REFERENCES

1. G. Adomian and R. Rach, Noise terms in decomposition solutions, series,

Comput. Math. Appl., 24(11),(1992) 61-64.

2. J.H. Ahlberg, E.N. Nilson and J.L. Walsh, The Theory of Splines and their

Applications, Academic Press, New York (1967).

3. E.L. Albasiny and W.D.Hoskins, Cubic spline solutions to two point boundary

value problems, Comput. J., 12 (1969) 151-153.

4. J. Albrecht, Zum differenzenverfahren bei parabolischen differential-

gleichungen, Z. Angew. Math, and Mech., 37(5-6) (1957) 202-212.

5. E.A.Al-Said, Spline solutions for system of second order boundary value

problems, Int .J. Comput. Math., 62(1996)143-154.

6. K. Amaratunga and J. Williams, Wavelet-Galerkin solutions for one-

dimensional partial differential equations. Int. J. Numer. Methods in Eng.,

37(1994)2703-2716.

7. Archer, An 0(h'') cubic spline collocation method for quasilinear parabolic

equations, SIAM J. Numer. Anal., 14(1977) 620-637.

8. U. M. Ascher, R.M. Mettheij and R.D.RusseU, Numerical Solution of

Boundary-Value Problems for Ordinary Differential Equations, Prentice Hall,

Englewood Cliff, New Jersey(1988).

9. T. Aziz, Numerical methods for differential equations using spline function

approximations, Ph.D Thesis, IIT Delhi (1983).

10. T. Aziz and A. Khan, A spline method for second order singularly-perturbed

boundary value problems, J. Comput. Appl. Math.,147(2) (2002) 445-452.

11. T. Aziz and A. Khan, Quintic Spline Approach to the Solution of a Singularly-

Perturbed Boundary Value Problems, Journal of Optimization Theory and

89

Applications, 112 (2002) 517-527.

12. T. Aziz and M. Kumar, A fourth order finite difference method based on non­

uniform mesh for a class of singular two point boundary- value problems,

Journal of Comput. Appl. Math, 136(2001) 337-342.

13. C. Baiocchi and A. Capelo , Variational and Quasi- Variational Inequalities,

John Wiley and Sons, New York (1984).

14. G. Beylkin, R. Coifman and V. Rokhlin, Wavelets in numerical analysis, in M.

Ruskai at al., eds., Wavelets and Their Applications, Jones and Bartlett, Bostan,

(1992)181-102.

15. W. G. Bickley, Piecewise cubic interpolation and two point boundary value

problems, Comput. J., 11(1968) 206-208.

16. M. M. Chawla, A fourth order finite difference method based on uniform mesh

for singular two-point boundary-value problems, J. Comput. App.Math.,17

(1987)359-364.

17. M. M. Chawla and C. P. Katti, Finite difference methods and their

convergence for a class of singular two-point boundary-value problems, Numer.

Math., 39(1982)341-350.

18. M. Chawla and R. Subramanian, A new fourth order cubic spline method for

non-linear two point boundary value problems. Intern. J. Comput. Math.

22(1987)321-341.

19. , A fourth order spline method for singular two point boundary value

problems, J. Comput. App.Math., 21(1988) 189-202.

20. , High accuracy quintic spline solution of fourth order two point

boundary value problems, Intem. J. Comput. Math., 31(1989) 87-94.

21. R.C.Y. Chin and R. Krasny, A hybrid asymptotic finite element method for

stiff two-point boundary value problems, SIAM J. Sci. Stat. Comput.,

90

4(1983)229-243.

22. C. Chui, An Introduction to Wavelets, Academic Press, Boston, 1992a.

23. L. CoUatz, The Numerical Treatment of Differential Equations, Springer-

Verlag, Berlin (1966).

24. , Hermitian methods for initial value problems in partial

differential equations, Topics in numerical analysis (Ed J. J.H. Miller),

Academic Press, (1973) 41-61.

25. S.D. Conte, A stable implicit finite difference approximation to a fourth

order parabolic equation, J. Assoc. Comput .Mech. 4 (1957)18-23.

26. S.D. Conte and W.C. Royster, Convergence of finite difference solutions

to a solution of the equation of the vibrating rod ,Proc. Amer. Math. Soc,

7(1956)742-749.

27. R.W. Cottle, F. Giannessi and J.L. Lions, Variational Inequalities and

Complimentary Problems: Theory and Applications, Oxford, U.K. (1984)

28. S. H. Crandall, Numerical treatment of a fourth order partial differential

equations, J. Assoc.Comput.Mech.,l(1954)l 11-118.

29. J.Crank, Free and Moving Boundary Problems, Clarendon Press, Oxford

U.K.(1984).

30. J.W. Daniel and B.K. Swartz, Extrapolated collocation for two point

boundary value problems using cubic splines, J. Inst. Math. Applies.,

16(1975) 161-174.

31. P. J. Davis, Interpolation and Approximation, Blaisdell, New York (1963).

32. C. De Boor, The method of projections as applied to the numerical solution of

two point boundary value problems using cubic splines, Ph.D

Thesis,University of Michigan (1966).

33. , On uniform approximations by splines, J. Approx. Theory, 1

91

(1968)219-235.

34. jPractical guide to Splines, Springer- Verlag, Berlin New York

(1978).

35. E.P.Doolan, J.J. H.Miller and W.H.A.Schilders, Uniform Numerical

Methods for Problems with Initial and Boundary Layers, Boole Press,

Dublin, Ireland (1980).

36. J. Douglas, The solution of the diffusion equation by a high order correct

difference equation, J. Math. Phys.,35 (1956) 145-151.

37. D.J. Evans, A stable explicit method for the finite difference solution of a

fourth order parabolic partial differential equation, Comput. J., 8 (1965)

280-287.

38. D.J. Evans and W.S. Yousif, A note on solving the fourth order parabolic

equation by the AGE method, Int. J.Comput. Math., 40 (1991) 93-97.

39. G. Fairweather and A. R. Gourlay, Some stable difference approximations

to a fourth order parabolic partial differential equation, Math. Comput.,

21(1967)1-11.

40. G. Fairweather and D. Meade, A survey of spline collocation methods for

the numerical solution of differential equations. Mathematics for Large

Scale Computing (J.C. Diaz ed.) Lecture Notes in Pure and Applied

Maths, New York, Marcel Dekker, 120(1989) 297-341.

41. J. Ferguson and S. Pruess, Shape preserving interpolation by parametric

piecewise cubic polynomials, Comput. Aided Design, 23(1991) 498-505.

42. L. Fox, The Numerical Solution of two point Boundary-Value Problems in

Ordinary Differential Equations, Oxford University Press(1957).

43. C.Froberg, Numerical Mathematics, Theory and Computer Applications,

Benjamin/Cummings, Reading ,MA (1985).

92

44. D.J. Fyfe , The use of cubic splines in the solution of two point boundary-

value problems, Comput. J., 12(1969)188-192.

45. D. J. Fyfe, The use of cubic splines in the solution of two-point boundary-value

problems, Computer Journal, 12 (1973) 328-344.

46. , Linear dependence relations connecting equal interval N th

degree splines and their derivatives, J. Inst. Math. Applics.,7(1971) 398-

406.

47. R. Glowinski, J.L.Lions and R. Tremolieres, Numerical Analysis of

Variational Inequalities, North-Holland, Amsterdam(1981).

48. T.N.E. Greville, Introduction to Spline Functions In : Theory and

Application of Spline Functions, Academic Press, New York (1969).

49. B. Gustafsson, A numerical method for solving singular boundary-value

problems, Numer. Math., 21(1973) 328-344.

50. P. Henrici, Discrete Variable Methods in Ordinary Differential Equations,

John Wiley, New York (1962).

51. E.N. Houstis, E.A. Vavalis and J.R. Rice, Convergence of O (h )̂ cubic

spline collocation methods for elliptic partial differential equations, SIAM

J.Numer. Anal, 25(1988).

52. M. Irodotou-EUina and E.N. Houstis, An 0(h*) quintic spline collocation

method for singular two-point boundary-value problems, BIT 28(1988)

288-301.

53. S.R.K. Iyengar and P. Jain, Spline finite difference methods for two-point

boundary-value problems, Numer. Math., 50 (1987) 363-387.

54. M.K. Jain, Numerical Solution of Differential Equations, Wiley Eastern

Ltd.,NewDelhi(1984).

93

55. M.K. Jain and T. Aziz, Numerical solution of stiff and convection-

diffusion equations using adaptive spline function approximation, App.

Math. Modelling, 7 (1983)57 - 63.

56. , Spline function approximation for differential equations,

Comput. Method Appl. Mech. Engg., 26(1981)129-143.

57. , Cubic spline solution of two-point boundary-value problems with

significant first derivatives, Comput. Method Appl. Mech. Engg., 39(1983)

83-91.

58. M.K. Jain, S.R.K. Iyengar and R.K. Jain, Numerical methods for scientific

and engineering computation. New Age Inter.(P) Ltd. Publ. (1997).

59. M.K.Jain, S.R.K. Iyengar and A.G. Lone, Higher order difference formulas

for a fourth order parabolic partial differential equation, Int. J. Numer.

Methods Engg., 10(1976) 1357-1367.

60. P. Jamet, On the convergence of finite difference approximations to one-

dimensional singular boundary-value problems, Numer. Math.,14(1970)355-

378.

61. M. K. Kadalbajoo and R. K.Bawa, Variable-mesh difference scheme for

singularly - perturbed boundary - value problems using splines. Journal

of Optim. Theory and Applies., 90( 2 )(1996) 405 -416.

62. M. K. Kadalbajoo and K.C. Patidar, Variable mesh spline in compression for

the numerical solution of singular perturbation problems. Int. J. Comp.

Math.,80(l) (2003) 83-93.

94

63. M. K. Kadalbajoo and K.C. Patidar, A survey of numerical techniques for

solving singularly-perturbed ordinary differential equations, Appl. Math.

Comput., 130 (2002) 457-510.

64. M.K. Kadalbajoo and Y.N. Reddy, Asymptotic and numerical analysis of

singular perturbation problems :A survey, Appl. Math. Comput., 30(1989)

223-259.

65. J. Kevorkin and J.D. Cole, Multiple scale and singular perturbation methods,

Springer-Verlag, New York (1996).

66. A.K. Khalifa and M.A.Noor, Quintic spline solutions of a class of contact

problems , Math. Comput. Modelling, 13(1990)51-58.

67. A.Q.M. Khaliq and E.H. Twizell, A family of second order methods for

variable coefficient fourth order parabolic partial differential equation. Intern. J.

Computer Math, 23(1987) 63-76.

68. A. Khan, Spline Solution of Differential Equations, PhD Thesis, Aligarh

Muslim University, Aligarh, India, 2002.

69. A. Khan and T. Aziz, Variable-mesh spline difference method for

singularly perturbed boundary-value problems, Journal of Indian Math.

Soc.,68 (2001).

70. A. Khan and T. Aziz, Spline method for the solution of singular two-point

boundary-value problems, Math. Sci. Res. Hot-Line, 5 (2001) 11-23.

71. N. Kikuchi and J.T. Oden, Contact Problems in Elasticity, SIAM

Publishing Co., Philadelphia (1988).

72. M. Kumar, A second-order spline finite-difference method for singular two-

point boundary-value problems, Appl. Math. Comput., 142 (2003) 283-290.

73. M. Lees, Alternating direction and semi-explicit difference methods for

95

parabolic partial differential equations, Numer. Math., 3(1961) 398-412.

74. H. Lewy and G. Stampacchia, On the regularity of the solutions of the

variational inequalities, Commun. Pure and Appl. Math., 22(1969)153-188.

75. F.R. Loscalzo, An introduction to the application of spline functions to

initial value problems. Theory and Applications of Spline Functions, edited

by T.N.E. Greville, Academic Press, New York (1969) 37-64.

76. F.R. Loscalzo and I.J. Schoenberg, On the use of spline functions for the

approximation of solution of ordinary differential equations , Tech. Summary

Report no.723, Math. Research Centre, University of Wisconsin,

Madison(1967).

77. F.R. Loscalzo and T.D. Talbot, Spline function approximations for solution

of ordinary differential equations, Bull.Amer.Math. Soc.,73(1967)438-442.

78. , Spline function approximations for solution of ordinary

differential equations, SIAM J. Numer. Anal.,4(1967)433-445.

79. M. R. Majer, Numerical solution of singularly perturbed boundary value

problems using a collocation method with tension splines, Progress in

Scientific Computing, Birkhauser, Boston, Massachusetts,5(1985)206 - 223.

80. G.L Marchuk, Methods of Numerical Mathematics, second edition.

Springer- Verlag, NewYork (1982).

81. G.Micula, Approximation solution of the differential equation with spline

functions. Math. Comput., 27(1973) 807-816.

82. G.Micula and Sanda Micula, Hand Book of Splines, Kluwer Academic

Publisher's (1999).

83. J.J.H.Miller, E.O'Riordan and G.LShishkin, Fitted numerical methods for

singular perturbation problems, World scientific,Singapore(1996).

96

84. S. Natesan and N. Ramanujam, A " Booster method" for singular

perturbation problems arising in chemical reactor theory, Appl. Math.

Comput.,100(1999) 27-48.

85. S. Natesan and N. Ramanujam, Improvement of numerical solution of self-

adjoint singular-perturbation problems by incorporation of asymptotic

approximations, Appl. Math. Comput, 98(1999) 119-137.

86. M.A. Noor and A.K. Khalifa, Cubic splines collocation methods for

unilateral problems. Int. J. Engg. Sci., 25 (1987) 1527-1530.

87. M.A. Noor, K.I. Noor and Th. Rassias, Some aspects of variational

inequalities ,J. Comput. Appl. Math.,47(l993)285-312.

88. M.A. Noor and S.I.A. Tirmizi, Finite difference techniques for solving

obstacle problems, Appl. Math. Letters, 1(1988) 267-271.

89. R. K. Pandey, On the convergence of a finite difference method for a class of

singular two-point boundary value problems. Int. J.Comput. Math.,42(1992)

237-241.

90. F. Patricio, A numerical method for solving initial value problems with

spline functions, BIT, 19(1979) 489-494.

91. , Cubic spline functions and initial value problems, BIT, 18

(1978)342-347.

92. P.M. Prenter, Splines and Variational Methods, John Wiley & Sons

INC.(1975).

93. J. Rashidinia, Applications of splines to the numerical solution of

differential equations, Ph.D Thesis ,A.M.U.Aligarh,(1994).

97

94. W. G. Reddien and L. L. Schumaker, On a collocation method for singular

two-point boundary- value problems, Numer. Math., 25 (1973) 427-432.

95. J.N. Reddy, An introduction to the Finite Element Method, Second Ed., Mc

Graw-Hill, Inc. New York (1993).

96. E.L.Reiss, A.J. Callegari and D.S. Ahluwalia, Ordinary Differential

Equations with Applications, Holt, Rinehart and Winston, NewYork(1976).

97. R.D. Richtmyer and K.W.Mortan, Difference methods for initial value

problems, second edition, Interscience, NewYork (1967).

98. J.F. Rodrigues , Obstacle Problems in Mathematical Physics ,North-

Holland, Amsterdam (1987).

99. H.G.Roos, M.Stynes and L.Tobiska, Numerical Methods for Singularly

Perturbed Differential Equations, Springer, New York (1996).

100. S.G. Rubin and R.A. Graves Jr., Viscous flow solutions with a cubic spline

approximation, Computers and Fluids, 3(1975) 1-36.

101. S.G. Rubin and P.K. Khosla, Higher order numerical solution using cubic

splines, AIAA .J., 14 (1976) 851-858.

102. R.D. Russell and L.F. Shampine, A collocation method for boundary value

problems, Numer. Math., 19(1972) 1-28.

103. M. Sakai, Piecewise cubic interpolation and two point boundary value

problems, Publ. RIMS., Kyoto Univ. 7(1972) 345-362.

104. , Numerical solution of boimdary-value problems for second

order functional differential equations by the use of cubic splines, Mem.

Fac. Sci, Kyushu Univ.,29(l975) 113-122.

105. , Two sided quintic spline approximations for two point

boundary-value problems, Rep. Fac. Sci., Kagoshima Univ., (Math., Phys.,

Chem.)no. 10.(1977) 1-17.

98

106. , On consistency relations for cubic spline-on-spline and their

symptotic error estimates, J. Approx. Theory, 55(1985) 195-200.

107. I.J. Schoenberg, Contribution to the problem of approximation of equidistant

data by analytic functions. Quart. Appl. Math., 4(1946)345-369.

108. L.L. Schumaker, Spline Functions: Basic Theory, John Wiley,NewYork,

(1981).

109. M. Stojanovic, Spline difference methods for a singular perturbation

problem, Applied Numer.Math., 21(3)(1996) 321-333.

110. , A difference method for non-self- adjoint singular perturbation

problem of second order, App. Math. Modelling, 15(1991)381-385.

111. K.Surla, D.Herceg and L. Cvetkovic, A family of exponential spline

difference schemes, Univ. u. Novom Sadu, Zb.Rad.Prirod. Mat. Fak. Ser.

Mat.,19(2) (1991)12-23.

112. K. Surlaand M. Stojanovic, Solving singularly perturbed boundary value

problems by spline in tension. Journal Comput. Appl. Math., 24(1988)355-

363.

113. K. Surla and Z. Uzelec, Some imiformly convergent spline difference

schemes for singularly perturbed boundary-value problems, IMA J. Numer.

Anal., 10(1990) 209-222.

114. K.Surla and V.Vukoslavcevic, A spline difference scheme for boundary

-value problems with a small parameter, Univ.u. Novom Sadu, Zb.Rad.

Zb. Rad. Prirod. Mat. Fak. Ser.Mat.,25(2),( 1995)159 -166.

115. O.A. Taiwo and D.J. Evans, On the solution of singularly perturbed

boundary value problem by a chebyshev method, Intern. J. comput. Math.,

71(1999)521-529.

116. R.P. Tewarson, On the use of splines for the numerical solution of non

99

linear two point boundary value problems, BIT, 20 (1980) 223-232.

117. R.P. Tewarson and Y. Zhang, Solution of two point boundary value

problems using splines, Intern. J.Numer. Method Engg., 23(1986)707-710.

118. J. Todd, A direct approach to the problem of stability in the numerical

solution of partial differential equations, Commim. Pure and Appl. Math.,

9(1956)597-612.

119. E.H. Twizell and S.I.A. Tirmizi, Multiderivative methods for linear fourth

order boundary value problems. Tech. Rep TR/06, Dept. of Mathematics

and Statistics, Brunei University (1984).

120. R.A. Usmani, Bounds for the solution of a second order differential

equation with mixed boundary conditions, J. Engg. Math.,9 (1975)159-164.

121. ,Discrete methods for boundary value problems with applications

in plate deflection theory, ZAMP, 30(1979)87-99.

122. , Spline solutions for nonlinear two point boundary value problems.

Intern. J. Math, and Math. Sci., 3(1980) 151-167.

123. , The use of quartic splines in the numerical solution of a fourth-

order boundary value problem, J. Comput. Appl. Math., 44(1992) 187-199.

124. R.A. Usmani and M. Sakai, A connection between quartic spline solution

and Numerov solution of a boundary value problem, Int. J. Comput. Math.,

26(1989)263-273.

125. R.A. Usmani and S.A. Warsi, Quintic spline solution of boundary value

problems, Comput. Math. Appl., 6 (1980)197-203.

126. R.S. Varga, Error bounds for spline interpolation, In approximations with

special emphasis on spline functions, (I.J. Schoenberg, ed.),Academic Press,

New York (1969)367-388.

127. F. Villagio ,The Ritz method in solving unilateral problems in elasticity,

100

MECCANICA, 123-127(1981).

128. A.M. Wazwaz, On the solution of the fourth order parabolic equation by

the decomposition method, hit. J.comput. Math., 57(1995)213-217.

101