A recent survey on computational techniques for solving singularly perturbed boundary value problems

This survey paper contains a surprisingly large amount of material and indeed can serve as an introduction to some of ideas and methods of singular perturbation theory. In continuation of a survey performed earlier, this paper limits its coverage to some standard numerical methods developed by numerous researchers between 2000 and 2005. A summary of the results of some recent methods is presented and this leads to conclusions and recommendations regarding methods to use on singular perturbation problems. Because of space constraints, we considered one-dimensional singularly perturbed boundary value problems only.

[1]  S. Natesan,et al.  "Shooting method" for the solution of singularly perturbed two-point boundary-value problems having less severe boundary layer , 2002, Appl. Math. Comput..

[2]  W. Eckhaus,et al.  Theory and Applications of Singular Perturbations , 1982 .

[3]  Rajesh K. Bawa,et al.  Spline based computational technique for linear singularly perturbed boundary value problems , 2005, Appl. Math. Comput..

[4]  Srinivasan Natesan,et al.  A computational method for self-adjoint singular perturbation problems using quintic spline , 2005 .

[5]  Zorica Uzelac,et al.  A numerical-asymptotic solution for singular perturbation problems , 1991, Int. J. Comput. Math..

[6]  J. Mackenzie,et al.  On a uniformly accurate finite difference approximation of a singulary peturbed reaction-diffusion problem using grid equidistribution , 2001 .

[7]  John J. H. Miller.  Fitted Numerical Methods for Singular Perturbation Problems: Error Estimates in the Maximum Norm for Linear Problems in One and Two Dimensions , 1996 .

[8]  M. D. Ardema.  Singular perturbations in systems and control , 1983 .

[9]  N. Ramanujam,et al.  A boundary value technique for boundary value problems for singularly perturbed fourth-order ordinary differential equations , 2004 .

[10]  M. Stojanović.  Global convergence method for singularly perturbed boundary value problems , 2005 .

[11]  V. Shanthi,et al.  Asymptotic numerical methods for singularly perturbed fourth order ordinary differential equations of convection-diffusion type , 2002, Appl. Math. Comput..

[12]  Zorica Uzelac,et al.  A uniformly accurate spline collocation method for a normalized flux , 2004 .

[13]  Gabil M. Amiraliyev,et al.  A finite difference method for the singularly perturbed problem with nonlocal boundary condition , 2005, Appl. Math. Comput..

[14]  Jesús Vigo-Aguiar,et al.  An efficient numerical method for singular perturbation problems , 2006 .

[15]  Y. N. Reddy,et al.  Numerical patching method for singularly perturbed two-point boundary value problems using cubic splines , 2004, Appl. Math. Comput..

[16]  Lutz Tobiska,et al.  Numerical Methods for Singularly Perturbed Differential Equations , 1996 .

[17]  S. Valarmathi,et al.  An asymptotic numerical fitted mesh method for singularly perturbed third order ordinary differential equations of reaction-diffusion type , 2002, Appl. Math. Comput..

[18]  V. Shanthi,et al.  Computational methods for reaction-diffusion problems for fourth order ordinary differential equations with a small parameter at the highest derivative , 2004, Appl. Math. Comput..

[19]  N. Ramanujam,et al.  An asymptotic-numerical method for singularly perturbed Robin problems-I , 2002, Appl. Math. Comput..

[20]  V. Shanthi,et al.  A numerical method for boundary value problems for singularly perturbed fourth-order ordinary differential equations , 2002, Appl. Math. Comput..

[21]  C. Pearson.  On a Differential Equation of Boundary Layer Type , 1968 .

[22]  M. Holmes.  Introduction to Perturbation Methods , 1995 .

[23]  Alan E. Berger,et al.  Generalized OCI schemes for boundary layer problems , 1980 .

[24]  M. K. Kadalbajoo,et al.  Variable mesh spline approximation method for solving singularly perturbed turning point problems having boundary layer(s) , 2001 .

[25]  M. Çakir,et al.  Numerical solution of the singularly perturbed problem with nonlocal boundary condition , 2002 .

[26]  Fluid mechanics and singular perturbations , 1967 .

[27]  D. K. Babu,et al.  A similarity solution to a nonlinear diffusion equation of the singular type: a uniformly valid solution by perturbations , 1979 .

[28]  George Beckett,et al.  Convergence analysis of finite difference approximations on equidistributed grids to a singularly perturbed boundary value problem , 2000 .

[29]  Rajesh Sharma,et al.  Asymptotic analysis , 1986 .

[30]  J.I. Ramos.  Linearization techniques for singularly-perturbed initial-value problems of ordinary differential equations , 2005, Appl. Math. Comput..

[31]  R. K. Mohanty,et al.  A class of variable mesh spline in compression methods for singularly perturbed two point singular boundary value problems , 2005, Appl. Math. Comput..

[32]  R. K. Mohanty,et al.  A family of non-uniform mesh tension spline methods for singularly perturbed two-point singular boundary value problems with significant first derivatives , 2006, Appl. Math. Comput..

[33]  F. Howes.  Singular perturbations and differential inequalities , 1976 .

[34]  Weigao Ge,et al.  Singular perturbations for third-order nonlinear multi-point boundary value problem , 2005 .

[35]  P. Zweifel.  Advanced Mathematical Methods for Scientists and Engineers , 1980 .

[36]  Natalia Kopteva,et al.  Numerical analysis of a singularly perturbed nonlinear reaction-diffusion problem with multiple solutions , 2004 .

[37]  M. K. Kadalbajoo,et al.  Numerical treatment for singularly perturbed nonlinear differential difference equations with negative shift , 2005 .

[38]  R. Bellman.  Perturbation techniques in mathematics, physics, and engineering , 1972 .

[39]  M. K. Kadalbajoo,et al.  Exponentially fitted spline in compression for the numerical solution of singular perturbation problems , 2003 .

[40]  Mohan K. Kadalbajoo,et al.  Fitted mesh B-spline collocation method for solving self-adjoint singularly perturbed boundary value problems , 2005, Appl. Math. Comput..

[41]  M. K. Kadalbajoo,et al.  A survey of numerical techniques for solving singularly perturbed ordinary differential equations , 2002, Appl. Math. Comput..

[42]  Discrete approximations of solutions and derivatives for a singularly perturbed parabolic convection-diffusion equation , 2004 .

[43]  J. J. Miller,et al.  Fitted Numerical Methods for Singular Perturbation Problems , 1996 .

[44]  Kailash C. Patidar,et al.  High order fitted operator numerical method for self-adjoint singular perturbation problems , 2005, Appl. Math. Comput..

[45]  R. K. Mohanty,et al.  A new fourth order discretization for singularly perturbed two dimensional non-linear elliptic boundary value problems , 2006, Appl. Math. Comput..

[46]  J. Lubuma,et al.  Uniformly convergent non-standard finite difference methods for self-adjoint singular perturbation problems , 2006 .

[47]  Kapil K. Sharma,et al.  ε-Uniformly convergent non-standard finite difference methods for singularly perturbed differential difference equations with small delay , 2006, Appl. Math. Comput..

[48]  K. O. Friedrichs,et al.  Singular perturbations of non-linear oscillations , 1946 .

[49]  M. K. Kadalbajoo,et al.  Numerical analysis of singularly perturbed delay differential equations with layer behavior , 2004, Appl. Math. Comput..

[50]  M. A. Heaslet,et al.  DIFFUSION FROM A FIXED SURFACE WITH A CONCENTRATION DEPENDENT COEFFICIENT , 1961 .

[51]  H. Weitzner,et al.  Perturbation Methods in Applied Mathematics , 1969 .

[52]  José M. Ferrándiz,et al.  A General Procedure For the Adaptation of Multistep Algorithms to the Integration of Oscillatory Problems , 1998 .

[53]  R. Serfling,et al.  Asymptotic Expansions—I , 2006 .

[54]  P. Hartman.  Ordinary Differential Equations , 1965 .

[55]  J. Parlange,et al.  Approximate Analytical Solution of the Nonlinear Diffusion Equation for Arbitrary Boundary Conditions , 1998 .

[56]  A. Quarteroni,et al.  Numerical Approximation of Partial Differential Equations , 2008 .

[57]  S. Valarmathi,et al.  A computational method for solving boundary value problems for third-order singularly perturbed ordinary differential equations , 2002, Appl. Math. Comput..

[58]  Tao Tang,et al.  Numerical solution of a singularly perturbed two-point boundary value problem using equidistribution: analysis of convergence , 2000 .

[59]  L. Segel,et al.  Introduction to Singular Perturbations. By R. E. O'MALLEY, JR. Academic Press, 1974. $ 16.50. , 1975, Journal of Fluid Mechanics.

[60]  Spline Techniques for the Numerical Solution of Singular Perturbation Problems , 2002 .

[61]  V. Shanthi,et al.  Asymptotic numerical method for boundary value problems for singularly perturbed fourth-order ordinary differential equations with a weak interior layer , 2006, Appl. Math. Comput..

[62]  Yao-Lin Jiang,et al.  A general method for solving singular perturbed impulsive differential equations with two-point boundary conditions , 2005, Appl. Math. Comput..

[63]  Gabil M. Amiraliyev,et al.  Uniform difference method for a parameterized singular perturbation problem , 2006, Appl. Math. Comput..

[64]  Ralph A. Willoughby,et al.  Stiff differential systems , 1974 .

[65]  R. B. Kellogg,et al.  Numerical analysis of singular perturbation problems , 1983 .

[66]  M. K. Kadalbajoo,et al.  Asymptotic and numerical analysis of singular perturbation problems: A survey , 1989 .

[67]  O. Axelsson,et al.  Analytical and Numerical Approaches to Asymptotic Problems in Analysis. , 1983 .

[68]  F. A. Howes,et al.  Nonlinear Singular Perturbation Phenomena , 1984 .

[69]  N. Ramanujam,et al.  Asymptotic numerical methods for singularly perturbed fourth-order ordinary differential equations of reaction-diffusion type , 2003 .

[70]  H. Weinberger,et al.  Maximum principles in differential equations , 1967 .

[71]  Wil H. A. Schilders,et al.  Uniform Numerical Methods for Problems with Initial and Boundary Layers , 1980 .

[72]  Sanda Micula,et al.  Handbook of Splines , 1998 .

[73]  S. Valarmathi,et al.  Computational methods for solving two-parameter singularly perturbed boundary value problems for second-order ordinary differential equations , 2003, Appl. Math. Comput..

[74]  Vimal Singh,et al.  Perturbation methods , 1991 .

[75]  J. L. Walsh,et al.  The theory of splines and their applications , 1969 .

[76]  R. Bruce Kellogg,et al.  Corner singularities and boundary layers in a simple convection–diffusion problem☆ , 2005 .

[77]  Gabil M. Amiraliyev,et al.  A note on a parameterized singular perturbation problem , 2005 .

[78]  Jesús Vigo-Aguiar,et al.  A numerical algorithm for singular perturbation problems exhibiting weak boundary layers , 2003 .

[79]  W. Tollmien,et al.  Über Flüssigkeitsbewegung bei sehr kleiner Reibung , 1961 .

[80]  M. K. Kadalbajoo,et al.  Numerical treatment of a mathematical model arising from a model of neuronal variability , 2005 .

[81]  Juan I. Ramos.  A smooth locally-analytical technique for singularly perturbed two-point boundary-value problems , 2005, Appl. Math. Comput..

[82]  A. Erdélyi.  MATCHED ASYMPTOTIC EXPANSIONS AND SINGULAR PERTURBATIONS , 1974 .

[83]  Tariq Aziz,et al.  A spline method for second-order singularly perturbed boundary-value problems , 2002 .

[84]  S. Valarmathi,et al.  An asymptotic numerical method for singularly perturbed third-order ordinary differential equations of convection-diffusion type , 2002 .

[85]  Lei Wang.  A novel method for a class of nonlinear singular perturbation problems , 2004, Appl. Math. Comput..

[86]  John J. H. Miller,et al.  Grid approximation of a singularly perturbed boundary value problem modelling heat transfer in the case of flow over a flat plate with suction of the boundary layer , 2004 .

[87]  N. Ramanujam,et al.  An asymptotic initial value method for boundary value problems for a system of singularly perturbed second order ordinary differential equations , 2004, Appl. Math. Comput..

[88]  W. Eckhaus.  Matched Asymptotic Expansions and Singular Perturbations , 1973 .

[89]  M. K. Kadalbajoo,et al.  Numerical solution of singularly perturbed two-point boundary value problems by spline in tension , 2002, Appl. Math. Comput..

[90]  L. Shampine.  Some Singular Concentration Dependent Diffusion Problems , 1973 .

[91]  A. Macgillivray,et al.  AN INTRODUCTION TO SINGULAR PERTURBATIONS , 2000 .

[92]  J. Salomon,et al.  A new method for the analytical solution of a degenerate diffusion equation , 2005 .

[93]  Muhammed I. Syam,et al.  Numerical solution of singularly perturbed fifth order two point boundary value problem , 2005, Appl. Math. Comput..