424
Views
30
CrossRef citations to date
0
Altmetric
Original Articles

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

, &
Pages 1439-1463 | Received 16 Jun 2006, Accepted 18 Feb 2007, Published online: 24 Sep 2007

References

  • Prandtl , L. 1905 . “ U¨ber Flüssigkeitsbewegung bei sehr kleiner Reibung ” . In Verhandlung des dritten inter-nationalen Mathematiker-Kongresses , 484 – 491 . Leipzig : Tübner .
  • Friedrichs , K. O. and Wasow , W. 1946 . Singular perturbations of non-linear oscillations . Duke Mathematical Journal , 13 : 367 – 381 .
  • Wasow , W. 1942 . “ On boundary layer problems in the theory of ordinary differential equations, Doctoral Dissertation ” . New York University .
  • Pearson , C. E. 1968 . On a differential equation of boundary layer type . Journal of Mathematics and Physics , 47 : 134 – 154 .
  • Ardema , M. D. 1983 . Singular Perturbations in Systems and Control , Edited by: Ardema , M. D. New York : Springer .
  • Axelsson , O. , Frank , L. S. and Vander Sluis , A. 1981 . Analytical and Numerical Approaches to Asymptotic Problems in Analysis , Amsterdam : North-Holland .
  • Bellman , R. 1964 . Perturbation Techniques in Mathematics, Physics and Engineering , New York : Holt, Rine Hart & Ulinston .
  • Bender , C. M. and Orszag , S. A. 1978 . Advanced Mathematical Methods for Scientists and Engineers , New York : McGraw-Hill .
  • Carrier , G. F. and Pearson , C. E. 1968 . Ordinary Differential Equations , Waltham, MA : Blaisdell .
  • Chang , K. W. and Howes , F. A. 1984 . Non-Linear Singular Perturbation Phenomena: Theory and Application , New York : Springer .
  • Cole , J. D. and Kevorkian , J. 1981 . Perturbation Methods in Applied Mathematics , New York : Springer .
  • Van Dyke , M. 1964 . Perturbation Methods in Fluid Mechanics , New York : Academic Press .
  • Eckhaus , W. 1973 . Matched Asymptotic Expansions and Singular Perturbations , Amsterdam : North-Holland .
  • Eckhaus , W. and deJager , E. M. 1982 . Theory and Applications of Singular Perturbations, Lecture Notes in Mathematics 942 , Edited by: Eckhaus , W. and deJager , E. M. Berlin : Springer-Verlag .
  • Erdelyi , A. 1956 . Asymptotic Expansions , New York : Dover Publications .
  • Hemker , P. W. and Miller , J. J.H. 1979 . Numerical Analysis of Singular Perturbation Problems , New York : Academic Press .
  • Holmes , M. H. 1995 . Introduction to Perturbation Methods , Berlin : Springer-Verlag .
  • Kaplan , S. 1967 . Fluid Mechanics and Singular Perturbations , New York : Academic Press .
  • Kevorkian , J. and Cole , J. D. 1981 . Perturbation Methods in Applied Mathematics , New York : Springer .
  • O'Malley , R. E. 1974 . Introduction to Singular Perturbations , New York : Academic Press .
  • Miller , J. J. H. 1993 . Application of Advanced Computational Methods for Boundary and Interior Layers , Edited by: Miller , J. J. H. Dublin : Boole Press .
  • Morton , K. W. 1996 . Numerical Solution of Convection–Diffusion Problems , London : Chapman & Hall .
  • Nayfeh , A. H. 1973 . Perturbation Methods , New York : John Wiley .
  • Quarteroni , A. and Valli , A. 1994 . Numerical Approximation of Partial Differential Equations , Berlin : Springer-Verlag .
  • Verhulst , F. Asymptotic Analysis, Lecture Notes in Mathematics 711 , Berlin : Springer-Verlag .
  • Willoughby , R. A. 1974 . Stiff Differential Systems , Edited by: Willoughby , R. A. New York : Plenum Press .
  • Kadalbajoo , M. K. and Reddy , Y. N. 1989 . Asymptotic and numerical analysis of singular perturbation problems: a survey . Applied Mathematics and Computation , 30 : 223 – 259 .
  • Kadalbajoo , M. K. and Patidar , K. C. 2002 . A survey of numerical techniques for solving singularly perturbed ordinary differential equations . Applied Mathematics and Computation , 130 : 457 – 510 .
  • Bawa , R. K. and Natesan , S. 2005 . A computational method for self adjoint singular perturbation problems using quintic spline . International Journal of Computers and Mathematics with Applications , 50 : 1371 – 1382 .
  • Stojanovic , M. 2005 . Global convergence method for singularly perturbed boundary value problems . Journal of Computational and Applied Mathematics , 181 : 326 – 335 .
  • Bawa , R. K. 2005 . Spline based computational technique for linear singularly perturbed boundary value problems . Applied Mathematics and Computation , 167 : 225 – 236 .
  • Doolan , E. P. , Miller , J. J.H. and Schilder , W. H.A. 1980 . Uniform Numerical Methods for Problems with Initial and Boundary Layers , Dublin : Boole Press .
  • Aziz , T. and Khan , A. 2002 . A spline method for second order singularly perturbed boundary value problems . Journal of Computational and Applied Mathematics , 147 : 445 – 452 .
  • Kadalbajoo , M. K. and Patidar , K. C. 2002 . Spline techniques for the numerical solution of singular perturbation problems . Journal of Optimization Theory and Applications , 112 : 575 – 594 .
  • Wang , X.-Y. and Jiang , Y.-L. 2005 . A general method for solving singularly perturbed impulsive differential equations with two-point boundary conditions . Applied Mathematics and Computation , 171 : 775 – 806 .
  • Mohanty , R. K. and Arora , U. 2006 . A family of non-uniform mesh tension spline methods for singularly perturbed two-point singular boundary value problems with significant first derivatives . Applied Mathematics and Computation , 172 : 531 – 544 .
  • Reddy , Y. N. and Pramod Chakravarthy , P. 2004 . Numerical patching method for singularly perturbed two-print boundary value problems using cubic splines . Applied Mathematics and Computation , 149 : 441 – 468 .
  • Wang , L. 2004 . A novel method for a class of non-linear singular perturbation problems . Applied Mathematics and Computation , 156 : 847 – 856 .
  • Kadalbajoo , M. K. and Patidar , K. C. 2003 . Exponentially fitted spline in compression for the numerical solution of singular perturbation problems . International Journal of Computers and Mathematics with Applications , 46 : 751 – 767 .
  • Berger , A. E. , Solomon , J. M. , Ciment , M. , Leventhal , S. H. and Weinberg , B. C. 1980 . Generalized operator compact implicit schemes for boundary layer problems . Journal of Mathematical Computation , 35 : 695 – 731 .
  • Valarmathi , S. and Ramanujam , N. 2003 . Computational methods for solving two parameter singularly perturbed boundary value problems for second order ordinary differential equations . Applied Mathematics and Computation , 136 : 415 – 441 .
  • Kadalbajoo , M. K. and Patidar , K. C. 2002 . Numerical solution of singularly perturbed two-point boundary value problems by spline in tension . Applied Mathematics and Computation , 131 : 299 – 320 .
  • Beckett , G. and Mackenzie , J. A. 2001 . On a uniformly accurate finite difference approximation of a singularly perturbed reaction-diffusion problem using grid equidistribution . Journal of Computational and Applied Mathematics , 131 : 381 – 405 .
  • Vigo-Aguiar , J. and Natesan , V. 2006 . An efficient numerical method for singular perturbation problems . Journal of Computational and Applied Mathematics , 192 : 132 – 141 .
  • Roos , H. G. , Stynes , M. and Tobiska , L. 1996 . Numerical Method for Singularly Perturbed Differential Equations , New York : Springer .
  • Vigo-Aguiar , J. and Ferrandiz , J. M. 1998 . A general procedure for the adaptation of multi-step algorithms to the integration of oscillatory problems . SIAM Journal of Numerical Analysis , 35 : 1684 – 1708 .
  • Mohanty , R. K. and Jha , N. 2005 . A class of variable mesh spline in compression methods for singularly perturbed two-point singular boundary value problems . Applied Mathematics and Computation , 168 : 704 – 716 .
  • Vigo-Aguiar , J. , Natesan , S. and Ramanujam , N. 2003 . A numerical algorithm for singular perturbation problems exhibiting weak boundary layers . International Journal of Computers and Mathematics with Applications , 45 : 469 – 479 .
  • Natesan , S. and Ramanujam , N. 2002 . ‘Shooting method’ for the solution of singularly perturbed two-point boundary value problems having less severe boundary layer . Applied Mathematics and Computation , 133 : 623 – 641 .
  • Vercelj , V. , Adzic , N. and Uzeloc , Z. 1991 . A numerical asymptotic solution for singular perturbation problems . International Journal of Computational Mathematics , 39 : 229 – 238 .
  • Natesan , S. and Ramanujam , N. 2002 . An asymptotic numerical method for singularly perturbed Robin problem I . Applied Mathematics and Computation , 126 : 97 – 107 .
  • Kadalbajoo , M. K. and Patidar , K. C. 2001 . Variable mesh spline approximation method for solving singularly perturbed turning point problems having boundary layer(s) . International Journal of Computers and Mathematics with Applications , 42 : 1439 – 1453 .
  • Zhou , M. , Ge , W. and Du , Z. 2005 . Singular perturbations for third-order non-linear multi-point boundary value problem . Journal of Differential Equations , 218 : 69 – 90 .
  • Valarmathi , S. and Ramanujam , N. 2002 . A computational method for solving boundary value problems for third-order singularly perturbed ordinary differential equations . Applied Mathematics and Computation , 129 : 345 – 373 .
  • Valarmathi , S. and Ramanujam , N. 2002 . An asymptotic numerical method for singularly perturbed third-order ordinary differential equation of convection- diffusion type . International Journal of Computers and Mathematics with Applications , 44 : 693 – 710 .
  • Valarmathi , S. and Ramanujam , N. 2002 . An asymptotic numerical fitted mesh method for singularly perturbed third-order ordinary differential equation of reaction-diffusion type . Applied Mathematics and Computation , 132 : 87 – 104 .
  • Miller , J. J.H. , O'Riordan , E. and Shishkin , G. I. 1996 . “ Fitted numerical methods for singular perturbation problems ” . In Error Estimates in the Maximum Norm for Linear Problems in One and Two Dimensions , Singapore : World Scientific . In:
  • Lubuma , J. M.S. and Patidar , K. C. 2007 . Uniformly convergent non-standard finite difference methods for self adjoint singular perturbation problems . Journal of Computational and Applied Mathematics , in press
  • Kadalbajoo , M. K. and Aggarwal , V. K. 2005 . Fitted mesh B-spline collocation method for solving self adjoint singularly perturbed boundary value problems . Applied Mathematics and Computation , 161 : 973 – 987 .
  • Potter , M. H. and Weinberger , H. F. 1967 . Maximum Principles in Differential Equations , Englewood Cliffs, NJ : Prentice Hall .
  • Ahlberg , J. M. , Nilson , E. N. and Walsh , J. L. 1967 . The Theory of Splines and their Applications , New York : Academic Press .
  • Micula , G. 1999 . Handbook of Splines , Dordrecht : Kluwer Academic .
  • Schumaker , L. L. 1981 . Spline Functions: Basis Theory , Melbourne, FL : Krieger .
  • Patidar , K. C. 2005 . High order fitted operator numerical method for self- adjoint singular perturbation problems . Applied Mathematics and Computation , 171 : 547 – 566 .
  • Ramanujam , N. and Shanthi , V. 2006 . Asymptotic numerical method for boundary value problems for singularly perturbed fourth-order ordinary differential equations with a weak interior layer . Applied Mathematics and Computation , 172 : 252 – 266 .
  • Ramanujam , N. and Shanthi , V. 2004 . Computational methods for reaction–diffusion problems for fourth-order ordinary differential equations with a small parameter at the highest derivative . Applied Mathematics and Computation , 147 : 97 – 113 .
  • Ramanujam , N. and Shanthi , V. 2004 . A boundary value technique for boundary value problems for singularly perturbed fourth-order ordinary differential equations . International Journal of Computers and Mathematics with Applications , 47 : 1673 – 1688 .
  • Shanthi , V. and Ramanujam , N. 2003 . Asymptotic numerical methods for singularly perturbed fourth-order ordinary differential equations of reaction–diffusion type . International Journal of Computers and Mathematics with Applications , 46 : 463 – 478 .
  • Shanthi , V. and Ramanujam , N. 2002 . Asymptotic numerical methods for singularly perturbed fourth-order ordinary differential equations of convection–diffusion type . Applied Mathematics and Computation , 133 : 559 – 579 .
  • Shanthi , V. and Ramanujam , N. 2002 . A numerical method for boundary value problems for singularly perturbed fourth-order ordinary differential equations . Applied Mathematics and Computation , 129 : 269 – 294 .
  • Kadalbajoo , M. K. and Sharma , K. K. 2005 . Numerical treatment for singularly perturbed non-linear differential difference equations with negative shift . Nonlinear Analysis , 63 : e1909 – e1924 .
  • Howes , F. A. 1976 . “ Singular perturbations and differential inequalities ” . In Memoirs of the American Mathematical Society 168 , Providence, RI : American Mathematical Society .
  • Kadalbajoo , M. K. and Sharma , K. K. 2004 . Numerical analysis of singularly perturbed delay differential equations with layer behavior . Applied Mathematics and Computation , 157 : 11 – 28 .
  • Ramos , J. I. 2005 . A smooth locally-analytical technique for singularly perturbed two-point boundary value problems . Applied Mathematics and Computation , 163 : 1123 – 1142 .
  • Surla , K. and Uzelac , Z. 2004 . A uniformly accurate spline collocation method for a normalized flux . Journal of Computational and Applied Mathematics , 166 : 291 – 305 .
  • Amiraliyev , G. M. and Cakir , M. 2005 . A finite difference method for the singularly perturbed problem with non-local boundary condition . Applied Mathematics and Computation , 160 : 539 – 549 .
  • Amiraliyev , G. M. 1990 . Difference method for the solution of one problem of the theory of dispersive waves . Differential Equations , 26 : 2146 – 2154 . (Russian)
  • Amiraliyev , G. M. and Cakir , M. 2000 . A uniformly convergent difference scheme for a singularly perturbed problem with convective term and zeroth-order reduced equation . International Journal of Applied Mathematics , 2 : 1407 – 1419 .
  • Amiraliyev , G. M. and Cakir , M. 2002 . Numerical solution of the singularly perturbed problem with non-local boundary condition . Applied Mathematics and Mechanics (English edn) , 23 : 755 – 764 .
  • Prasad , S. N. and Salomon , J. B. 2005 . A new method for the analytical solution of a degenerate diffusion equation . Advances in Water Resources , 28 : 1091 – 1101 .
  • Shampine , L.F. 1973 . Some singular concentration dependent diffusion problems . ZAMM—Journal of Applied Mathematics and Mechanics , 53 : 421 – 422 .
  • Heaslet , M. A. and Alksne , A. 1961 . Diffusion from a fixed surface with a concentration dependent coefficient . Journal of the Society of Industrial and Applied Mathematics , 9 : 584 – 596 .
  • Babu , D. K. and Van Genuchten , M. A. 1979 . Similarity solution to a non-linear diffusion equation of the singular type: a uniformly valid solution by perturbations . Quarterly Journal of Applied Mathematics , 37 : 11 – 21 .
  • Parlange , J.-Y. , Hogarth , W. L. , Parlange , M. B. , Haverkamp , R. , Barry , D. A. , Ross , P. J. and Steenhuis , T. S. 1998 . Approximate analytical solution of the non-linear diffusion equation for arbitrary boundary conditions . Transport Porous Medium , 30 : 45 – 55 .
  • Syam , M. I. and Attili , B. S. 2005 . Numerical solution of singularly perturbed fifth-order two-point boundary value problem . Applied Mathematics and Computation , 170 : 1085 – 1094 .
  • Amiraliyev , G. M. and Duru , H. 2005 . A note on a parameterized singular perturbation problem . Journal of Computational and Applied Mathematics , 182 : 233 – 242 .
  • Kadabajoo , M. K. and Sharma , K. K. 2005 . Numerical treatment of a mathematical model arising from a model of neuronal variability . Journal of Mathematical Analysis , 307 : 606 – 627 .
  • Patidar , K. C. and Sharma , K. K. 2007 . ϵ-Uniformly convergent non-standard finite difference methods for singularly perturbed differential difference equations with small delay . Applied Mathematics and Computation , in press
  • Mickens , R. E. 1994 . Non-Standard Finite Difference Models of Differential Equations , Singapore : World Scientific .
  • Amiraliyev , G. M. , Kudu , M. and Duru , H. 2007 . Uniform difference method for a parameterized singular perturbation problem . Applied Mathematics and Computation , in press
  • Kellogg , R. B. and Stynes , M. 2005 . Corner singularities and boundary layers in a simple convection–diffusion problem . Journal of Differential Equations , 213 : 81 – 120 .
  • Valanarasu , T. and Ramanujam , N. 2004 . An asymptotic initial-value method for boundary value problems for a system of singularly perturbed second order ordinary differential equations . Applied Mathematics and Computation , 147 : 227 – 240 .
  • Miller , J. J.H. , Shishkin , G. I. , Koren , B. and Shishkina , L. P. 2004 . Grid approximation of a singularly perturbed boundary value problem modeling heat transfer in the case of flow over a flat plate with suction of the boundary layer . Journal of Computational and Applied Mathematics , 166 : 221 – 232 .
  • Shishkin , G. I. 2004 . Discrete approximation of solutions and derivatives for a singularly perturbed parabolic convection-diffusion equation . Journal of Computational and Applied Mathematics , 166 : 247 – 266 .
  • Kopteva , N. and Stynes , M. 2004 . Numerical analysis of a singularly perturbed non-linear reaction-diffusion problem with multiple solutions . Applied Numerical Mathematics , 51 : 273 – 288 .
  • Beckett , G. and Mackenzie , J. A. 2000 . Convergence analysis of finite difference approximations on equidistributed grids to a singularly perturbed boundary value problem . Applied Numerical Mathematics , 35 : 87 – 109 .
  • Mohanty , R. K. and Singh , S. 2006 . A new fourth-order discretization for singularly perturbed two-dimensional non-linear elliptic boundary value problems . Applied Mathematics and Computation , 175 : 1400 – 1414 .
  • Qiu , Y. , Sloan , D. M. and Tang , T. 2000 . Numerical solution of a singularly perturbed two- point boundary value problem using equidistribution: analysis of convergence . Journal of Computational and Applied Mathematics , 116 : 121 – 143 .
  • Ramos , J. I. 2005 . Linearization techniques for singularly perturbed initial-value problems of ordinary differential equations . Applied Mathematics and Computation , 163 : 1143 – 1163 .

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.