Abstract
In this chapter we try to give a list of references to books and papers, the contents of which will reflect as far as possible the sentiments and mathematical simplicity of this book. Occasionally a topic or topics we consider important may have no such simple written publicity, in which case we may have to give a reference involving more elaborate mathematics.
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References
Chapter 1
H.T.H. Piaggio (1971), An Elementary Treatise on Differential Equations and Their Applications
C.G. Lambe and C.J. Tranter (1961), Differential Equations for Engineers and Scientists, English Universities Press.
K.F. Riley (1983), Mathematical Methods for the Physical Sciences, Cambridge University Press.
D.W. Jordan and F. Smith (1971), Nonlinear Ordinary Differential Equations, Oxford University Press.
L.M. Milne-Thomson (1933), The Calculus of Finite Differences, Macmillan.
L. Fox and D.F. Mayers (1981), On the numerical solution of implicit “ ordinary differential equations. I.M.A.J. Num. Anal., 1, 377–401.
A. Jepson and A. Spence (1984), On implicit ordinary differential equations. I.M.A.J. Num. Anal., 4, 253–274.
C.W. Gear and L.R. Petzold (1982), Differential algebraic systems, pp. 75–89, Lecture notes in mathematics No. 973. Springer.
Chapter 2
L. Fox and D.F. Mayers (1968), Computing Methods for Scientists and Engineers, Clarendon Press, Oxford
F.W.J. Olver (1964), Error analysis of Miller’s recurrence algorithm.Math. Comp ., 18, 65–74.
Chapters 3 and 4
J.D. Lambert (1973), Computational Methods in Ordinary Differential Equations, Wiley
C.W. Gear (1971), Numerical Initial-value Problems in Ordinary Differential Equations, Prentice-Hall.
L.F. Shampine and M.K. Gordon (1975), Computer Solution of Ordinary Differential Equations: The Initial-value Problem, Freeman.
G. Hall and J.M. Watt (Eds.) (1976), Modern Numerical Methods for Ordinary Differential Equations (proceedings of Liverpool-Manchester Summer School), Clarendon Press, Oxford
I. Gladwell and D. Sayers (Eds.) (1980), Computational Techniques for Ordinary Differential Equations (proceedings of an I.M.A. Symposium), Academic Press.
A.R. Curtis (1980), The FACSIMILE numerical integrator for stiff initial- value problems.
D. Barton, J.M. Willers and R.V.M. Zahar (1972), Taylor series method for ordinary differential equations (in Mathematical Software, Ed. J.R. Rice), Academic Press.
D. Barton (1980), On Taylor series and stiff systems. TOMS, 6, 280–294.
G. Corliss and Y.F. Chang (1982) Solving ordinary differential equations using Taylor series. TOMS, 8, 114–144.
J.R. Cash (1983), Block Runge-Kutta methods for the numerical integration of initial-value problems in ordinary differential equations. I The non-stiff case. Math. Comp ., 40, 175–191.
J.R. Cash (1983), Block Runge-Kutta methods for the numerical integration of initial-value problems in order differential equations. II The stiff case. Math. Comp ., 40, 193–206.
J.C. Butcher, K. Burrage and F.H. Chipman (1979), STRIDE: Stable Runge-Kutta integration for differential equations. Report Series No. 150 (Computational Mathematics No. 20), Dept. of Mathematics, University of Auckland.
L. Fox and E.T. Goodwin (1949), Some new methods for the numerical integration of ordinary differential equations. Proc. Camb. Phil Soc ., 45, 373–388.
R.K. Jain, N.S. Kambo and R. Goel (1984), A sixth-order P-stable symmetric multi-step method for periodic initial-value problems of second- order differential equations. I.M.A.J. Num. Anal., 4, 117–125.
L. Fox, D.F. Mayers, J.R. Ockendon and A.B. Taylor (1971), On a functional differential equation. J. Inst. Maths. Applies., 8, 271–307.
C.W. Cryer (1972), Numerical methods for functional differential equations, pp. 97–102 of Delay and Functional Differential Equations and Their Applications. (Ed. K. Schmitt), Academic Press.
E. Hansen (Ed.) (1969), Topics in Interval Analysis, Clarendon Press, Oxford.
Kruckeberg (1969), Ordinary differential equations (from reference [27]).
K.L.E. Nickel (1980),Interval Mathematics. Academic Press.
Chapters 5 and 6
M.R. Osborne (1969), On shooting methods for boundary-value problems. J. Math. Anal. Appl., 27, 417–433
P. Deuflhard (1980), Recent advances in multiple shooting techniques
G.H. Meyer (1973), Initial-value Methods for Boundary-value Problems. Academic Press, New York
L. Fox (1957), The Numerical Solution of Two-point Boundary-value Problems in Ordinary Differential Equations. Clarendon Press, Oxford.
L. Fox (1980), Numerical methods for boundary-value problems.
V. Pereyra (1978), PASVA3. An adaptive finite-difference Fortran program for first-order non-linear boundary-value problems.
M. Lentini and V. Pereyra (1977), An adaptive finite-difference solver for nonlinear two-point boundary-value problems with mild boundary layers. SIAM J. Num. Anal., 14, 91–111.
V. Pereyra (1973), High-order finite-difference solution of differential equations. Comp. Sci. Dept. Stanford University report STAN-CA-73–348.
J.W. Daniel and A.J. Martin (1977), Numerov’s method with deferred corrections for two-point boundary-value problems. SIAM J. Num. Anal., 14, 1033–1050
C.W. Gear (1980), Initial-value problems: practical theoretical developments.
R.E. Scheid (1984), The accurate solution of highly oscillatory ordinary differential equations. Math. Comp., 41, 487–509.
W.L. Miranker (1981), Numerical Methods for Stiff Equations and Singular Perturbation Problems, Reidel, Dordrecht.
B. Kreiss and H-O Kreiss (1981), Numerical methods for singular perturbation problems. SI AM J. Num. Anal., 18, 262–276.
H-O Kreiss, N.K. Nichols and D.L. Brown (1983), Numerical methods for stiff two-point boundary-value problems. University of Wisconsin Math. Res. Center Summary Report 2599.
E.P. Doolan, J.J.H. Miller and W.H.A. Schilders (1980), Uniform Numerical Methods for Problems with Initial and Boundary Layers. Boole Press, Dublin.
H.B. Keller (1976), Numerical solution of two-point boundary-value problems. Regional Conference Series in Applied Mathematics 24, SIAM, Philadelphia
P.B. Bailey, M.K. Gordon and L.F. Shampine (1978), Automatic solution of the Sturm-Liouville problem. TOMS, 4, 193–208
J.K. Lund and B.V. Riley (1984), A sinc-collocation method for the computation of the eigenvalues of the radial Schrödinger equation. I.M.A.J. Num. Anal., 4, 83–98.
D.F. Mayers (1964),The deferred approach to the limit in Ordinary differential equations. Comp. J., 7, 54–57.
M.M Chawla (1978)A fourth-order tridiagonal finite-difference method for general two-point boundary-value problems With nonlinear boundary conditions. J.I.M.A., 22, 89–97.
E. Hansen (1969), On solving two-point boundary-value problems using interval arithmetic. From reference [27].
L. Fox and M.R. Valenca (1980), Some experiments with interval methods for two-point boundary-value problems in ordinary differential equations. B.I.T., 20, 67–82.
Chapter 7
C. Lanczos (1957), Applied Analysis. Prentice-Hall, New York; Pitman, London
L. Fox and I.B. Parker (1968), Chebyshev Polynomials in Numerical Analysis. Oxford Mathematical Handbooks (Eds. J. Crank and C.C. Ritchie), Oxford University Press
C.W. Clenshaw (1957), The numerical solution of linear differential equations in Chebyshev series. Proc. Camb. Phil. Soc., 53, 134–149
C.W. Clenshaw and H.J. Norton (1963), The solution of nonlinear ordinary differential equations in Chebyshev series. Computer J., 6, 88–92.
E.L. Ortiz, W.F.C. Purser and F.J. Rodriguez (1972), Automation of the r method. Royal Irish Academy Conference on Numerical Analysis, Dublin.
K. Wright (1964), Chebyshev collocation methods for ordinary differential equations. Computer J., 6, 358–363
V. Ascher, J. Christiansen and R.D. Russell (1978), COLSYS - A collocation code for boundary-value problems.
Chapter 8 and General
W.H. Enright and T.E. Hull (1976), Test results on initial-value methods for non-stiff ordinary differential equations. SIAM J. Num. Anal., 13, 944–961
W.K. Enright, T.E. Hull and B. Lindberg (1975), Comparing numerical methods for stiff systems of ordinary differential equations, B.I.T., 15, 10–48.
T.E. Hull (1980), Comparison of algorithms for initial-value problems.
B. Childs, M. Scott, T.W. Daniel, E. Denman and P. Nelson (1978), Codes for boundary-value problems in ordinary differential equations. Lecture Notes in Computer Science 76, Springer-Verlag, Berlin
I. Gladwell (1980), A survey of sub-routines for solving boundary-value problems in ordinary differential equations.
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Fox, L., Mayers, D.F. (1987). Further notes and bibliography. In: Numerical Solution of Ordinary Differential Equations. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-3129-9_9
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