muhammad Bilal Riaz Dumitru Baleanu Adil Jhangeer Naseem Abbas Nonlinear self-adjointness, conserved vectors, and traveling wave structures for the kinetics of phase separation dependent on ternary alloys in iron (Fe-Cr-Y (Y = Mo, Cu)) Results in Physics Convective-diffusive Cahn-Hilliard equation Nonlinear self-adjointness Direct method Similarity reduction Solitary wave solutions |
| title | Nonlinear self-adjointness, conserved vectors, and traveling wave structures for the kinetics of phase separation dependent on ternary alloys in iron (Fe-Cr-Y (Y = Mo, Cu)) |
| title_full | Nonlinear self-adjointness, conserved vectors, and traveling wave structures for the kinetics of phase separation dependent on ternary alloys in iron (Fe-Cr-Y (Y = Mo, Cu)) |
| title_fullStr | Nonlinear self-adjointness, conserved vectors, and traveling wave structures for the kinetics of phase separation dependent on ternary alloys in iron (Fe-Cr-Y (Y = Mo, Cu)) |
| title_full_unstemmed | Nonlinear self-adjointness, conserved vectors, and traveling wave structures for the kinetics of phase separation dependent on ternary alloys in iron (Fe-Cr-Y (Y = Mo, Cu)) |
| title_short | Nonlinear self-adjointness, conserved vectors, and traveling wave structures for the kinetics of phase separation dependent on ternary alloys in iron (Fe-Cr-Y (Y = Mo, Cu)) |
| title_sort | nonlinear self adjointness conserved vectors and traveling wave structures for the kinetics of phase separation dependent on ternary alloys in iron fe cr y y mo cu |
| topic | Convective-diffusive Cahn-Hilliard equation Nonlinear self-adjointness Direct method Similarity reduction Solitary wave solutions |
| url | http://www.sciencedirect.com/science/article/pii/S221137972100303X |
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