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Abstract

Partial differential equations arise frequently in the formulation of fundamental laws of nature and in the mathematical analysis of a wide variety of problems in applied mathematics, mathematical physics, and engineering science. This subject plays a central role in modern mathematical sciences, especially in physics, geometry, and analysis. Many problems of physical interest are described by partial differential equations with appropriate initial and/or boundary conditions. These problems are usually formulated as initial-value problems, boundary-value problems, or initial boundary-value problems. In order to prepare the reader for study and research in nonlinear partial differential equations, a broad coverage of the essential standard material on linear partial differential equations and their applications is required.

Keywords

Wave Equation Linear Partial Differential Equation Free Surface Elevation Telegraph Equation Fractional Diffusion Equation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Bibliography

  1. Courant, R. and Hilbert, D. (1953). Methods of Mathematical Physics, Vol. 1, Wiley-Interscience, New York. Google Scholar
  2. Debnath, L. (1989). The linear and nonlinear Cauchy–Poisson wave problems for an inviscid or viscous liquid, in Topics in Mathematical Analysis (ed. T.M. Rassias), World Scientific, Singapore, 123–155. Google Scholar
  3. Debnath, L. (1994). Nonlinear Water Waves, Academic Press, Boston. MATHGoogle Scholar
  4. Debnath, L. (1995). Integral Transforms and Their Applications, CRC Press, Boca Raton. MATHGoogle Scholar
  5. Debnath, L. (2003a). Fractional integral and fractional partial differential equation in fluid mechanics, Fract. Calc. Appl. Anal. 6, 119–155. MathSciNetGoogle Scholar
  6. Debnath, L. (2003b). Recent applications of fractional calculus to science and engineering, Int. J. Math. Sci. 54, 3413–3442. MathSciNetCrossRefGoogle Scholar
  7. Debnath, L. and Mikusinski, P. (1999). Introduction to Hilbert Spaces with Applications, 2nd edition, Academic Press, Boston. MATHGoogle Scholar
  8. Debnath, L. and Mikusinski, P. (2005). Introduction to Hilbert Spaces with Applications, 3rd edition, Elsevier Academic Press, London. Google Scholar
  9. Erdélyi, A. (1955). Higher Transcendental Functions, Vol. 3, McGraw-Hill, New York. MATHGoogle Scholar
  10. Gordon, W. (1926). Der Comptoneffekt nach der Schrödingerchen Theorie, Zeit. Für Physik 40, 117–133. CrossRefGoogle Scholar
  11. Klein, O. (1927). Elektrodynamik und Willenmechanik vom Standpunkt des Korrespondenzprinzips, Zeit. Für Physik 41, 407–442. CrossRefGoogle Scholar
  12. Lamb, H. (1904). On the propagation of tremors over the surface of elastic solid, Philos. Trans. R. Soc. Lond. A203, 1–42. Google Scholar
  13. Mainardi, F. (1994). On the initial-value problem for the fractional diffusion–wave equation, in Waves and Stability in Continuous Media (eds. S. Rionero, and T. Ruggeri), World Scientific, Singapore, 246–251. Google Scholar
  14. Mainardi, F. (1995). The time fractional diffusion–wave equation, Radiofisika 38, 20–36. MathSciNetGoogle Scholar
  15. Mainardi, F. (1996a). Fractional relaxation–oscillation and fractional diffusion–wave phenomena, Chaos Solitons Fractals 7, 1461–1477. MathSciNetMATHCrossRefGoogle Scholar
  16. Mainardi, F. (1996b). The fundamental solitons for the fractional diffusion–wave equations, Appl. Math. Lett. 9, 23–28. MathSciNetMATHCrossRefGoogle Scholar
  17. Nigmatullin, R.R. (1986). The realization of the generalized transfer equation in a medium with fractal geometry, Phys. Status Solidi (b) 133, 425–430. CrossRefGoogle Scholar
  18. Schneider, W.R. and Wyss, W. (1989). Fractional diffusion and wave equations, J. Math. Phys. 30, 134–144. MathSciNetMATHCrossRefGoogle Scholar
  19. Sen, A.R. (1963). Surface waves due to blasts on and above liquids, J. Fluid Mech. 16, 65–81. MATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Texas, Pan AmericanEdinburgUSA

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