The natural extension to PDEs of Lie’s reduction of order algorithm for ODEs
In this paper, we further consider the symmetry-based method for seeking nonlocally related systems for partial differential equations. In particular, we show that the symmetry-based method for partial differential equations is the natural extension of Lie’s reduction of order algorithm for ordinary...
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| Published in | Communications in nonlinear science & numerical simulation Vol. 140; p. 108438 |
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| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
Elsevier B.V
01.01.2025
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| Online Access | Get full text |
| ISSN | 1007-5704 1878-7274 |
| DOI | 10.1016/j.cnsns.2024.108438 |
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| Abstract | In this paper, we further consider the symmetry-based method for seeking nonlocally related systems for partial differential equations. In particular, we show that the symmetry-based method for partial differential equations is the natural extension of Lie’s reduction of order algorithm for ordinary differential equations by looking at this algorithm from a different point of view. Many examples exhibit various situations that can arise.
•Lie’s reduction method extended to partial differential equations.•Use of symmetry-based method.•Illustratives examples. |
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| AbstractList | In this paper, we further consider the symmetry-based method for seeking nonlocally related systems for partial differential equations. In particular, we show that the symmetry-based method for partial differential equations is the natural extension of Lie’s reduction of order algorithm for ordinary differential equations by looking at this algorithm from a different point of view. Many examples exhibit various situations that can arise.
•Lie’s reduction method extended to partial differential equations.•Use of symmetry-based method.•Illustratives examples. |
| ArticleNumber | 108438 |
| Author | Bluman, George W. de la Rosa, Rafael |
| Author_xml | – sequence: 1 givenname: George W. surname: Bluman fullname: Bluman, George W. email: bluman@math.ubc.ca organization: Department of Mathematics, University of British Columbia, Vancouver, British Columbia V6T 1Z2, Canada – sequence: 2 givenname: Rafael surname: de la Rosa fullname: de la Rosa, Rafael email: rafael.delarosa@uca.es organization: Departamento de Matemáticas, Universidad de Cádiz, 11510 Puerto Real, Cádiz, Spain |
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| Cites_doi | 10.1063/1.5004755 10.1137/0150100 10.1063/1.527974 10.1016/j.jmaa.2020.124354 10.1063/1.5122319 10.1140/epjp/s13360-020-00530-5 10.1016/j.jmaa.2006.10.091 10.2307/1967773 10.1016/0022-247X(89)90322-3 10.1063/1.2349488 10.1063/1.4819724 10.1063/1.531866 10.1063/1.527659 10.1007/s10440-005-9001-6 |
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| Keywords | Symmetry-based method Nonlocal symmetries Nonlocally related systems Lie’s reduction of order algorithm |
| Language | English |
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| References | Bluman, Cheviakov, Anco (b9) 2010 Bianchi (b37) 1918 Bluman, Anco (b15) 2002 Cantwell (b33) 2002 Bluman, Yüzbaşi (b14) 2020; 491 Anco, The (b6) 2005; 89 Miller (b16) 1977 Olver (b32) 1995 Lie S. In: Teubner BG, editor. Theorie der Transformationsgruppen, III. Leipzig; 1893. Anco, Bluman (b5) 1997; 38 Ovsiannikov (b34) 1959; 125 Satapathy, Sekhar (b12) 2018; 59 Page (b19) 1897 Ovsiannikov (b23) 1982 Stephani (b31) 1989 Bluman (b39) 1990; 50 Sil, Sekhar (b13) 2020; 135 Bluman, de la Rosa, Bruzón, Gandarias (b36) 2020; 61 Eisenhart (b38) 1933 Bluman, Cheviakov (b8) 2007; 333 Bluman, Cole (b22) 1974 Yang (b11) 2013 Lie (b1) 1881; 6 Bluman, Kumei, Reid (b3) 1988; 29 Bluman, Kumei (b2) 1987; 28 Bluman, de la Rosa, Bruzón, Gandarias (b17) 2021; 477 Bluman, Kumei (b40) 1989; 138 Ovsiannikov (b30) 1962 Bluman (b35) 1967 Friedrichs (b21) 1965 Dickson (b26) 1924; 25 Bluman, Kumei (b4) 1989 Bluman, Yang (b10) 2013; 54 Bluman, Yang (b28) 2014; vol. 73 Cohen (b20) 1911 Hydon (b25) 2000 Bluman, de la Rosa, Bruzón, Gandarias (b18) 2021; 477 Olver (b24) 1986 Wang, Zhao, Li (b29) 2023 Bluman, Cheviakov, Ivanova (b7) 2006; 47 Bluman (10.1016/j.cnsns.2024.108438_b8) 2007; 333 Bluman (10.1016/j.cnsns.2024.108438_b14) 2020; 491 Yang (10.1016/j.cnsns.2024.108438_b11) 2013 Anco (10.1016/j.cnsns.2024.108438_b5) 1997; 38 Bluman (10.1016/j.cnsns.2024.108438_b36) 2020; 61 Cohen (10.1016/j.cnsns.2024.108438_b20) 1911 Wang (10.1016/j.cnsns.2024.108438_b29) 2023 Miller (10.1016/j.cnsns.2024.108438_b16) 1977 Hydon (10.1016/j.cnsns.2024.108438_b25) 2000 10.1016/j.cnsns.2024.108438_b27 Bluman (10.1016/j.cnsns.2024.108438_b15) 2002 Dickson (10.1016/j.cnsns.2024.108438_b26) 1924; 25 Bluman (10.1016/j.cnsns.2024.108438_b28) 2014; vol. 73 Sil (10.1016/j.cnsns.2024.108438_b13) 2020; 135 Lie (10.1016/j.cnsns.2024.108438_b1) 1881; 6 Ovsiannikov (10.1016/j.cnsns.2024.108438_b30) 1962 Bluman (10.1016/j.cnsns.2024.108438_b9) 2010 Stephani (10.1016/j.cnsns.2024.108438_b31) 1989 Bluman (10.1016/j.cnsns.2024.108438_b40) 1989; 138 Bluman (10.1016/j.cnsns.2024.108438_b4) 1989 Cantwell (10.1016/j.cnsns.2024.108438_b33) 2002 Friedrichs (10.1016/j.cnsns.2024.108438_b21) 1965 Page (10.1016/j.cnsns.2024.108438_b19) 1897 Bluman (10.1016/j.cnsns.2024.108438_b35) 1967 Anco (10.1016/j.cnsns.2024.108438_b6) 2005; 89 Bluman (10.1016/j.cnsns.2024.108438_b7) 2006; 47 Ovsiannikov (10.1016/j.cnsns.2024.108438_b23) 1982 Bluman (10.1016/j.cnsns.2024.108438_b3) 1988; 29 Eisenhart (10.1016/j.cnsns.2024.108438_b38) 1933 Bluman (10.1016/j.cnsns.2024.108438_b39) 1990; 50 Bluman (10.1016/j.cnsns.2024.108438_b18) 2021; 477 Satapathy (10.1016/j.cnsns.2024.108438_b12) 2018; 59 Bluman (10.1016/j.cnsns.2024.108438_b10) 2013; 54 Bluman (10.1016/j.cnsns.2024.108438_b22) 1974 Olver (10.1016/j.cnsns.2024.108438_b32) 1995 Olver (10.1016/j.cnsns.2024.108438_b24) 1986 Bluman (10.1016/j.cnsns.2024.108438_b2) 1987; 28 Bluman (10.1016/j.cnsns.2024.108438_b17) 2021; 477 Ovsiannikov (10.1016/j.cnsns.2024.108438_b34) 1959; 125 Bianchi (10.1016/j.cnsns.2024.108438_b37) 1918 |
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| Title | The natural extension to PDEs of Lie’s reduction of order algorithm for ODEs |
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