Spline solution of non-linear singular boundary value problems

References

• Hiltmann , P. and Lory , P. 1983 . On oxygen diffusion in a spherical cell with Michaelis–Menten oxygen uptake kinetics . Bulletin of Mathematical Biology , 45 : 661 – 664 .
   PubMed Web of Science ®Google Scholar
• Anderson , N. and Arthurs , A. M. 1983 . Analytical diffusion in a spherical cell with Michalelis–Menten oxygen uptake kinetics . Bulletin of Mathematical Biology , 47 : 145 – 153 .
   Google Scholar
• Adam , J. A. 1986 . A simplified mathematical model of tumor growth . Mathematical Biosciences , 81 : 224 – 229 .
   Web of Science ®Google Scholar
• Adam , J. A. 1987 . A mathematical model of tumor growth II: effect of geometry and spatial non-uniformity on stability . Mathematical Biosciences , 86 : 183 – 211 .
   Web of Science ®Google Scholar
• Adam , J. A. and Maggelakis , S. A. 1989 . Mathematical model of tumor growth IV: effect of necrotic core . Mathematical Biosciences , 97 : 121 – 136 .
   PubMed Web of Science ®Google Scholar
• Burton , A. C. 1966 . Rate of growth of solid tumor as a problem of diffusion . Growth , 30 : 157 – 176 .
   PubMedGoogle Scholar
• Greenspan , H. P. 1972 . Models for the growth of solid tumor as a problem by diffusion . Studies in Applied Mathematics , 52 : 317 – 340 .
   Google Scholar
• Asaithambi , N. S. and Goodman , J. B. 1989 . Point wise bounds for a class of singular diffusion problems in physiology . Applied Mathematics and Computation , 30 : 215 – 222 .
   Web of Science ®Google Scholar
• Lin , H. S. 1976 . Oxygen diffusion in a spherical cell with nonlinear oxygen uptake kinetics . Journal of Theoretical Biology , 60 : 449 – 457 .
   PubMed Web of Science ®Google Scholar
• McElwain , D. L.S. 1978 . A re-examination of oxygen diffusion in a spherical cell with Michaelis–Menten oxygen uptake kinetics . Journal of Theoretical Biology , 71 : 255 – 263 .
   PubMed Web of Science ®Google Scholar
• Rashevsky , N. 1960 . Mathematical Biophysics , Vol. 1 , New York : Dover .
   Google Scholar
• Flesch , U. 1975 . The distribution of heat sources in the human head: a theoretical consideration . Journal of Theoretical Biology , 54 : 285 – 287 .
   PubMed Web of Science ®Google Scholar
• Garner , J. B. and Shivaji , R. 1990 . Diffusion problems with mixed nonlinear boundary condition . Journal of Mathematic Analysis and Applications , 148 : 422 – 430 .
   Web of Science ®Google Scholar
• Gray , B. F. 1980 . The distribution of heat sources in the human head: a theoretical consideration . Journal of Theoretical Biology , 82 : 473 – 476 .
   PubMed Web of Science ®Google Scholar
• Duggan , R. C. and Goodman , A. M. 1989 . Point wise bounds for nonlinear heat conduction model for the human head . Bulletin of Mathematical Biology , 48 ( 2 ) : 229 – 236 .
   Google Scholar
• Pandey , R. K. and Singh , Arvind K. 2004 . On the convergence of a finite difference method for weakly regular singular boundary value problems . International Journal of Computer Mathematics , 81 : 227 – 238 .
   Web of Science ®Google Scholar
• Pandey , R. K. and Singh , Arvind K. 2004 . On the convergence of a finite difference method for a class of singular boundary value problems arising in physiology . Journal of Computational and Applied Mathematics , 166 : 553 – 564 .
   Web of Science ®Google Scholar
• Bickley , W. G. 1968 . Piecewise cubic interpolation and two point boundary problems . Computer Journal , 11 : 206 – 208 .
   Web of Science ®Google Scholar
• Chawla , M. M. and Katti , C. P. 1982 . Finite difference methods and their convergence for a class of singular two point boundary value problems . Numerical Mathematics , 39 : 341 – 350 .
   Web of Science ®Google Scholar
• Chawla , M. M. , Mckee , S. and Shaw , G. 1986 . Order h 2 method for a singular two-point boundary value problem . BIT , 26 : 326 – 381 .
   Google Scholar
• El-Gebeily , M. A. and Abu-Zaid , I. T. 1998 . On a finite-difference method for singular two-point boundary value problems . IMA Journal of Numerical Analysis , 18 : 179 – 190 .
   Google Scholar
• Pandey , R. K. 1997 . A finite difference method for a class of singular two point boundary value problems arising in physiology . International Journal of Computer Mathematics , 65 : 131 – 140 .
   Web of Science ®Google Scholar
• Pandey , R. K. 1997 . On a class of regular singular two point boundary value problems . Journal of Mathematic Analysis and Applications , 208 : 388 – 403 .
   Web of Science ®Google Scholar
• Pandey , R. K. 1996 . On a class of weakly regular singular two point boundary value problems, I . Nonlinear Analysis Theory, Methods and Applications , 27 ( 1 ) : 1 – 12 .
   Google Scholar
• Pandey , R. K. 1996 . On a class of weakly regular singular two point boundary value problems, II . Journal of Differential Equations , 127 : 110 – 123 .
   Web of Science ®Google Scholar
• Sakai , M. and Usmani , R. A. 1989 . An application of Chawla's identity to a different scheme for singular problems . BIT , 29 : 566 – 568 .
   Google Scholar
• Ravi Kanth , A. S.V. and Bhattacharya , V. 2006 . Cubic spline for a class of non-linear singular boundary value problems arising in physiology . Applied Mathematics and Computation , 174 : 768 – 774 .
   Web of Science ®Google Scholar
• Rashidinia , J. 1990 . “ Applications of spline to numerical solution of differential equations ” . India : AMU . M. Phil. dissertation
   Google Scholar
• Siraj-ul-Islam , H. and Tirmizi , I. A. 2006 . Non-polynomial spline approach to the solution of a system of second-order boundary-value problems . Applied Mathematics and Computation , 173 : 1208 – 1218 .
   Web of Science ®Google Scholar
• Van Daele , M. , Vanden Berghe , G. and De Meyer , H. 1994 . A smooth approximation for the solution of a fourth-order boundary value problem based on non-polynomial splines . Journal of Computational and Applied Mathematics , 51 : 383 – 394 .
   Web of Science ®Google Scholar
• Ahlberg , J. H. , Nilson , E. N. and Walsh , J. L. 1967 . The Theory of Splines and Their Applications , New York : Academic Press .
   Google Scholar
• Jain , M. K. 1984 . Numerical Solution of Differential Equations , 2 , India : Wiley Eastern Limited .
   Google Scholar
• Aziz , Tariq and Khan , Arshad . 2002 . A spline method for second-order singularly perturbed boundary value problems . Journal of Computational and Applied Mathematics , 147 : 445 – 452 .
   Web of Science ®Google Scholar
• Henrici , P. 1962 . Discrete Variable Methods in Ordinary Differential Equations , New York : Wiley .
   Google Scholar