Numerical solution of a third-order nonlinear boundary-value problem by automatic differentiation

References

• Adomian , G. 1984 . A new approach to nonlinear partial differential equations . J. Math. Anal. Appl. , 102 : 420 – 434 .
   Web of Science ®Google Scholar
• Adomian , G. 1988 . A review of the decomposition method in applied mathematics . J. Math. Anal. Appl. , 135 : 501 – 544 .
   Web of Science ®Google Scholar
• Adomian , G. 1994 . Solving Frontier Problems of Physics: The Decomposition Method , Dordrecht : Kluwer Academic Publishers .
   Google Scholar
• Adomian , G. and Adomian , G. E. 1984 . A global method for solution of complex systems . Math. Model. , 5 : 251 – 263 .
   Google Scholar
• Adomian , G. and Rach , R. 1985 . Polynomial nonlinearities in differential equations . J. Math. Anal. Appl. , 109 : 90 – 95 .
   Google Scholar
• Al-Said , E. A. and Noor , M. A. 2003 . Cubic splines method for a system of third-order boundary-value problems . Appl. Math. Comput. , 42 : 195 – 204 .
   Google Scholar
• Asaithambi , A. 2005 . Solution of the Falkner–Skan equation by recursive evaluation of Taylor coefficients . J. Comput. Appl. Math. , 176 : 203 – 214 .
   Google Scholar
• Asaithambi , A. 2006 . A shooting method for a nonlinear diffusion problem using automatic differentiation . Appl. Math. Comput. , 180 : 264 – 269 .
   Google Scholar
• Asaithambi , A. 2007 . Numerical solution of Stefan problems using automatic differentiation . Appl. Math. Comput. , 189 : 943 – 948 .
   Google Scholar
• Babolian , E. and Javadi , Sh. 2004 . New method for calculating Adomian polynomials . Appl. Math. Comput. , 153 : 253 – 259 .
   Google Scholar
• Bartholomew-Biggs , M. C. , Brown , S. , Christianson , B. and Dixon , L. C.W. 2000 . Automatic differentiation of algorithms . J. Comput. Appl. Math. , 124 : 171 – 190 .
   Web of Science ®Google Scholar
• Bendtsen , C. and Stauning , O. 1996 . “ FADBAD: A flexible C++ package for automatic differentiation ” . IMM Department of Mathematical Modelling, Technical University of Denmark . Technical Report
   Google Scholar
• Bendsten , C. and Stauning , O. 1997 . “ TADIFF, A flexible C++ package for automatic differentiation using Taylor series expansion ” . IMM Department of Mathematical Modelling, Technical University of Denmark . Technical Report
   Google Scholar
• Berz , M. , Bischof , C. H. , Corliss , G. F. and Griewank , A. 1996 . Computational Differentiation: Techniques, Applications, and Tools , Philadelphia, PA : SIAM Publications .
   Google Scholar
• Berz , M. , Makino , K. , Shamseddine , K. , Hoffstätter , G. H. and Wan , W. 1996 . COSY INFINITY and its applications in nonlinear dynamics, Computational Differentiation: Techniques, Applications, and Tools , Edited by: Berz , M. , Bischof , C. , Corliss , G. and Griewank , A. Philadelphia, PA : SIAM Publications .
   Google Scholar
• Caglar , H. N. , Caglar , S. H. and Twizell , E. H. 1999 . The numerical solution of third-order boundary-value problems with fourth-degree B-spline functions . Int. J. Comput. Math. , 71 : 373 – 381 .
   Web of Science ®Google Scholar
• Charpentier , I. and Utke , J. 2009 . Rapsodia: User Manual , Argonne National Laboratory . Technical Report
   Google Scholar
• Cherruault , Y. 1989 . Convergence of Adomian's method . Kybernetes Cybern. Gen. Syst. , 18 : 31 – 38 .
   Web of Science ®Google Scholar
• Cherruault , Y. and Adomian , G. 1993 . Decomposition methods: A new proof of convergence . Math. Comput. Model. , 18 : 103 – 106 .
   Google Scholar
• Chiu , C.-H. and Chen , C.-K. 2002 . A decomposition method for solving the convective longitudinal fins with variable thermal conductivity . Int. J. Heat Mass Transf. , 45 : 2067 – 2075 .
   Web of Science ®Google Scholar
• Dehghan , M. and Tatari , M. 2010 . Finding approximate solutions for a class of third-order non-linear boundary value problems via the decomposition method of Adomian . Int. J. Comput. Math. , 87 : 1256 – 1263 .
   Web of Science ®Google Scholar
• El-Danaf , T. S. 2008 . Quartic nonpolynomial spline solutions for third order two-point boundary value problem . World Acad. Science Eng. Technol. , 45 : 453 – 456 .
   Google Scholar
• Gear , C. W. 1971 . Numerical Initial Value Problems in Ordinary Differential Equations , New Jersey : Prentice Hall .
   Google Scholar
• Ghorbani , A. 2009 . Beyond Adomian polynomials: He polynomials . Chaos, Solitons Fractals , 39 : 1486 – 1492 .
   Web of Science ®Google Scholar
• Griewank , A. and Walther , A. 2008 . Evaluating Derivatives: Principles and Techniques of Algorithmic Differentiation, Other Titles in Applied Mathematics , 2 , Vol. 105 , Philadelphia, PA : SIAM Publications .
   Google Scholar
• Griewank , A. , Juedes , D. and Utke , J. 1996 . Algorithm 755: ADOL-C: A package for the automatic differentiation of algorithms written in C/C++ . ACM Trans. Math. Softw. , 22 : 131 – 167 .
   Web of Science ®Google Scholar
• He , J. H. 2000 . Variational iteration method for autonomous ordinary differential systems . Appl. Math. Comput. , 114 : 115 – 123 .
   Web of Science ®Google Scholar
• He , J. H. 2004 . Variational principles for some nonlinear partial differential equations with variable coefficients . Chaos, Solitons Fractals , 19 : 847 – 851 .
   Web of Science ®Google Scholar
• He , J. H. 2006 . Some asymptotic methods for strongly nonlinear equations . Int. J. Mod. Phys. B , 20 : 1141 – 1199 .
   Web of Science ®Google Scholar
• He , J. H. 2007 . Variational iteration method–Some recent results and new interpretations . J. Comput. Appl. Math. , 207 : 3 – 17 .
   Web of Science ®Google Scholar
• Howarth , L. 1938 . On the solution of the laminar boundary layer equations . Proc. R. Soc. Lond. A , 164 : 547 – 579 .
   Google Scholar
• Islam , S-U. and Tirmizi , I. A. 2006 . A smooth approximation for the solution of special nonlinear third-order boundary-value problems based on non-polynomial splines . Int. J. Comput. Math. , 83 : 397 – 407 .
   Google Scholar
• Khan , A. and Aziz , T. 2003 . The numerical solution of third-order boundary-value problems using quintic splines . Appl. Math. Comput. , 137 : 253 – 260 .
   Web of Science ®Google Scholar
• Lesnic , D. 2002 . Convergence of Adomian's decomposition method: Periodic temperatures . Comput. Math. Appl. , 44 : 13 – 24 .
   Web of Science ®Google Scholar
• Moore , R. E. 1979 . Methods and Applications of Interval Analysis , Philadelphia, PA : SIAM Publications .
   Google Scholar
• Na , T. Y. 1979 . Computational Methods in Engineering Boundary Value Problems , New York and London : Academic Press .
   Google Scholar
• Rall , L. B. 1981 . Automatic Differentiation: Techniques and Applications , Berlin : Springer-Verlag . Lecture Notes in Computer Science, Vol. 120
   Google Scholar
• Saadatmandi , A. and Razzaghi , M. 2007 . The numerical solution of third-order boundary value problems using Sinc-collocation method . Commun. Numer. Meth. Eng. , 23 : 681 – 689 .
   Web of Science ®Google Scholar
• Schlichting , H. 1986 . Boundary Layer Theory , 6 , New York, NY : McGraw Hill .
   Google Scholar
• Shampine , L. , Ketzscher , R. and Forth , S. 2005 . Using AD to solve BVPs in MATLAB . ACM Trans. Math. Softw. , 31 : 79 – 94 .
   Google Scholar
• Shiriaev , D. 1993 . “ Fast automatic differentiation for vector processors and reduction of the spatial complexity in a source translation environment ” . Fakultat¨ für Mathematik der Universität Karlsruhe . PhD Thesis
   Google Scholar
• Tirmizi , S. I.A. 1991 . On the numerical solution of third-order boundary-value problems . Commun. Appl. Numer. Meth. , 7 : 309 – 313 .
   Web of Science ®Google Scholar
• Tirmizi , S. I.A. 1994 . Higher-order numerical methods for third-order non-linear boundary-value problems . Communications in Numer. Methods Eng. , 10 : 934 – 948 .
   Google Scholar
• Tuck , E. O. and Schwartz , L. W. 1990 . A numerical and asymptotic study of some third order ordinary differential equations relevant to draining and coating flows . SIAM Review , 32 : 453 – 549 .
   Web of Science ®Google Scholar
• Usmani , R. A. and Sakai , M. 1984 . Quartic spline solution for two point boundary value problems involving third-order differential equation . J. Math. Phys. Sci. , 18 : 365 – 380 .
   Google Scholar
• Verma , A. 1996 . On the efficient methods to solve ODEs and BVPs using automatic differentiation . Cornell Theory Center Technical Report , : 96 – 253 . Available at http://ecommons.library.cornell.edu/handle/1813/5584
   Google Scholar
• von Hippel , G. M. 2007 . New version announcement for TaylUR, an arbitrary-order diagonal automatic differentiation package for Fortran 95 . Comput. Phys. Commun. , 176 : 710 – 711 .
   Google Scholar
• Wazwaz , A.-M. 1998 . A comparison between Adomian decomposition method and Taylor series method in the series solutions . Appl. Math. Comput. , 97 : 37 – 44 .
   Web of Science ®Google Scholar
• Wazwaz , A.-M. 1999 . A reliable modification of Adomian decomposition method . Appl. Math. Comput. , 102 : 77 – 86 .
   Web of Science ®Google Scholar
• Wazwaz , A.-M. 2000 . A new algorithm for calculating Adomian polynomials for nonlinear operators . Appl. Math. Comput. , 111 : 53 – 69 .
   Web of Science ®Google Scholar
• Wazwaz , A.-M. 2000 . Approximate solutions to boundary value problems of higher order by the modified decomposition method . Comput. Math. Appl. , 40 : 679 – 691 .
   Web of Science ®Google Scholar
• Wazwaz , A.-M. 2001 . Exact solutions to nonlinear diffusion equations obtained by the decomposition method . Appl. Math. Comput. , 123 : 109 – 122 .
   Web of Science ®Google Scholar