Access the full text.
Sign up today, get DeepDyve free for 14 days.
N. Ivanova (2006)
Conservation laws of multidimensional diffusion--convection equationsNonlinear Dynamics, 49
(1983)
On Lie–Bäcklund groups admitted by the heat equation with a source
R. Popovych, M. Kunzinger, H. Eshraghi (2006)
Admissible Transformations and Normalized Classes of Nonlinear Schrödinger EquationsActa Applicandae Mathematicae, 109
R. Popovych, N. Ivanova (2004)
Potential equivalence transformations for nonlinear diffusion–convection equationsJournal of Physics A, 38
N. Ivanova, R. Popovych (2006)
Equivalence of Conservation Laws and Equivalence of Potential SystemsInternational Journal of Theoretical Physics, 46
E. Pucci, G. Saccomandi (1993)
Contact symmetries and solutions by reduction of partial differential equationsJournal of Physics A, 27
R. Zhdanov, V. Lahno (2005)
Group Classification of the General Evolution Equation: Local and Quasilocal SymmetriesSymmetry Integrability and Geometry-methods and Applications, 1
R.W. Atherton, G.M. Homsy (1975)
On the existence and formulation of variational principles for nonlinear differential equationsStud. Appl. Math., 54
I. Cherkasov (1957)
On the Transformation of the Diffusion Process to a Wiener ProcessTheory of Probability and Its Applications, 2
S. Spichak, V. Stognii (1999)
Symmetry classification and exact solutions of the one-dimensional Fokker-Planck equation with arbitrary coefficients of drift and diffusionJournal of Physics A, 32
S. Anco, Dennis The (2005)
Symmetries, Conservation Laws, and Cohomology of Maxwell's Equations Using PotentialsActa Applicandae Mathematica, 89
S.V. Spichak, V.I. Stognii (1999)
Symmetric classification of the one-dimensional Fokker–Planck–Kolmogorov equation with arbitrary drift and diffusion coefficientsNeliniĭni Kolyv., 2
Cox Pdf (1977)
The Theory Of Stochastic ProcessesThe Mathematical Gazette, 61
P. Olver (1995)
Equivalence, Invariants, and Symmetry
I. Johnpillai, F. Mahomed (2001)
Singular invariant equation for the (1 + 1) Fokker–Planck equationJournal of Physics A: Mathematical and General, 34
G. Caviglia (1986)
Conservation laws for the Navier-Stokes equationsInternational Journal of Engineering Science, 24
N. Ibragimov (2002)
Laplace Type Invariants for Parabolic EquationsNonlinear Dynamics, 28
R. Khamitova (1982)
Group structure and the basis of conservation lawsTheoretical and Mathematical Physics, 52
Thomas Wolf (2002)
A comparison of four approaches to the calculation of conservation lawsEuropean Journal of Applied Mathematics, 13
R. Popovych, N. Ivanova (2003)
New results on group classification of nonlinear diffusion–convection equationsJournal of Physics A, 37
F. Estabrook, H. Wahlquist (1975)
Prolongation structures of nonlinear evolution equationsJournal of Mathematical Physics, 17
D.G.B. Edelen (1980)
Isovector Methods for Equations of Balance
N.M. Ivanova (2006)
Local and nonlocal conservation laws of diffusion–convection equationsCollect. Works Inst. Math. (Kyiv), 3
G. Bluman (1980)
ON THE TRANSFORMATION OF DIFFUSION PROCESSES INTO THE WIENER PROCESSSiam Journal on Applied Mathematics, 39
M. MacCallum (1986)
TRANSFORMATION GROUPS APPLIED TO MATHEMATICAL PHYSICS (Mathematics and Its Applications (Soviet Series))Bulletin of The London Mathematical Society, 18
G. Bluman (1990)
Simplifying the form of Lie groups admitted by a given differential equationJournal of Mathematical Analysis and Applications, 145
W. Feller (1951)
Diffusion Processes in Genetics
G. Bluman, V. Shtelen (2004)
Nonlocal transformations of Kolmogorov equations into the backward heat equationJournal of Mathematical Analysis and Applications, 291
I.S. Akhatov, R.K. Gazizov, N.K. Ibragimov (1989)
Itogi Nauki i Tekhniki, Current Problems in Mathematics. Newest Results
G. Bluman, S. Anco (2002)
Symmetry and Integration Methods for Differential Equations
M. Crum (1999)
Associated Sturm-Liouville systems
S. Sharma, H. Patel (2010)
The Fokker-Planck Equation
W. Shtelen, V. Stogny (1989)
Symmetry properties of one- and two-dimensional Fokker-Planck equationsJournal of Physics A, 22
(2005)
Normalized classes of nonlinear Schroedinger equations Lie theory and its application to physics
P. Morse, H. Feshbach (1955)
Methods of theoretical physics
A.N. Kolmogorov (1938)
On analytic methods in probabilityUspekhi Mat. Nauk, 5
O. Morozov (2004)
Contact-equivalence problem for linear hyperbolic equationsJournal of Mathematical Sciences, 135
C.W. Gardiner (1985)
Handbook of Stochastic Methods. For Physics, Chemistry and the Natural Sciences
P. Basarab-Horwath, V. Lahno, O. Magda (2000)
The Structure of Lie Algebras and the Classification Problem for Partial Differential EquationsActa Applicandae Mathematica, 69
A. Claus, D. Kleitman (1975)
Heuristic Methods for Solving Large Scale Network Routing Problems: The Telpaking ProblemStudies in Applied Mathematics, 54
G. Cicogna, D. Vitali (1990)
Classification of the extended symmetries of Fokker-Planck equationsJournal of Physics A, 23
I. Gihman, A. Skorohod (1974)
The theory of stochastic processes
E. Pucci, G. Saccomandi (1993)
Potential Symmetries of Fokker-Planck Equations
V.V. Zharinov (1986)
Conservation laws of evolution systemsTeor. Mat. Fiz., 68
(1917)
Wiss. Phys. Math. K1
N. Ibragimov (2007)
A new conservation theoremJournal of Mathematical Analysis and Applications, 333
(2006)
Local and nonlocal conservation laws of diffusion-convection equations, Collection of Works of Institute of Mathematics, Kyiv
(2004)
Symmetry analysis of evolution type equations, RCD, Moscow– Izhevsk
W. Fushchych, W. Shtelen, M. Serov, R. Popovych (2002)
Q-conditional symmetry of the linear heat equation
R.O. Popovych (2006)
Classification of admissible transformations of differential equationsCollect. Works Inst. Math. (Kyiv), 3
V. Zharinov (1986)
Conservation laws of evolution systemsTheoretical and Mathematical Physics, 68
S. Spichak, V. Stognii (2000)
One-Dimensional Fokker-Planck Equation Invariant under Four- and Six-Parametrical Group
(1987)
Group classification of equation of nonlinear filtration Dokl
(1881)
Über die Integration durch bestimmte Integrale von einer Klasse linear partieller Differentialgleichung
(1953)
Métodes variationnelles en théorie des collisions
R.O. Popovych (2006)
Normalized classes of nonlinear Schroedinger equationsBulg. J. Phys., 33
C. Gardiner (1983)
Handbook of Stochastic Methods
B. Finlayson (1972)
Existence of Variational Principles for the Navier‐Stokes EquationPhysics of Fluids, 15
S. Anco, G. Bluman (2001)
Direct construction method for conservation laws of partial differential equations Part I: Examples of conservation law classificationsEuropean Journal of Applied Mathematics, 13
E. Tonti (1973)
On the variational formulation for linear initial value problemsAnnali di Matematica Pura ed Applicata, 95
(1938)
On analytic methods in probability, Uspehi Mat
N. Ibragimov, M. Torrisi, A. Valenti (2012)
Modern Group Analysis: Advanced Analytical and Computational Methods in Mathematical Physics
R. Liboff (1966)
Introduction to the theory of kinetic equations
G. Bluman, G. Reid, S. Kumei (1988)
New classes of symmetries for partial differential equationsJournal of Mathematical Physics, 29
P. Olver (1995)
Equivalence, Invariants, and Symmetry: References
J. Kingston, C. Sophocleous (1998)
On form-preserving point transformations of partial differential equationsJournal of Physics A, 31
A. Vinogradov (1984)
The b-spectral sequence, Lagrangian formalism, and conservation laws. II. The nonlinear theoryJournal of Mathematical Analysis and Applications, 100
A. Bocharov, I. Krasilʹshchik, A. Vinogradov (1999)
Symmetries and conservation laws for differential equations of mathematical physics
A. Nikitin, W. Fushchich, V. Fushchich (1994)
Symmetries of Equations of Quantum Mechanics
Admissible point transformations of nonlinear Schrodinger equations, math-ph/0611061
C. Sophocleous (1996)
Potential symmetries of nonlinear diffusion-convection equationsJournal of Physics A, 29
P.M. Morse, H. Feshbach (1953)
Methods of Theoretical Physics, vol. 1
R.O. Popovych (1995)
On the symmetry and exact solutions of a transport equationUkr. Math. J., 47
C. Gardiner (1986)
Handbook of stochastic methods - for physics, chemistry and the natural sciences, Second Edition
D. Edelen (1980)
Isovector Methods for Equations of Balance: With Programs for Computer Assistance in Operator Calculations and an Exposition of Practical Topics of the Exterior Calculus
M. Fels, P. Olver (1999)
Moving Coframes: II. Regularization and Theoretical FoundationsActa Applicandae Mathematica, 55
V.I. Lahno, S.V. Spichak, V.I. Stognii (2004)
Symmetry Analysis of Evolution Type Equations
Valery Stohny (1997)
Symmetry Properties and Exact Solutions of the Fokker-Planck EquationJournal of Nonlinear Mathematical Physics, 4
O. Morozov (2003)
Contact Equivalence Problem for Linear Parabolic EquationsarXiv: Mathematical Physics
(2006)
No-go theorem on reduction operators of linear second-order parabolic equations
P. Olver (1986)
Applications of lie groups to differential equationsActa Applicandae Mathematica, 20
G. Saccomandi (1997)
Potential symmetries and direct reduction methods of order twoJournal of Physics A, 30
T. Tsujishita (1982)
On variation bicomplexes associated to differential equationsOsaka Journal of Mathematics, 19
S. Anco, G. Bluman (1997)
Nonlocal symmetries and nonlocal conservation laws of Maxwell’s equationsJournal of Mathematical Physics, 38
J. Meinhardt (1981)
Symmetries and differential equationsJournal of Physics A, 14
A. Kiselev, J. Leur (1982)
Group analysis of differential equations
R. Popovich (1995)
On the symmetry and exact solutions of a certain transport equationUkrainian Mathematical Journal, 47
R.S. Khamitova (1982)
The structure of a group and the basis of conservation lawsTeor. Mat. Fiz., 52
W. Buckland, J. Neyman (1952)
Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability.Biometrika, 115
M. Prokhorova (2005)
The structure of the category of parabolic equationsarXiv: Analysis of PDEs
R. Popovych, N. Ivanova (2004)
Hierarchy of conservation laws of diffusion-convection equationsJournal of Mathematical Physics, 46
C. Sastri, K. Dunn (1985)
Lie symmetries of some equations of the Fokker–Planck typeJournal of Mathematical Physics, 26
(1989)
Nonlocal symmetries. A heuristic approach, Itogi Nauki i Tekhniki, Current problems in mathematics
G. Bluman, Temuerchaolu, S. Anco (2006)
New conservation laws obtained directly from symmetry action on a known conservation lawJournal of Mathematical Analysis and Applications, 322
M. Fels, P. Olver (1998)
Moving Coframes: I. A Practical AlgorithmActa Applicandae Mathematica, 51
S. Anco, G. Bluman (2001)
Direct construction method for conservation laws of partial differential equations Part II: General treatmentEuropean Journal of Applied Mathematics, 13
N. Ibragimov, T. Kolsrud (2004)
Lagrangian Approach to Evolution Equations: Symmetries and Conservation LawsNonlinear Dynamics, 36
A. Fokker
Die mittlere Energie rotierender elektrischer Dipole im StrahlungsfeldAnnalen der Physik, 348
N. Ibragimov (1984)
Transformation groups applied to mathematical physicsActa Applicandae Mathematica, 6
(1994)
Ibragimov: Lie S. On integration of a class of linear partial differential equations by means of definite integrals
G. Bluman (2005)
Connections Between Symmetries and Conservation LawsSymmetry Integrability and Geometry-methods and Applications, 1
V. Matveev, M. Salle (1992)
Darboux Transformations and Solitons
We carry out an extensive investigation of conservation laws and potential symmetries for the class of linear (1+1)-dimensional second-order parabolic equations. The group classification of this class is revised by employing admissible transformations, the notion of normalized classes of differential equations and the adjoint variational principle. All possible potential conservation laws are described completely. They are in fact exhausted by local conservation laws. For any equation from the above class the characteristic space of local conservation laws is isomorphic to the solution set of the adjoint equation. Effective criteria for the existence of potential symmetries are proposed. Their proofs involve a rather intricate interplay between different representations of potential systems, the notion of a potential equation associated with a tuple of characteristics, prolongation of the equivalence group to the whole potential frame and application of multiple dual Darboux transformations. Based on the tools developed, a preliminary analysis of generalized potential symmetries is carried out and then applied to substantiate our construction of potential systems. The simplest potential symmetries of the linear heat equation, which are associated with single conservation laws, are classified with respect to its point symmetry group. Equations possessing infinite series of potential symmetry algebras are studied in detail.
Acta Applicandae Mathematicae – Springer Journals
Published: Nov 27, 2007
Read and print from thousands of top scholarly journals.
Already have an account? Log in
Bookmark this article. You can see your Bookmarks on your DeepDyve Library.
To save an article, log in first, or sign up for a DeepDyve account if you don’t already have one.
Copy and paste the desired citation format or use the link below to download a file formatted for EndNote
Access the full text.
Sign up today, get DeepDyve free for 14 days.
All DeepDyve websites use cookies to improve your online experience. They were placed on your computer when you launched this website. You can change your cookie settings through your browser.