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Singular nonlinear three point BVPs arising in thermal explosion in a cylindrical reactor

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Abstract

In this work we deal with a nonlinear three point singular boundary value problems (SBVPs), when the nonlinearity depends upon derivative. We establish the maximum principles for linear model. Prove some new inequalities based on Bessel and modified Bessel functions. Finally by using the Monotone Iterative Technique, we obtain some new existence results with well order and reverse order upper and lower solutions. The method developed in this paper can be used in computer algebra to compute solutions of real life problems.

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References

  1. P.L. Chamber, On the solution of the Poisson–Boltzmann equation with the application to the theory of thermal explosions. J. Chem. Phys. 20, 1795–1797 (1952)

    Article  Google Scholar 

  2. S. Chandrasekhar, Introduction to the Study of Stellar Structure (Dover, New York, 1967)

    Google Scholar 

  3. J.B. Keller, I. Electrohydrodynamics, The equilibrium of a charged gas in a container. J. Rational Mech. Anal. 5, 715–724 (1956)

    Google Scholar 

  4. R.D. Russell, L.F. Shampine, Numerical methods for singular boundary value problems. SlAM J. Numer. Anal. 12, 13–36 (1975)

    Article  Google Scholar 

  5. M.M. Chawla, P.N. Shivkumar, On the existence of solutions of a class of singular nonlinear two-point boundary value problems. J. Comput. Appl. Math. 19, 379–388 (1987)

    Article  Google Scholar 

  6. R.K. Pandey, A.K. Verma, Existence-uniqueness results for a class of singular boundary value problems-II. J. Math. Anal. Appl. 338(2), 1387–1396 (2008)

    Article  Google Scholar 

  7. R.K. Pandey, A.K. Verma, A note on existence-uniqueness results for a class of doubly singular boundary value problems. Nonlinear Anal. Theory Methods Appl. 71(7), 3477–3487 (2009)

    Article  Google Scholar 

  8. A.K. Verma, Monotone iterative method and zero’s of Bessel functions for nonlinear singular derivative dependent BVP in the presence of upper and lower solutions. Nonlinear Anal. 74(14), 4709–4717 (2011)

    Article  Google Scholar 

  9. A.K. Verma, L. Verma, Nonlinear singular BVP of limit circle type and the presence of reverse-ordered upper and lower solutions. Int. J. Differ. Equ. (2011). doi:10.1155/2011/986948

  10. A.K. Verma, R.P. Agarwal, Upper and lower solutions method for regular singular differential equations with quasi-derivative boundary conditions. Commun. Nonlinear Sci. Numer. Simul. 17, 4551–4558 (2012)

    Article  Google Scholar 

  11. A.K. Verma, R.P. Agarwal, L. Verma, Bessel functions and singular BVPs arising in physiology in the presence of upper and lower solutions in reverse order. J. Appl. Math. Comput. 39, 445–457 (2012)

    Article  Google Scholar 

  12. A.K. Verma, The Monotone iterative method and regular singular nonlinear BVP in the presence of reverse ordered upper and lower solutions, Electronic. J. Differ. Equ. 2012, 1–10 (2012)

    Google Scholar 

  13. F. Li, M. Jia, X. Liu, C. Li, G. Li, Existence and uniqueness of solutions of second-order three-point boundary value problems with upper and lower solutions in the reversed order. Nonlinear Anal. 68, 2381–2388 (2008)

    Article  Google Scholar 

  14. M. Singh, A.K. Verma, Picard type iterative scheme with initial iterates in reverse order for a class of nonlinear three point BVPs. Int. J. Differ. Equ. (2013). doi:10.1155/2013/728149

  15. M. Singh, A.K. Verma, On a monotone iterative method for a class of three point nonlinear nonsingular BVPs with upper and lower solutions in reverse order. J. Appl. Math. Comput. 43, 99–114 (2013)

    Article  Google Scholar 

  16. A.K. Verma, M. Singh, Existence of Solutions for Three-Point Bvps Arising in Bridge Design. Electron. J. Differ. Equ. 2014(173), 1–11 (2014)

    Google Scholar 

  17. J.R.L. Webb, Existence of positive solutions for a thermostat model. Nonlinear Anal. Real World Appl. 13, 923–938 (2012)

    Article  Google Scholar 

  18. M. Gregus, F. Neumann, F.M. Arscott, Three-point boundary value problems for differential equations. J. London Math. Soc. 3, 429–436 (1971)

    Article  Google Scholar 

  19. R. Ma, Existence of solutions of nonlinear m-point boundary-value problems. J. Math. Anal. Appl. 256, 556–567 (2001)

    Article  Google Scholar 

  20. J. Nieto, An abstract monotone iterative technique. Nonlinear Anal. 28, 1923–1933 (1997)

    Article  Google Scholar 

  21. J. Henderson, B. Karna, C.C. Tisdell, Existence of solutions for three-point boundary value problems for second order equations. Proc. Amer. Math. Soc. 133, 1365–1369 (2004)

  22. A.K. Verma, M. Singh, Maximum principle and nonlinear three point singular boundary value problems arising due to spherical symmetry. Commun. Appl. Anal. (in press)

  23. M. Singh, A.K. Verma, R.P. Agarwal, Maximum and anti-maximum principles for three point SBVPs and nonlinear three point SBVPs. J. Appl. Math. Comput. doi:10.1007/s12190-014-0773-6

  24. A. Erdrlyi (ed.), Higher Transcendental Functions, Vol. II (McGraw-Hill, New York, 1953)

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Acknowledgments

This work is partially supported by Grant provided by UGC, New Delhi, India, File No. F.4-1/2006 (BSR)/7-203/2009(BSR).

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Authors and Affiliations

  1. Department of Mathematics, BITS Pilani, Pilani, 333031, Rajasthan, India

    Amit K. Verma & Mandeep Singh

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  1. Amit K. Verma
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  2. Mandeep Singh
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Correspondence to Amit K. Verma.

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Verma, A.K., Singh, M. Singular nonlinear three point BVPs arising in thermal explosion in a cylindrical reactor. J Math Chem 53, 670–684 (2015). https://doi.org/10.1007/s10910-014-0447-5

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  • DOI: https://doi.org/10.1007/s10910-014-0447-5

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