Some recent developments of Numerov’s method. (English) Zbl 1002.65082
The authors survey some recent developments of Numerov’s method covering existence-uniqueness, computation, and discuss some recent developments.
Reviewer: J.D.P.Donnelly (Oxford)
MSC:
| 65L10 | Numerical solution of boundary value problems involving ordinary differential equations |
| 65L12 | Finite difference and finite volume methods for ordinary differential equations |
| 34B15 | Nonlinear boundary value problems for ordinary differential equations |
References:
| [1] | Henrici, P., Discrete Variable Methods in Ordinary Differential Equations (1962), John Wiley: John Wiley New York · Zbl 0112.34901 |
| [2] | Lees, M., Discrete methods for nonlinear two-point boundary value problems, (Bramble, J. H., Numerical Solution of Partial Differential Equations (1966), Academic Press: Academic Press New York), 59-72 · Zbl 0148.39206 |
| [3] | Numerov, B. V., A method of extrapolation of perturbations, Roy. Ast. Soc. Monthly Notices, 84, 592-601 (1924) |
| [4] | Fischer, C. F.; Usmani, R. A., Properties of some tridiagonal matrices and their applications to boundary value problems, SIAM J. Numer. Anal, 6, 127-132 (1969) · Zbl 0176.46802 |
| [5] | Fox, L., The Numerical Solution of Two-Point Boundary Value Problems in Ordinary Differential Equations (1957), Oxford University Press: Oxford University Press London · Zbl 0077.11202 |
| [6] | Froberg, G., Introduction to Numerical Analysis (1969), Addison-Wesley: Addison-Wesley Reading, MA |
| [7] | Ralston, W., Mathematical Methods for Digital Computers (1960), John Wiley: John Wiley New York · Zbl 0089.12602 |
| [8] | Usmani, R. A., Boundary value techniques for the numerical solution of certain initial value problems in ordinary differential equations, J. Assoc. Comput. Mach., 13, 287-295 (1966) · Zbl 0137.33202 |
| [9] | Usmani, R. A., Integration of second order linear differential equations with mixed boundary conditions, Intern. J. Computer Math., 3, 389-397 (1973) · Zbl 0269.65043 |
| [10] | Agarwal, R. P., Difference Equations and Inequalities (1992), Marcel Dekker: Marcel Dekker New York · Zbl 0784.33008 |
| [11] | Agarwal, R. P., On multi-point boundary value problems for discrete equations, J. Math. Anal. Appl., 96, 520-534 (1983) · Zbl 0539.39005 |
| [12] | Usmani, R. P.; Agarwal, R. P., On the numerical solution of two point boundary value problems, Appl. Math. Comp., 25, 247-264 (1988) · Zbl 0664.65121 |
| [13] | Agarwal, R. P., On Numerov’s method for solving two point boundary value problems, Utilitas Math, 28, 159-174 (1985) · Zbl 0614.65085 |
| [14] | Wang, Y.-M., Monotone methods for a boundary value problem of second-order discrete equation, Computers Math. Applic., 36, 6, 77-92 (1998) · Zbl 0932.65130 |
| [15] | Wang, Y.-M.; Guo, B.-Y., On Numerov scheme for nonlinear two-points boundary value problem, J. Comp. Math, 16, 345-366 (1998) · Zbl 0921.65058 |
| [16] | Wang, Y.-M., Accelerated monotone iterative methods for a boundary value problem of second-order discrete equations, Computers Math. Applic., 39, 3/4, 85-94 (2000) · Zbl 0954.65063 |
| [17] | Chawla, M. M.; Shivakumar, P. N., Numerov’s method for non-linear two-point boundary value problems, Intern. J. Computer Math., 17, 167-176 (1985) · Zbl 0566.65064 |
| [18] | Wang, Y.-M.; Agarwal, R. P., Monotone methods for solving a boundary value problem of second order discrete system, Mathematical Problems in Engineering: Theory, Methods and Applications, 5, 291-315 (2000) · Zbl 0962.65087 |
| [19] | Guo, B.-Y.; Wang, Y.-M., An almost monotone approximation for a nonlinear two-point boundary value problem, Advances in Comp. Math., 8, 65-96 (1998) · Zbl 0892.65054 |
| [20] | Guo, B.-Y.; Wang, Y.-M., A new monotone approximation for nonlinear systems, (Chen, Z., Advances in Computational Mathematics, Lecture Notes in Pure and Applied Mathematics, 202 (1998), Marcel Dekker), 155-172 · Zbl 0931.65087 |
| [21] | Guo, B.-Y.; Wang, Y.-M., Almost monotone approximation to a nonlinear system, Comm. Appl. Anal., 2, 227-247 (1998) · Zbl 0896.65055 |
| [22] | Y.-M. Wang and B.-Y. Guo, Monotone approximation to a system without monotone nonlinearity, J. Comp. Math. (to appear).; Y.-M. Wang and B.-Y. Guo, Monotone approximation to a system without monotone nonlinearity, J. Comp. Math. (to appear). · Zbl 0987.65073 |
| [23] | Ladde, G. S.; Lakshmikantham, V.; Vatsala, A. S., Monotone Iterative Techniques for Nonlinear Differential Equations (1985), Pittman: Pittman Boston, MA · Zbl 0658.35003 |
| [24] | Pao, C. V., Nonlinear Parabolic and Elliptic Equations (1992), Plenum Press: Plenum Press New York · Zbl 0780.35044 |
| [25] | Pang, P. Y.H.; Agarwal, R. P., Periodic boundary value problems for first and second order discrete systems, Mathl. Comput. Modelling, 16, 10, 102-112 (1992) · Zbl 0767.65094 |
| [26] | Pang, P. Y.H.; Agarwal, R. P., On discrete boundary value problems arising in transport phenomena, Appl. Math. Comp., 60, 193-213 (1994) · Zbl 0789.65067 |
| [27] | Pang, P. Y.H.; Agarwal, R. P., Monotone iterative methods for a general class of discrete boundary value problems, Computers Math. Applic., 28, 1-3, 243-254 (1994) · Zbl 0805.65140 |
| [28] | Sheng, Q.; Agarwal, R. P., Monotone methods for higher-order partial difference equations, Computers Math. Applic., 28, 1-3, 291-307 (1994) · Zbl 0812.65130 |
| [29] | Wang, Y.-M., Monotone enclosure for a class of discrete boundary value problems without monotone nonlinearities, Computers Math. Applic., 35, 6, 51-60 (1998) · Zbl 0907.39005 |
| [30] | Zhuang, W.; Cheng, Y.; Cheng, S. S., Monotone methods for a discrete boundary problem, Computers Math. Applic., 32, 12, 41-49 (1996) · Zbl 0872.39005 |
| [31] | Pao, C. V., Monotone iterative methods for finite difference system of reaction-diffusion equations, Numer. Math., 46, 571-586 (1985) · Zbl 0589.65072 |
| [32] | Pao, C. V., Block monotone iterative methods for numerical solutions of nonlinear elliptic equations, Numer. Math., 72, 239-262 (1995) · Zbl 0838.65104 |
| [33] | Pao, C. V., Numerical analysis of coupled system of nonlinear parabolic equations, SIAM J. Numer. Anal., 19, 335-344 (1999) |
| [34] | Varga, R. S., Matrix Iterative Analysis (1962), Prentice-Hall: Prentice-Hall Englewood Cliffs, NJ · Zbl 0133.08602 |
| [35] | Berman, A.; Plemmons, R. J., Nonnegative Matrices in the Mathematical Sciences (1979), Academic Press: Academic Press New York · Zbl 0484.15016 |
| [36] | Ortega, J. M.; Rheinboldt, W. C., Iterative Solution of Nonlinear Equations in Several Variables (1970), Academic Press: Academic Press New York · Zbl 0241.65046 |
| [37] | Rabinowitz, P. H., A minimax principle and applications to elliptic partial differential equations, (Chadam, J. M., Lecture Notes in Mathematics, 648 (1970), Springer: Springer Berlin), 97-115 · Zbl 0377.35020 |
| [38] | Guo, B.-Y., Difference Methods for Partial Differential Equations (1988), Science Press: Science Press Beijing |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
