Nonlocal Extensions of Similarity Methods

References

• Akhatov , I , Gazizov , R and Ibragimov , N H . 1991 . Nonlocal symmetries. Heuristic approach . J. Sov. Math. , 55 : 1401 – 1450 .
   Google Scholar
• Ames , W , Lohner , R and Adams , E . 1981 . Group properties of u tt  = [f(u)u x]x . Int. J. Nonlin. Mech. , 16 : 447 – 439 .
   Web of Science ®Google Scholar
• Anco , S and Bluman , G . 1997 . Direct construction of conservation laws from field equations . Phys. Rev. Lett. , 78 : 2869 – 2873 .
   Web of Science ®Google Scholar
• Anco , S and Bluman , G . 2002 . Direct construction method for conservation laws of partial differential equations Part I: Examples of conservation law classifications; Part II: General treatment . EJAM , 13 : 545 – 566 . 567–585
   Google Scholar
• Anco , S , Bluman , G and Wolf , T . 2008 . Invertible mappings of nonlinear PDEs to linear PDEs through admitted conservation laws , To appear in Acta Applicandae Mathematica .
   Google Scholar
• Anderson , R , Kumei , S and Wulfman , C . 1972 . Generalization of the concept of invariance of differential equations . Phys. Rev. Lett. , 28 : 988 – 991 .
   Web of Science ®Google Scholar
• Bluman , G . 1967 . Construction of Solutions to Partial Differential Equations by the Use of Transformation Groups , California Institute of Technology . Ph.D. Thesis
   Google Scholar
• Bluman , G . 1980 . On the transformation of diffusion processes into the Wiener process . SIAM J. Appl. Math. , 39 : 238 – 247 .
   Web of Science ®Google Scholar
• Bluman , G . 1983 . On mapping linear partial differential equations to constant coefficient equations . SIAM J. Appl. Math. , 43 : 1259 – 1273 .
   Web of Science ®Google Scholar
• Bluman , G and Anco , S . 2002 . “ Symmetry and Integration Methods for Differential Equations ” . In Appl. Math Sci. , Vol. 154 , New York : Springer .
   Google Scholar
• Bluman , G and Cheviakov , A . 2005 . Framework for potential systems and nonlocal symmetries: Algorithmic approach . J. Math. Phys. , 46 : 123506
   Web of Science ®Google Scholar
• Bluman , G and Cheviakov , A . 2007 . Nonlocally related systems, linearization and nonlocal symmetries for the nonlinear wave equation . J. Math. Anal. Appl. , 333 : 93 – 111 .
   Web of Science ®Google Scholar
• Bluman , G , Cheviakov , A and Ivanova , N . 2006 . Framework for nonlocally related partial differential equations systems and nonlocal symmetries: Extension, simplification, and examples . J. Math. Phys. , 47 : 113505
   Web of Science ®Google Scholar
• Bluman , G and Cole , J . 1969 . The general similarity solution of the heat equation . J. Math. Mech. , 18 : 1025 – 1042 .
   Web of Science ®Google Scholar
• Bluman , G and Cole , J . 1974 . Similarity Methods for Differential Equations , Vol. 13 , New York : Springer . Appl. Math Sci. No
   Google Scholar
• Bluman , G and Doran-Wu , P . 1995 . The use of factors to discover potential systems or linearizations . Acta Appl. Math. , 41 : 21 – 43 .
   Web of Science ®Google Scholar
• Bluman , G and Kumei , S . 1980 . On the remarkable nonlinear diffusion equation [a(u+b 2)-2 u x ] x — u t  = 0 . J. Math. Phys. , 21 : 1019 – 1023 .
   Web of Science ®Google Scholar
• Bluman , G and Kumei , S . 1987 . On invariance properties of the wave equation . J. Math. Phys. , 28 : 307 – 318 .
   Web of Science ®Google Scholar
• Bluman , G and Kumel , S . 1989 . “ Symmetries and Differential Equations ” . In Appl. Math Sci. , Vol. 81 , New York : Springer .
   Google Scholar
• Bluman , G and Kumei , S . 1990 . Symmetry-based algorithms to relate partial differential equations. I. Local symmetries . EJAM , 1 : 189 – 216 .
   Google Scholar
• Bluman , G and Kumei , S . 1990 . Symmetry-based algorithms to relate partial differential equations. II. Linearization by nonlocal symmetries . EJAM , 1 : 217 – 223 .
   Google Scholar
• Bluman , G , Kumei , S and Reid , G . 1988 . New classes of symmetries for partial differential equations . J. Math. Phys. , 29 : 806 – 811 .
   Web of Science ®Google Scholar
• Bluman , G and Shtelen , V . 1996 . New classes of Schrödinger equations equivalent to the free particle equation through non-local transformations . J. Phys. A , 29 : 4473 – 4480 .
   Google Scholar
• Bluman , G and Shtelen , V . 2004 . Nonlocal transformations of Kolmogorov equations into the backward heat equation . J. Math. Anal. Appl. , 291 : 419 – 437 .
   Web of Science ®Google Scholar
• Bluman , G and Temuerchaolu . 2005 . Conservation laws for nonlinear telegraph equations . J. Math. Anal. Appl. , 310 : 459 – 476 .
   Web of Science ®Google Scholar
• Bluman , G , Temuerchaolu , S and Anco , S . 2006 . New conservation laws obtained directly from symmetry action on a known conservation law . J. Math. Anal. Appl. , 322 : 233 – 250 .
   Web of Science ®Google Scholar
• Bluman , G , Temuerchaolu , R and Sahadevan , R . 2005 . Local and nonlocal symmetries for nonlinear telegraph equations . J. Math. Phys. , 46 : 023505
   Web of Science ®Google Scholar
• Bluman , G and Yan , Z . 2005 . Nonclassical potential solutions of partial differential equations . EJAM , 16 : 239 – 261 .
   Google Scholar
• Buckingham , E . 1914 . On physically similar systems; illustrations of the use of dimension equations . Phys. Rev. , 4 : 345 – 376 .
   Web of Science ®Google Scholar
• Cantwell , B . 2002 . Introduction to Symmetry Analysis , Cambridge, UK : Cambridge University Press .
   Google Scholar
• http://www.math.ubc.ca/˜alexch/gemCheviakov A, GeM software package for computation of symmetries and conservation laws of differential equations, Comp. Phys. Comm. 176 (2007), 48–61. GeM package and documentation available at
   Google Scholar
• Clarkson , P and Kruskal , M . 1989 . New similarity solutions of the Boussinesq equation . J. Math. Phys. , 30 : 2201 – 2213 .
   Web of Science ®Google Scholar
• Clarkson , P and Mansfield , E . 1994 . Algorithms for the nonclassical method of symmetry reductions . SIAM J. Appl. Math. , 54 : 1693 – 1719 .
   Web of Science ®Google Scholar
• Fokas , A and Liu , Q . 1994 . Nonlinear interaction of traveling waves of nonintegrable equations . Phys. Rev. Lett. , 72 : 3293 – 3296 .
   PubMed Web of Science ®Google Scholar
• Hereman , W . 1997 . Review of symbolic software for Lie symmetry analysis . Math. Comp. Mod. , 25 : 115 – 132 .
   Web of Science ®Google Scholar
• Hydon , P . 2000 . Symmetry Methods for Differential Equations. A Beginner’s Guide , Cambridge , UK : Cambridge University Press .
   Google Scholar
• Kumei , S and Bluman , G . 1982 . When nonlinear differential equations are equivalent to linear differential equations . SIAM J. Appl. Math. , 42 : 1157 – 1173 .
   Web of Science ®Google Scholar
• Levi , D and Winternitz , P . 1989 . Non-classical symmetry reduction: example of the Boussinesq equation . J. Phys. A , 22 : 2915 – 2924 .
   Google Scholar
• Lie , S . 1881 . Über die Integration durch bestimmte Integrale von einer Klasse linearer partieller Differentialgleichungen . Arch. Math. , 6 : 328 – 268 .
   Google Scholar
• Ma , A . 1990 . Extended Group Analysis of the Wave Equation , Columbia : University of British . MSc thesis
   Google Scholar
• Mikhailov , A , Shabat , A and Sokolov , V . 1991 . The symmetry approach to classification of integrable equations, in What is Integrability? , Edited by: Zacharov , V . 115 – 182 . New York : Springer .
   Google Scholar
• Noether , E . 1918 . Invariante Variationsprobleme. Nachr. Konig. Gesell. Wissen, Gottingen . Math.-Phys. , Kl : 235 – 257 .
   Google Scholar
• Nucci , M and Clarkson , P . 1992 . The nonclassical method is more general than the direct method for symmetry reductions: an example of the Fitzhugh-Nagumo equation . Phys. Lett. , 164A : 49 – 56 .
   Google Scholar
• Olver , P . 1977 . Evolution equations possessing infinitely many symmetries . J. Math. Phys. , 18 : 1212 – 1215 .
   Web of Science ®Google Scholar
• Olver , P . 1986 . Applications of Lie Groups to Differential Equations , Vol. 107 , New York : Springer . Graduate Texts in Mathematics, No
   Google Scholar
• Ovsiannikov , L . 1962 . Group Properties of Differential Equations , Novosibirsk : Nauka . (in Russian)
   Google Scholar
• Ovsiannikov , L . 1982 . Group Analysis of Differential Equations , New York : Academic Press .
   Google Scholar
• Popovych , R and Ivanova , N . 2005 . Hierarchy of conservation laws of diffusion- convection equations . J. Math. Phys. , 46 043502
   Web of Science ®Google Scholar
• Sjöberg , A and Mahomed , F . 2004 . Non-local symmetries and conservation laws for one- dimensional gas dynamics equations . Appl. Math. Comput. , 150 : 379 – 397 .
   Web of Science ®Google Scholar
• Stephani , H . 1989 . Differential Equations: Their Solution Using Symmetries , Cambridge , UK : Cambridge University Press .
   Google Scholar
• Wolf , T . 2002 . A comparison of four approaches to the calculation of conservation laws . EJAM , 13 : 129 – 152 .
   Google Scholar
• Wolf , T . 2000 . Crack, LiePDE, ApplySym and ConLaw, Section 4.3.5 and computer program on CD-ROM , Edited by: Grabmeier , J. , Kaltofen , E. and Weispfenning , V. 465 – 468 . Computer Algebra Handbook, Springer .
   Google Scholar