Monotone methods for the Thomas-Fermi equation
Author:
J. W. Mooney
Journal:
Quart. Appl. Math. 36 (1978), 305-314
MSC:
Primary 80A30; Secondary 34B30
DOI:
https://doi.org/10.1090/qam/508774
MathSciNet review:
508774
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Abstract: The boundary-value problem for the ionized atom case of the Thomas-Fermi equation is transformed to a certain convex nonlinear boundary-value problem. Two iterative procedures, previously developed for such problems, are constructed for the ionized atom problem. A comparative analysis of the efficiency of the iteration schemes is presented. The existence and uniqueness of a solution is established and the solution is shown to have monotonic dependence on the boundary conditions. Numerical bounds are obtained for a specific problem.
- C. D. Luning and W. L. Perry, An iterative technique for solution of the Thomas-Fermi equation utilizing a nonlinear eigenvalue problem, Quart. Appl. Math. 35 (1977/78), no. 2, 257–268. MR 445056, DOI https://doi.org/10.1090/S0033-569X-1977-0445056-8
- J. W. Mooney and G. F. Roach, Iterative bounds for the stable solutions of convex non-linear boundary value problems, Proc. Roy. Soc. Edinburgh Sect. A 76 (1976/77), no. 2, 81–94. MR 440195, DOI https://doi.org/10.1017/S0308210500013901
- J. W. Mooney, A unified approach to the solution of certain classes of nonlinear boundary value problems using monotone iterations, Nonlinear Anal. 3 (1979), no. 4, 449–465. MR 537330, DOI https://doi.org/10.1016/0362-546X%2879%2990061-0
J. W. Mooney, Ph.D. Thesis, University of Strathclyde, 1976
- Donald S. Cohen, Positive solutions of a class of nonlinear eigenvalue problems, J. Math. Mech. 17 (1967), 209–215. MR 0213695, DOI https://doi.org/10.1512/iumj.1968.17.17010
D. D. Joseph, Nonlinear heat generation in conducting solids, Int. J. Heat and Mass Transfer 8, 281–288 (1965)
- D. D. Joseph and E. M. Sparrow, Nonlinear diffusion induced by nonlinear sources, Quart. Appl. Math. 28 (1970), 327–342. MR 272272, DOI https://doi.org/10.1090/S0033-569X-1970-0272272-0
- I. M. Gel′fand, Some problems in the theory of quasilinear equations, Amer. Math. Soc. Transl. (2) 29 (1963), 295–381. MR 0153960
D. A. Frank-Kamenetskii, Diffusion and heat transfer in chemical kinetics, Plenum, Princeton, 1969
R. Aris, Mathematical theory of diffusion and reaction, vols. I, II, Clarendon, Oxford, 1975
P. Csavinskzy, Calculation of diamagnetic susceptibilities of ions using a universal approximate analytical solution of the Thomas-Fermi equation, Bull. Amer. Phys. Soc. (2) 18, 726-727 (1973)
- L. F. Shampine, Some nonlinear eigenvalue problems, J. Math. Mech. 17 (1967/1968), 1065–1072. MR 0226190
- Herbert B. Keller and Donald S. Cohen, Some positone problems suggested by nonlinear heat generation, J. Math. Mech. 16 (1967), 1361–1376. MR 0213694
- David H. Sattinger, Topics in stability and bifurcation theory, Lecture Notes in Mathematics, Vol. 309, Springer-Verlag, Berlin-New York, 1973. MR 0463624
K. H. Meyn and B. Werner, Randmaximum und Monotonieprinzipien für elliptische Randwertaufgaben mit Gebietszerlegungen, Abh. Math. Sem. Univ. Hamburg (1976)
- J. P. Keener and H. B. Keller, Positive solutions of convex nonlinear eigenvalue problems, J. Differential Equations 16 (1974), 103–125. MR 346305, DOI https://doi.org/10.1016/0022-0396%2874%2990029-1
- Michael G. Crandall and Paul H. Rabinowitz, Some continuation and variational methods for positive solutions of nonlinear elliptic eigenvalue problems, Arch. Rational Mech. Anal. 58 (1975), no. 3, 207–218. MR 382848, DOI https://doi.org/10.1007/BF00280741
- Murray H. Protter and Hans F. Weinberger, Maximum principles in differential equations, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1967. MR 0219861
A. R. Forsyth, A treatise on differential equations, Macmillan, London, 1961
- F. W. J. Olver, Asymptotics and special functions, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1974. Computer Science and Applied Mathematics. MR 0435697
- Herbert Amann, Existence of multiple solutions for nonlinear elliptic boundary value problems, Indiana Univ. Math. J. 21 (1971/72), 925–935. MR 320517, DOI https://doi.org/10.1512/iumj.1972.21.21074
C. D. Luning and W. L. Perry, An iterative technique for solution of the Thomas-Fermi equation utilizing a nonlinear eigenvalue problem, Quart. Appl. Math. 35, 257–268 (1977)
[2[ J. W. Mooney and G. F. Roach, Iterative bounds for the stable solutions of convex nonlinear boundary value problems, Proc. Roy. Soc. Edin. (A) 76, 81–94 (1976)
J. W. Mooney, A unified approach to the solution of certain classes of nonlinear boundary value problems using monotone iterations. (to appear).
J. W. Mooney, Ph.D. Thesis, University of Strathclyde, 1976
D. S. Cohen, Positive solutions of a class of nonlinear eigenvalue problems, J. Math. Mech. 17, 209–215 (1967)
D. D. Joseph, Nonlinear heat generation in conducting solids, Int. J. Heat and Mass Transfer 8, 281–288 (1965)
D. D. Joseph and E. M. Sparrow, Nonlinear diffusion induced by nonlinear sources, Quart. Appl Math. 28, 327–342 (1970)
I. M. Gelfand, Some problems in theory of quasilinear equations, Amer. Math. Soc. Transl. 29, 295–381 (1963)
D. A. Frank-Kamenetskii, Diffusion and heat transfer in chemical kinetics, Plenum, Princeton, 1969
R. Aris, Mathematical theory of diffusion and reaction, vols. I, II, Clarendon, Oxford, 1975
P. Csavinskzy, Calculation of diamagnetic susceptibilities of ions using a universal approximate analytical solution of the Thomas-Fermi equation, Bull. Amer. Phys. Soc. (2) 18, 726-727 (1973)
L. F. Shampine, Some nonlinear eigenvalue problems, J. Math. Mech. 17, 1067–1072 (1968)
H. B. Keller and D. S. Cohen, Some positone problems suggested by nonlinear heat generation, J. Math. Mech. 16, 1361–1376 (1967)
D. Sattinger, Topics in stability and bifurcation theory, Lecture Notes 309, Springer, 1973
K. H. Meyn and B. Werner, Randmaximum und Monotonieprinzipien für elliptische Randwertaufgaben mit Gebietszerlegungen, Abh. Math. Sem. Univ. Hamburg (1976)
J. P. Keener and H. B. Keller, Positive solutions of convex nonlinear eigenvalue problems, J. Diff. Eqs. 16, 103–125 (1974)
M. G. Crandall and P. H. Rabinowitz, Some continuation and variational methods for positive solutions of nonlinear elliptic eigenvalue problems, Arch. Rat. Mech. Anal. 58, 207–218 (1975)
M. H. Protter and H. F. Weinberger, Maximum principles in differential equations, Prentice-Hall, Englewood Cliffs, 1967
A. R. Forsyth, A treatise on differential equations, Macmillan, London, 1961
F. W. J. Olver, Asymptotics and special functions, Academic Press, New York, 1974
H. Amann, Existence of multiple solutions for nonlinear elliptic boundary value problems, Indiana University Math. J. 21, 925–935 (1972)
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Article copyright:
© Copyright 1978
American Mathematical Society
