A uniformly convergent scheme for two-parameter problems having layer behaviour

References

• D. Avijit and S. Natesan, SDFEM for singularly perturbed boundary-value problems with two parameters, J. Appl. Math. Comput. 64 (2020), pp. 591–614.
   Web of Science ®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
• C. de Boor, A Practical Guide to Splines, Springer, New York, 2001.
   Google Scholar
• R.G. Durán 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
• P.A. Farrell, A.F. Hegarty, and J.J.H. Miller, et al. Robust Computational Techniques for Boundary Layers, Chapman & Hall, London, 2000.
   Google Scholar
• J. Goh, A.A. Majid, and A.I.M. Ismail, Numerical method using cubic B-spline for the heat and wave equation, Comput. Math. Appl. 62 (2011), pp. 4492–4498.
   Web of Science ®Google Scholar
• J.L. Gracia, E. O'Riordan, and M.L. Pickett, A parameter robust higher order numerical method for a singularly perturbed two-parameter problem, Appl. Numer. Math. 56 (2006), pp. 962–980.
   Web of Science ®Google Scholar
• P.W. Hemker, G.I. Shishkin, and L.P. Shishkina, ϵ-uniform schemes with high-order time-accuracy for parabolic singular perturbation problems, IMA J. Numer. Anal. 20 (2000), pp. 99–121.
   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
• M.K. Kadalbajoo and A.S. Yadaw, Finite difference, finite element and B-spline collocation methods applied to two parameter singularly perturbed boundary value problems, J. Numer. Anal. Ind. Appl. Math. 5 (2011), pp. 163–180.
   Google Scholar
• D. Kumar, A parameter-uniform method for singularly perturbed turning point problems exhibiting interior or twin boundary layers, Int. J. Comput. Math. 96 (2019), pp. 865–882.
   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, The necessity of Shishkin decompositions, Appl. Math. Lett. 14 (2001), pp. 891–896.
   Web of Science ®Google Scholar
• T. Linß, Layer-adapted meshes for one-dimensional reaction-convection-diffusion problems, J. Numer. Math. 12 (2004), pp. 193–205.
   Google Scholar
• T. Linß, A posteriori error estimation for a singularly perturbed problem with two small parameters, Int. J. Numer. Anal. Model. 7 (2010), pp. 491–506.
   Web of Science ®Google Scholar
• T. Linß, Layer-Adapted Meshes for Reaction-Convection-Diffusion Problems, Springer-Verlag, Berlin Heidelberg, 2010.
   Google Scholar
• T. Linß, and H.-G. Roos, Analysis of a finite-difference scheme for a singularly perturbed problem with two small parameters, J. Math. Anal. Appl. 289 (2004), pp. 355–366.
   Web of Science ®Google Scholar
• J.J.H. Miller, E. O'Riordan, and G.I. Shishkin, Fitted Numerical Methods for Singular Perturbation Problems: Error Estimates in the Maximum Norm for Linear Problems in One and Two Dimensions, World Scientific, Singapore, 1996.
   Google Scholar
• A.H. Nayfeh, Introduction to Perturbation Techniques, John Wiley and Sons, New York, 1993.
   Google Scholar
• M.J. Ng-Stynes, E. O'Riordan, and M. Stynes, Numerical methods for time-dependent convection-diffusion equations, J. Comput. Appl. Math. 21 (1988), pp. 289–310.
   Web of Science ®Google Scholar
• R.E. O'Malley Jr., Singular perturbations of boundary value problems for linear ordinary differential equations involving two-parameters, J. Math. Anal. Appl. 19 (1967), pp. 291–308.
   Web of Science ®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., Boundary value problems for linear systems of ordinary differential equations involving many small parameters, J. Math. Mech. 18 (1969), pp. 835–856.
   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
• R.E. O'Malley Jr., Singular perturbation methods for ordinary differential equations, In: Applied Mathematical Sciences, Springer, New York, 1991.
   Google Scholar
• E. O'Riordan and M.L. Pickett, Numerical approximations to the scaled first derivatives of the solution to a two parameter singularly perturbed problem, J. Comput. Appl. Math. 347 (2019), pp. 128–149.
   Web of Science ®Google Scholar
• E. O'Riordan, M.L. Pickett, and G.I. Shishkin, Singularly perturbed problems modelling reaction-convection-diffusion processes, Comput. Methods Appl. Math. 3 (2003), pp. 424–442.
   Google Scholar
• K.C. Patidar, A robust fitted operator finite difference method for a two-parameter singular perturbation problem, J. Diff. Equ. Appl. 14 (2008), pp. 1197–1214.
   Web of Science ®Google Scholar
• H.-G. Roos, M. Stynes, and L. Tobiska, Numerical Methods for Singular Perturbed Differential Equations: Convection-Diffusion and Flow Problems, Springer-Verlag, New York, 1996.
   Google Scholar
• H.-G. Roos, M. Stynes, and L. Tobiska, Robust numerical methods for singularly perturbed differential equations, Springer Series in Computational Mathematics Vol. 24, Springer-Verlag, Berlin, Heidelberg, 2008.
   Google Scholar
• H.-G. Roos, L. Teofanov, and Z. Uzelac, Graded meshes for higher order FEM, J. Comput. Math. 33 (2015), pp. 1–16.
   Web of Science ®Google Scholar
• H.-G. Roos and Z. Uzelac, The SDFEM for a convection-diffusion problem with two small parameters, Comput. Methods Appl. Math. 3 (2003), pp. 443–458.
   Google Scholar
• G.I. Shishkin, On finite difference fitted schemes for singularly perturbed boundary value problems with a parabolic boundary layer, J. Math. Anal. Appl. 208 (1997), pp. 181–204.
   Web of Science ®Google Scholar
• M. Shivhare, P.P. Chakravarthy, and D. Kumar, Quadratic B-spline collocation method for two-parameter singularly perturbed problem on exponentially graded mesh, Accepted in Int. J. Comput. Math..
   Google Scholar
• G. Singh and S. Natesan, Study of the NIPG method for two-parameter singular perturbation problems on several layer adapted grids, J. Appl. Math. Comput. 63 (2020), pp. 683–705.
   Web of Science ®Google Scholar
• M. Stynes and E. O'Riordan, Uniformly convergent difference schemes for singularly perturbed parabolic diffusion-convection problems without turning points, Numer. Math. 55 (1989), pp. 521–544.
   Web of Science ®Google Scholar
• K. Surla, Z. Uzelac, and L. Teofanov, The discrete minimum principle for quadratic spline discretization of a singularly perturbed problem, Math. Comput. Simul. 79 (2009), pp. 2490–2505.
   Web of Science ®Google Scholar
• J.M. Varah, A lower bound for the smallest singular value of a matrix, Linear Algebra Appl. 11 (1975), pp. 3–5.
   Web of Science ®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
• R. Vulanović, Special meshes and higher-order schemes for singularly perturbed boundary values problems, Novi Sad J. Math. 31 (2001), pp. 1–7.
   Google Scholar
• P.J.Y. Wong and R.P. Agarwal, Explicit error estimates for quintic and biquintic spline interpolation, Comput. Math. Appl. 18 (1989), pp. 701–722.
   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