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Several Families of New Exact Solutions for Wave-Like Equations with Variable Coefficients

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Proceedings of the Seventh International Conference on Mathematics and Computing

Part of the book series: Advances in Intelligent Systems and Computing ((AISC,volume 1412))

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Abstract

Several families of new exact solutions for second-order wave-like partial differential equations with variable coefficients are derived in this paper. To derive these solutions, certain recurrence relations for powers of the variables and coefficients are formed using the method of balancing powers of the variables. Solving these recurrence relations, several exact solutions are derived. We derive all polynomial solutions of these partial differential equations. Different families of new exact solutions of these equations which can be expressed in terms of hypergeometric functions are also constructed. Euler–Tricomi equation, Keldysh equation, and generalized Euler–Tricomi equations are special cases of this general equations.

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References

  1. Azad H, Laradji A, Mustafa MT (2011) Polynomial solutions of differential equations. Adv. Differ. Equ. 58:58

    Article  MathSciNet  Google Scholar 

  2. Chen GQ, Slemrod M, Wang D (2008) Vanishing viscosity method for transonic flow. D Arch Ration Mech Anal 189:159–188

    Article  MathSciNet  Google Scholar 

  3. Chen S (2009) The fundamental solution of the Keldysh type operator. Sci China Ser A-Math 52:1829–1843

    Article  MathSciNet  Google Scholar 

  4. Chen S (2012) A mixed equation of tricomi-keldysh type. J Hyper Differ Equ 09(3):545–553

    Article  MathSciNet  Google Scholar 

  5. Hayman WK, Shanidze ZG (1999) Polynomial solutions of partial differential equations. Methods Appl Anal 6:97–108

    Article  MathSciNet  Google Scholar 

  6. Joseph SP (2018) Polynomial solutions and other exact solutions of axisymmetric generalized Beltrami flows. Acta Mech 229, 2737–2750

    Google Scholar 

  7. Keşan C (20) Taylor polynomial solutions of second order linear partial differential equations. Appl Math Comput 152:29–41

    Google Scholar 

  8. Lashkarian E, Hejazi SR (2016) Polynomial and non-polynomial solutions set for wave equation using Lie point symmetries. Comput Methods Differ Equ 4(4):298–308

    MathSciNet  MATH  Google Scholar 

  9. Manwell AR (1979) Tricomi equation: with applications to the theory of plane transonic flow. Chapman and Hall/CRC Research Notes in Mathematics Series

    Google Scholar 

  10. Miles EP, Williams E (1955) A basic set of homogeneous harmonic polynomials in k variables. Proc Am Math Soc 6(2):191–194

    MathSciNet  MATH  Google Scholar 

  11. Miles EP, Williams E (1956) A basic set of polynomial solutions for the Euler-Poisson-Darboux and Beltrami equations. Am Math Mon 63:401–404

    Article  MathSciNet  Google Scholar 

  12. Miles EP, Young EC (1967) Basic sets of polynomials for generalized Beltrami and Euler-Poisson-Darboux equations and their iterates. Proc Am Math Soc 18:981–986

    MathSciNet  MATH  Google Scholar 

  13. Morawetz CS (2004) Mixed equations and transonic flow. J Hyper Differ Equ 01(1):1–26

    Article  MathSciNet  Google Scholar 

  14. Olde Daalhuis AB (2010) Confluent Hypergeometric Functions. In: Olver FWJ (ed) NIST handbook of mathematical functions. Springer, pp 322–349

    Google Scholar 

  15. Olde Daalhuis AB (2010) Hypergeometric function. In: Olver FWJ (ed) NIST handbook of mathematical functions. Springer, pp 383–401

    Google Scholar 

  16. Pedersen P (1996) A basis for polynomial solutions to systems of linear constant coefficient PDE’s. Adv Math 117(1):157–163

    Article  MathSciNet  Google Scholar 

  17. Polyanin AD (2002) Handbook of linear partial differential equations for engineers and scientists. Chapman and Hall

    Google Scholar 

  18. Otway TH (2012) The dirichlet problem for elliptic-hyperbolic equations of Keldysh Type. Springer

    Google Scholar 

  19. Otway TH (2015) Elliptic-Hyperbolic partial differential equations: a mini-course in geometric and quasilinear methods. Spinger

    Google Scholar 

  20. Reznick B (1996) Homogeneous polynomial solutions to constant coefficient PDE’s. Adv Math 117:179–192

    Article  MathSciNet  Google Scholar 

  21. Saad N, Hall RL, Trentona VA (2014) Polynomial solutions for a class of second-order linear differential equations. Appl Math Comput 226:615–634

    MathSciNet  Google Scholar 

  22. Shou D, He J (2008) Beyond Adomian method: the variational iteration method for solving heat-like and wave-like equations with variable coefficients. Phys Lett A 372:233–237

    Article  MathSciNet  Google Scholar 

  23. Smith SP (1992) Polynomial solutions to constant coefficient differential equations. Trans Am Math Soc 329(2):551–569

    Article  MathSciNet  Google Scholar 

  24. Yagdjian K (2004) A note on the fundamental solution for the Tricomi-type equation in the hyperbolic domain. J Differ Equ 206(1):227–252

    Article  MathSciNet  Google Scholar 

  25. Wazwaz AM, Gorguis A (2004) Exact solutions for heat-like and wave-like equations with variable coefficients. Appl Math Comput 149:15–29

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Government Engineering College, Wayanad, Thalapuzha P.O., Wayanad, KL, India

    Subin P. Joseph

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  1. Subin P. Joseph
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Correspondence to Subin P. Joseph .

Editor information

Editors and Affiliations

  1. Department of Information Technology, Maulana Abul Kalam Azad University of Technology, Haringhata, India

    Debasis Giri

  2. The University of Texas at San Antonio, San Antonio, TX, USA

    Kim-Kwang Raymond Choo

  3. Department of Mathematics, Indian Institute of Technology Madras, Chennai, India

    Saminathan Ponnusamy

  4. Technical University of Denmark, Kongens Lyngby, Denmark

    Weizhi Meng

  5. Department of Computer Engineering, Ondokuz Mayis University, Atakum, Turkey

    Sedat Akleylek

  6. Department of Information Technology, Indian Institute of Engineering Science and Technology, West Bengal, India

    Santi Prasad Maity

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Joseph, S.P. (2022). Several Families of New Exact Solutions for Wave-Like Equations with Variable Coefficients. In: Giri, D., Raymond Choo, KK., Ponnusamy, S., Meng, W., Akleylek, S., Prasad Maity, S. (eds) Proceedings of the Seventh International Conference on Mathematics and Computing . Advances in Intelligent Systems and Computing, vol 1412. Springer, Singapore. https://doi.org/10.1007/978-981-16-6890-6_67

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