Quadratic B-spline collocation method for two-parameter singularly perturbed problem on exponentially graded mesh

References

• R. Bellman, Perturbation Techniques in Mathematics, Physics and Engineering, New York: Holt Rinehart & Winston, 1964.
   Google Scholar
• C.M. Bender and S.A. Orszag, Advanced Mathematical Methods for Scientist and Engineers, McGraw-Hill, New York, 1978.
   Google Scholar
• P.H. Bhathawala and A.P. Verma, A two parameter singular perturbation solution of one dimension flow through unsaturated porous media, Appl. Math. 43 (1975), pp. 380–384.
   Google Scholar
• J. Bigge and E. Bohl, Deformations of the bifurcation diagram due to discretization, Math. Comp. 45 (1985), pp. 393–403.
   Web of Science ®Google Scholar
• M. Brdar and H. Zarin, A singularly perturbed problem with two parameters on a Bakhvalov-type mesh, J. Comput. Appl. Math. 292 (2016), pp. 307–319.
   Web of Science ®Google Scholar
• M. Brdar and H. Zarin, On graded meshes for a two-parameter singularly perturbed problem, Appl. Math. Comput. 282 (2016), pp. 97–107.
   Web of Science ®Google Scholar
• J. Chen and R.E. O'Malley Jr., On the asymptotic solution of a two-parameter boundary value problem of chemical reactor theory, SIAM J. Appl. Math. 26 (1974), pp. 717–729.
   Web of Science ®Google Scholar
• J.D. Cole and J. Kevorkian, Perturbation Methods in Applied Mathematics, Springer, New York, 1979.
   Google Scholar
• P. Constantinou and C. Xenophontos, Finite element analysis of an exponentially graded mesh for singularly perturbed problems, Comput. Methods Appl. Math. 15 (2015), pp. 135–143.
   Web of Science ®Google Scholar
• R.C. Di Prima, Asymptotic methods for an infinitely long slider squeeze-film bearing, J. Lubrication Tech. Jan. 90 (1968), pp. 173–183.
   Web of Science ®Google Scholar
• R.G. Duran and A.L. Lombardi, Finite element approximation of convection diffusion problems using graded meshes, Appl. Numer. Math. 56 (2006), pp. 1314–1325.
   Web of Science ®Google Scholar
• W. Eckhaus, Asymptotic Analysis of Singular Perturbations, North Holland, New York, 1979.
   Google Scholar
• P.A. Farrell, E. O'Riordan, and G.I. Shishkin, A class of singularly perturbed semilinear differential equations with interior layers, Math. Comp. 74 (2005), pp. 1759–1776.
   Web of Science ®Google Scholar
• J.L. Gracia, E. O'Riordan, and M.L. Pickett, A parameter robust second order numerical method for a singularly perturbed two-parameter problem, Appl. Numer. Math. 56 (2006), pp. 962–980.
   Web of Science ®Google Scholar
• M.K. Kadalbajoo and A.S. Yadaw, B-spline collocation method for a two parameter singularly perturbed convection-diffusion boundary value problems, Appl. Math. Comput. 201 (2008), pp. 504–513.
   Web of Science ®Google Scholar
• W.J. Kammerer, G.W. Reddien, and R.S. Varga, Quadratic interpolatory splines, Numer. Math. 22 (1974), pp. 241–259.
   Web of Science ®Google Scholar
• S.A. Khuri and A. Sayfy, Self-adjoint singularly perturbed second-order two-point boundary value problems: a patching approach, Appl. Math. Model. 38(11-12) (2014), pp. 2901–2914.
   Web of Science ®Google Scholar
• D. Kumar, A.S. Yadaw, and M.K. Kadalbajoo, A parameter-uniform method for two parameters singularly perturbed boundary value problems via asymptotic expansion, Appl. Math. Inf. Sci. 7 (2013), pp. 1525–1532.
   Google Scholar
• T. Linß, Layer-Adapted Meshes for Reaction-Convection-Diffusion Problems, Springer-Verlag, Berlin, Heidelberg, 2010.
   Google Scholar
• T. Linß, G. Radojev, and H. Zarin, Approximation of singularly perturbed reaction-diffusion problems by quadratic C1-splines, Numer. Algorithms 61 (2012), pp. 35–55.
   Web of Science ®Google Scholar
• T. Linß and H.-G. Roos, Analysis of a finite difference scheme for a singularly perturbed two-parameter problem, J. Math. Anal. Appl. 289 (2004), pp. 355–366.
   Web of Science ®Google Scholar
• M.J. Marsden, Quadratic spline interpolation, Bull. Amer. Math. Soc. 80 (1974), pp. 903–906.
   Web of Science ®Google Scholar
• A.H. Nayfeh, Perturbation Methods, Wiley, New York, 1973.
   Google Scholar
• R.E. O'Malley Jr., Two-parameter singular perturbation problems for second order equations, J. Math. Mech. 16 (1967), pp. 1143–1164.
   Google Scholar
• R.E. O'Malley Jr., Introduction to Singular Perturbations, Academic Press, New York, 1974.
   Google Scholar
• R.E. O'Malley Jr., Singular Perturbation Methods for Ordinary Differential Equations, Springer, New York, 1990.
   Google Scholar
• E. O'Riordan and M. Stynes, A uniformly accurate finite-element method for a singularly perturbed one-dimensional reaction-diffusion problem, Math. Comp. 47 (1986), pp. 555–570.
   Web of Science ®Google Scholar
• A.S.V. Ravi Kanth and P.M.M. Kumar, Numerical analysis of two-parameter singularly perturbed boundary value problems via fitted splines, Analysis 37 (2017), pp. 133–143.
   Google Scholar
• H.-G. Roos, Layer-adapted grids for singular perturbation problems, Z. Angew. Math. Mech. 78 (1998), pp. 291–309.
   Web of Science ®Google Scholar
• H.-G. Roos and T. Skalický, A comparison of the finite element method on Shishkin and Gartland-type meshes for convection-diffusion problems, CWI Q. 10 (1997), pp. 277–300.
   Google Scholar
• H.G. Roos and Z. Uzelac, The SDFEM for a convection-diffusion problem with two small parameters, Comput. Method Appl. Math. 3 (2003), pp. 443–458.
   Google Scholar
• M. Shivhare, P.C. Podila, and D. Kumar, A uniformly convergent quadratic B-spline collocation method for singularly perturbed parabolic partial differential equations with two small parameters, J. Math. Chem. 59 (2021), pp. 186–215.
   Web of Science ®Google Scholar
• S. Valarmathi and N. Ramanujam, Computational methods for solving two-parameter singularly perturbed boundary value problems for second-order ordinary differential equations, Appl. Math. Comput. 136 (2003), pp. 415–441.
   Web of Science ®Google Scholar
• M. Vandyke, Perturbation Methods in Fluid Mechanics, Parabolic, Stanford, CA: The Parabolic Press, 1975.
   Google Scholar
• R. Vulanović, Special meshes and higher-order schemes for singularly perturbed boundary value problems, Novi Sad J. Math. 31 (2001), pp. 1–7.
   Google Scholar
• R. Vulanović, A higher-order scheme for quasilinear boundary value problems with two small parameters, Computing 67 (2001), pp. 287–303.
   Web of Science ®Google Scholar
• Y. Wu, N. Zhang, and J. Yuan, A robust adaptive method for singularly perturbed convection-diffusion problem with two small parameters, Comput. Math. Appl. 66 (2013), pp. 996–1009.
   Web of Science ®Google Scholar
• W.K. Zahra and A.M. El Mhlawy, Numerical solution of two-parameter singularly perturbed boundary value problems via exponential spline, J. King Saud. Univ. Sci. 25 (2013), pp. 201–208.
   Google Scholar
• H. Zarin, Exponentially graded mesh for singularly perturbed problem with two small parameters, Appl. Numer. Math. 120 (2017), pp. 233–242.
   Web of Science ®Google Scholar