A class of partial differential equations (a conservation law and four balance laws), with four independent variables and involving sixteen arbitrary continuously differentiable functions, is considered in the framework of equivalence transformations. These are point transformations of differential equations involving arbitrary elements and live in an augmented space of independent, dependent and additional variables representing values taken by the arbitrary elements. Projecting the admitted infinitesimal equivalence transformations into the space of independent and dependent variables, we determine some finite transformations mapping the system of balance laws to an equivalent one with the same differential structure but involving different arbitrary elements; in particular, the target system we want to recover is an autonomous system of conservation laws. An application to a physical problem is considered.
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Dafermos, C.M.: Hyperbolic Conservation Laws in Continuum Physics. Springer, Berlin (2000)
Bressan, A.: Hyperbolic Systems of Conservation Laws. Oxford University Press, Oxford (2000)
LeVeque, R.J.: Numerical Methods for Conservation Laws. Birkhäuser, Basel (1992)
Ovsiannikov, L.V.: Group Analysis of Differential Equations. Academic Press, New York (1982)
Ibragimov, N.H.: Transformation Groups Applied to Mathematical Physics. Reidel, Dordrecht (1985)
Olver, P.J.: Applications of Lie Groups to Differential Equations. Springer, New York (1986)
Ibragimov, N.H.: CRC Handbook of Lie Group Analysis of Differential Equations: Symmetries, Exact Solutions and Conservation Laws. CRC Press, Boca Raton (1994, 1995, 1996)
Olver, P.J.: Equivalence, Invariants, and Symmetry. Cambridge University Press, New York (1995)
Baumann, G.: Symmetry Analysis of Differential Equations with Mathematica. Springer, New York (2000)
Meleshko, S.V.: Methods for Constructing Exact Solutions of Partial Differential Equations. Springer, New York (2005)
Bluman, G.W., Cheviakov, A.F., Anco, S.C.: Applications of Symmetry Methods to Partial Differential Equations. Springer, New York (2009)
Oliveri, F., Speciale, M.P.: Exact solutions to the equations of ideal gas-dynamics by means of the substitution principle. Int. J. Non-Linear Mech. 33, 585–592 (1998)
Oliveri, F., Speciale, M.P.: Exact solutions to the equations of perfect gases through Lie group analysis and substitution principles. Int. J. Non-Linear Mech. 34, 1077–1087 (1999)
Oliveri, F., Speciale, M.P.: Exact solutions to the unsteady equations of perfect gases through Lie group analysis and substitution principles. Int. J. Non-Linear Mech. 37, 257–274 (2002)
Oliveri, F., Speciale, M.P.: Exact solutions to the ideal magneto-gas-dynamics equations through Lie group analysis and substitution principles. J. Phys. A, Math. Gen. 38, 8803–8820 (2005)
Margheriti, L., Speciale, M.P.: Unsteady solutions of Euler equations generated by steady solutions. Acta Appl. Math. 113, 289–303 (2011)
Kumei, S., Bluman, G.W.: When nonlinear differential equations are equivalent to linear differential equations. SIAM J. Appl. Math. 42, 1157–1173 (1982)
Donato, A., Oliveri, F.: Reduction to autonomous form by group analysis and exact solutions of axi-symmetric MHD equations. Math. Comput. Model. 18, 83–90 (1993)
Donato, A., Oliveri, F.: Linearization procedure of nonlinear first order systems of PDE’s by means of canonical variables related to Lie groups of point transformations. J. Math. Anal. Appl. 188, 552–568 (1994)
Donato, A., Oliveri, F.: When nonautonomous equations are equivalent to autonomous ones. Appl. Anal. 58, 313–323 (1995)
Donato, A., Oliveri, F.: How to build up variable transformations allowing one to map nonlinear hyperbolic equations into autonomous or linear ones. Transp. Theory Stat. Phys. 25, 303–322 (1996)
Currò, C., Oliveri, F.: Reduction of nonhomogeneous quasilinear 2×2 systems to homogeneous and autonomous form. J. Math. Phys. 49, 103504-1–103504-11 (2008)
Oliveri, F.: Lie symmetries of differential equations: classical results and recent contributions. Symmetry 2, 658–706 (2010)
Oliveri, F.: General dynamical systems described by first order quasilinear PDEs reducible to homogeneous and autonomous form. Int. J. Non-Linear Mech. 47, 53–60 (2012)
Ibragimov, N.H., Torrisi, M., Valenti, A.: Preliminary group classification of equations v tt =f(x,v x )v xx +g(x,v x ). J. Math. Phys. 32, 2988–2995 (1991)
Lisle, I.G.: Equivalence transformations for classes of differential equations. PhD dissertation, University of British Columbia, Vancouver, BC, Canada, 1992
Ibragimov, N.H., Torrisi, M.: Equivalence groups for balance equations. J. Math. Anal. Appl. 184, 441–452 (1994)
Torrisi, M., Tracinà, R.: Equivalence transformations for system of first order quasilinear partial differential equations. In: Ibragimov, N.H., Mahomed, F.H. (eds.) Modern Group Analysis VI, Developments in Theory, Computations and Applications, pp. 115–135. New Age International, New Delhi (1997)
Meleshko, S.V.: Generalization of the equivalence transformations. J. Nonlinear Math. Phys. 3, 170–174 (1996)
Torrisi, M., Tracinà, R.: Equivalence transformations and symmetries for a heat conduction model. Int. J. Non-Linear Mech. 33, 473–486 (1998)
Özer, S., Suhubi, E.: Equivalence groups for first-order balance equations and applications to electromagnetism. Theor. Math. Phys. 137, 1590–1597 (2003)
Oliveri, F., Speciale, M.P.: Equivalence transformations of quasilinear first order systems and reduction to autonomous and homogeneous form. Acta Appl. Math. 122, 447–460 (2012)
Oliveri, F., Speciale, M.P.: Reduction of balance laws to conservation laws by means of equivalence transformations. J. Math. Phys. 54, 041506 (2013)
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Gorgone, M., Oliveri, F. & Speciale, M.P. Reduction of Balance Laws in (3+1)-Dimensions to Autonomous Conservation Laws by Means of Equivalence Transformations. Acta Appl Math 132, 333–345 (2014). https://doi.org/10.1007/s10440-014-9929-5
- Systems of balance laws
- Equivalence transformations
- Derivation of autonomous and homogeneous conservation laws