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The Error Analysis of Finite Difference Approximation for a System of Singularly Perturbed Semilinear Reaction-Diffusion Equations with Discontinuous Source Term

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Finite Difference Methods. Theory and Applications (FDM 2018)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11386))

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Abstract

We consider a coupled system of two singularly perturbed semilinear reaction-diffusion equations with a discontinuous source term. The leading term in each equation is multiplied by a small positive parameter, but these parameters have different order of magnitude. The solution of these system of equations have overlapping and interacting boundary and interior layers. Based on the discrete Green’s function theory, the properties of the discretized operator are established. The error estimates are derived in the maximum norm for a central difference scheme on layer-adapted meshes, and the method is proved to be almost second order uniformly convergent independently of both the perturbation parameters. Numerical results validate the theoretical results.

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References

  1. Andreev, V.B.: On the uniform convergence of a classical difference scheme on a nonuniform mesh for the one-dimensional singularly perturbed reaction-diffusion equation. Comput. Math. Phys. 44, 449–464 (2001)

    Google Scholar 

  2. Boglaev, I., Pack, S.: A uniformly convergent method for a singularly perturbed semilinear reaction-diffusion problem with discontinuous data. Appl. Math. Comput. 182, 244–257 (2006)

    MathSciNet  MATH  Google Scholar 

  3. Farrell, P.A., O’Riordan, E., Shishkin, G.I.: A class of singularly perturbed semilinear differential equations with interior layers. Math. Comput. 74, 1759–1776 (2005)

    Article  MathSciNet  Google Scholar 

  4. Gracia, J.L., Lisbona, F., Madaune-Tort, M., O’Riordan, E.: A system of singularly perturbed semilinear equations. In: Hegarty, A., Kopteva, N., O’Riordan, E., Stynes, M. (eds.) BAIL 2008 - Boundary and Interior Layers. LNCSE, vol. 69, pp. 163–172. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-00605-0_12

    Chapter  Google Scholar 

  5. Linß, T., Madden, N.: Layer-adapted meshes for a system of coupled singularly perturbed reaction-diffusion problems. IMA J. Numer. Anal. 29, 109–125 (2009)

    Article  MathSciNet  Google Scholar 

  6. Rao, S.C.S., Chawla, S.: Interior layers in coupled system of two singularly perturbed reaction-diffusion equations with discontinuous source term. In: Dimov, I., Faragó, I., Vulkov, L. (eds.) NAA 2012. LNCS, vol. 8236, pp. 445–453. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-41515-9_50

    Chapter  Google Scholar 

  7. Rao, S.C.S., Chawla, S.: Second order uniformly convergent numerical method for a coupled system of singularly perturbed reaction-diffusion problems with discontinuous source term. In: Knobloch, P. (ed.) BAIL 2014. LNCSE, vol. 108, pp. 233–244. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-25727-3_18

    Chapter  Google Scholar 

  8. Chandra Sekhara Rao, S., Chawla, S.: Numerical solution for a coupled system of singularly perturbed initial value problems with discontinuous source term. In: Agrawal, P., Mohapatra, R., Singh, U., Srivastava, H. (eds.) Mathematical Analysis and its Applications. Springer Proceedings in Mathematics & Statistics, vol. 143, pp. 753–764. Springer, New Delhi (2015). https://doi.org/10.1007/978-81-322-2485-3_60

    Chapter  Google Scholar 

  9. Rao, S.C.S., Chawla, S.: Numerical solution of singularly perturbed linear parabolic system with discontinuous source term. Appl. Numer. Math. 127, 249–265 (2018)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Indian Institute of Technology Delhi, Hauz Khas, New Delhi, 110 016, India

    S. Chandra Sekhara Rao & Sheetal Chawla

Authors
  1. S. Chandra Sekhara Rao
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  2. Sheetal Chawla
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Correspondence to S. Chandra Sekhara Rao .

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Editors and Affiliations

  1. IICT, Bulgarian Academy of Sciences, Sofia, Bulgaria

    Ivan Dimov

  2. Eötvös Loránd University, Budapest, Hungary

    István Faragó

  3. University of Rousse, Rousse, Bulgaria

    Lubin Vulkov

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Rao, S.C.S., Chawla, S. (2019). The Error Analysis of Finite Difference Approximation for a System of Singularly Perturbed Semilinear Reaction-Diffusion Equations with Discontinuous Source Term. In: Dimov, I., Faragó, I., Vulkov, L. (eds) Finite Difference Methods. Theory and Applications. FDM 2018. Lecture Notes in Computer Science(), vol 11386. Springer, Cham. https://doi.org/10.1007/978-3-030-11539-5_18

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  • DOI: https://doi.org/10.1007/978-3-030-11539-5_18

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-11538-8

  • Online ISBN: 978-3-030-11539-5

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