Numerical solution of nonlinear two-point boundary problems by finite difference methods

ABSTRACT

Solution of nonlinear two-point boundary-value problems is often an extremely difficult task. Quite apart from questions of reality and uniqueness, there is no established numerical technique for this problem. At present, shooting techniques are the easiest method of attacking these problems. When these fail, the more difficult method of finite differences can often be used to obtain a solution. This paper gives examples and discusses the finite difference method for nonlinear two-point boundary-value problems.

REFERENCES

• 1: BERLYAND, O. S., GAVRILOVA, R. I., AND PRUDNIKOV, A. P. Tables of Integral Error Functions and Hermite Polynomials. Macmillan Co., New York, 1962.
• 2: COLLATZ, L. The Numerical Treatment of Differential Equations. Springer, Berlin, 3d ed., 1960.
• 3: FORSYTHE, G. E., AND WASOW, W. R. Finite-Difference Methods for Partial Differential Equations. Wiley, 1960.
• 4: FOX, L. The Numerical Solution of Two-Point Boundary Problems. Oxford, 1957.
• 5: ____. Numerical Solutions of Ordinary and Partial Differential Equations. Oxford, 1960.
• 6: GOODMAN, T. R., AND LANCE, G. N. The numerical integration of two-point boundary value problems. Math. Tables Other Aids Comput. 10 (1956), 82-86.
• 7: HENRICI, P. Error Propagation for Difference Methods. Wiley, 1963.
• 8: HOLT, J. F. SHARE Subroutine RWLE4F, Linear Equation Solver of Band Matrices, IBM Corp., White Plains, N. Y., Aug. 1960.
• 9: KING, W. S., AND LEWELLEN, W. S. Boundary-layer similarity solutions for rotating flows with and without magnetic interaction. Ref. ATN-63(9227)-6, Aerodynamics and Propulsion Res. Lab., Aerospace Corp., Los Angeles, Calif., July 1963.
• 10: LEVINSKY, S., AND YOSHIHARA, H. Rarefied hypersonic flow over a sphere. ARS International Hypersonic Conf., Cambridge, Mass., Aug. 1961, 89.
• 11: David D. Morrison , James D. Riley , John F. Zancanaro, Multiple shooting method for two-point boundary value problems, Communications of the ACM, v.5 n.12, p.613-614, Dec. 1962 [doi>10.1145/355580.369128]
• 12: Tables of the Error Function and Its Derivatives. Nat. Bur. Standards, U. S. Printing Off., Washington, D. C., 1954.
• 13: WHITTAKER, E. T., AND WATSON, G. N. Modern Analysis. Ch. X, Cambridge U. Press, 4th ed., 1952.