zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
A novel approach for the solution of a class of singular boundary value problems arising in physiology. (English) Zbl 1201.65135
Summary: A new approach implementing a modified decomposition method in combination with the cubic B-spline collocation technique is introduced for the numerical solution of a class of singular boundary value problems arising in physiology. The domain of the problem is split into two subintervals; a modified decomposition procedure based on a special integral operator is implemented in the vicinity of the singular point and outside this domain the resulting boundary value problem is tackled by applying the B-spline scheme. Performance of this method is examined numerically; the examples reveal that the current approach converges to the exact solution rapidly and with O(h 2 ) accuracy. Results show that the method yields a numerical solution in very good agreement with the existing exact and approximate solutions.
MSC:
65L60Finite elements, Rayleigh-Ritz, Galerkin and collocation methods for ODE
92C30Physiology (general)
References:
[1]Lin, Hs.: Oxygen diffusion in a spherical cell with nonlinear oxygen uptake kinetics, J. theor. Biol. 60, 449-457 (1976)
[2]Mcelwain, Dls.: A re-examination of oxygen diffusion in a spherical cell with michaelis-menten oxygen uptake kinetics, J. theor. Biol. 71, 255-263 (1978)
[3]Flesch, U.: The distribution of heat sources in the human head: a theoretical consideration, Journal of theor. Biol. 54, 285-287 (1975)
[4]Garner, J. B.; Shivaji, R.: Diffusion problems with mixed nonlinear boundary condition, Journal of math. Anal. appl. 148, 422-430 (1990) · Zbl 0712.34029 · doi:10.1016/0022-247X(90)90010-D
[5]Kanth, A. S. V. Ravi; Bhattacharya, Vishnu: Cubic spline for a class of non-linear singular boundary value problems arising in physiology, Appl. math. Comput. 174, No. 1, 768-774 (2006) · Zbl 1089.65075 · doi:10.1016/j.amc.2005.05.022
[6]Pandey, R. K.; Singh, Arvind K.: On the convergence of a finite difference method for a class of singular boundary value problems arising in physiology, J. comput. Appl. math. 166, 553-564 (2004) · Zbl 1086.65083 · doi:10.1016/j.cam.2003.09.053
[7]Caglar, Hikmet; Caglar, Nazan; Ozer, Mehmet: B-spline solution of non-linear singular boundary value problems arising in physiology, Chaos solitons fractals 39, 1232-1237 (2009) · Zbl 1197.65107 · doi:10.1016/j.chaos.2007.06.007
[8]Asaithambi, N. S.; Garner, J. B.: Pointwise solution bounds for a class of singular diffusion problems in physiology, Appl. math. Comput. 30, No. 3, 215-222 (1989) · Zbl 0669.92004 · doi:10.1016/0096-3003(89)90053-2
[9]Gustaffsson, B.: A numerical method for solving singular boundary value problems, Numer. math. 21, 328-344 (1973) · Zbl 0255.65032 · doi:10.1007/BF01436387
[10]Kanth, A. S. V. Ravi; Reddy, Y. N.: The method of inner boundary condition for singular boundary value problems, Appl. math. Comput. 139, No. 2–3, 429-436 (2003) · Zbl 1027.65102 · doi:10.1016/S0096-3003(02)00209-6
[11]Kanth, A. S. V. Ravi; Reddy, Y. N.: A numerical method for singular two point boundary value problems via Chebyshev economizition, Appl. math. Comput. 146, No. 2–3, 691-700 (2003) · Zbl 1035.65084 · doi:10.1016/S0096-3003(02)00613-6
[12]Hiltmann, P.; Lory, P.: On oxygen diffusion in a spherical cell with michaelis-menten oxygen uptake kinetics, Bull. math. Biol. 45, 661-664 (1983) · Zbl 0512.92008
[13]Adomian, G.: Solving frontier problems of physics: the decomposition method, (1994)
[14]Deeba, E.; Khuri, S. A.: Nonlinear equations, Wiley encyclopedia of electrical and electronics engineering 14, 562-570 (1999)
[15]Deeba, E. Y.; Khuri, S. A.: A decomposition method for solving the nonlinear Klein-Gordon equation, J. comput. Phys. 124, 442-448 (1996) · Zbl 0849.65073 · doi:10.1006/jcph.1996.0071
[16]Khuri, S. A.: A new approach to bratu’s problem, Appl. math. Comput. 147, 131-136 (2004) · Zbl 1032.65084 · doi:10.1016/S0096-3003(02)00656-2
[17]Khuri, S. A.: A numerical algorithm for solving the troesch’s problem, Int. J. Comput. math. 80, No. 4, 493-498 (2003) · Zbl 1022.65084 · doi:10.1080/0020716022000009228
[18]Khuri, S. A.: On the solution of coupled H-like equations of Chandrasekhar, Appl. math. Comput. 133, No. 2-3, 479-485 (2002) · Zbl 1018.85005 · doi:10.1016/S0096-3003(01)00251-X
[19]Khuri, S. A.: An alternative solution algorithm for the nonlinear generalized Emden-Fowler equation, Int. J. Nonlinear sci. Numeri. simul. 2, 299-302 (2001) · Zbl 1072.34503 · doi:10.1515/IJNSNS.2001.2.3.299
[20]Khuri, S. A.: On the decomposition method for the approximate solution of nonlinear ordinary differential equations, Int. J. Math. ed. Science tech. 32, No. 4, 525-539 (2001) · Zbl 1019.34005 · doi:10.1080/00207390110038286
[21]Khuri, S. A.: A Laplace decomposition algorithm applied to a class of nonlinear differential equations, J. appl. Math. 1, No. 4, 141-155 (2001) · Zbl 0996.65068 · doi:10.1155/S1110757X01000183