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Meccanica

, Volume 54, Issue 1–2, pp 7–18 | Cite as

Functional solutions for problems of heat and mass transfer

  • Giovanni CimattiEmail author
Article
  • 104 Downloads

Abstract

We prove the existence and, in certain cases, the uniqueness of functional solutions for boundary value problems of systems of P.D.E. in divergence form with constant boundary conditions. We are motivated by various problems of heat and mass transfer. After giving a suitable definition of functional solutions, we reformulate the boundary value problem as non-standard one-dimensional two point problem for a system of O.D.E coupled with a mixed problem for the laplacian. If \({{{\mathcal {C}}}}_F\) and \({{{\mathcal {C}}}}\) denote respectively the set of functional and classical solutions of the starting problem we settle, in simple cases, the question if \({{{\mathcal {C}}}}_F={{{\mathcal {C}}}}\).

Keywords

Mass and heat transfer Existence and uniqueness Two-point problem for O.D.E Systems of P.D.E in divergence form 

Mathematics Subject Classification

34L99 35J66 

Notes

Acknowledgements

Several pertinent and useful suggestions of the Referees are gratefully acknowledged.

References

  1. 1.
    Ambrosetti A, Prodi G (1993) A primer in non-linear analysis. Cambridge University Press, CambridgezbMATHGoogle Scholar
  2. 2.
    Bear J (1988) Dynamics of fluids in porous media. Dover, New YorkzbMATHGoogle Scholar
  3. 3.
    Cafiero F (1947) Su un problema ai limiti relativo all’equazione \(y^{\prime }=f(x, y,\lambda )\). Giorn Mat Battaglini 77:145–163MathSciNetzbMATHGoogle Scholar
  4. 4.
    Cimatti G (2011) Remark on the existence, uniqueness and semi-explicit solvability of systems of autonomous partial differential equations in divergence form with constant boundary conditions. Proc R Soc Edinb 141:481–495MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Cimatti G (2010) On the functional solutions of a system of partial differential equations relevant in mathematical physics. Riv Mat Univ Parma 14:423–439MathSciNetzbMATHGoogle Scholar
  6. 6.
    Cimatti G (2018) The formal analogy between the stationary axisymmetrical Einstein–Maxwell equations and the equations of electrical heating of conductors. Le Matematiche 73:89–98MathSciNetzbMATHGoogle Scholar
  7. 7.
    Cimatti G (2018) A nonlinear boundary value problem relevant in general relativity and in the theory of electrical heating of conductors. Boll Unione Mat Ital 11:191–204MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Cimatti G (2012) Functional solutions for a plane problem in magnetohydrodynamics. Rend Sem Mat Univ Padova 127:57–74MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Cimatti G (2017) Existence and nonexistence of functional solutions for the equations of axially symmetric gravitational fields in general relativity. Riv Math Univ Parma 2:287–306 NSMathSciNetzbMATHGoogle Scholar
  10. 10.
    Cimatti G (2012) Existence and uniqueness for a two-point problem with an application to the electrical heating in an electrolyte. Quart Appl Math 70:383–392MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Cimatti G (1989) Remark on the existence and uniqueness for the thermistor problem under mixed boundary conditions. Quart Appl Math 47:117–121MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Hikosaka-Noboru H (1929) Untersuchung Ueber die Unitaet der Loesung der Differentialgleichung \(\frac{dy}{dx}=\xi f(x, y)\). Proc Phys Math Jpn 2:72–83Google Scholar
  13. 13.
    Gilbarg D, Trudinger NS (1988) Elliptic partial differential equations of second order. Academic Press, New YorkzbMATHGoogle Scholar
  14. 14.
    Miranda C (1970) Partial differential equations of elliptic type. Spinger, New YorkCrossRefzbMATHGoogle Scholar
  15. 15.
    Nield DA, Bejan A (1999) Convection in porous media. Springer, New YorkCrossRefzbMATHGoogle Scholar
  16. 16.
    Protter MH, Weinberger HF (1967) Maximum principles in differential equations. Prentice-Hall, Englewood CliffszbMATHGoogle Scholar
  17. 17.
    Sansone G (1941) Equazioni differenziali nel campo reale, Cap. VIII. Zanichelli editore, Bologna, pp 105–109zbMATHGoogle Scholar
  18. 18.
    Zawischa K (1930) Ueber die Differentialgleichung \(y = kf(x, y)\) deren Loesungskurve durch zwei gegebene Punkte hindurchgehen soll. Monatsh Math Phys 37:103–124MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Zwirner G (1946) Sull’equazione \(y^{\prime }=\lambda f(x, y)\). Rend Sem Mat Univ Padova 15:33–39MathSciNetzbMATHGoogle Scholar
  20. 20.
    Zwirner G (1946–1947) Alcuni teoremi sulle equazioni differenziali dipendenti da un parametro. Ann Univ Trieste 2:145–150Google Scholar

Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of PisaPisaItaly

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