A mini-review of numerical methods for high-order problemsFootnote†

†This paper is dedicated to Professor E. H. Twizell.

References

• Zill , D. G. and Cullen , M. R. 1997 . Differential Equations with Boundary-value Problems , 176 – 192 . New York : Brooks Cole Publishing Company . chapter 5
   Google Scholar
• Greenberg , M. D. 2001 . Differential Equations and Linear Algebra , 344 – 351 . Engelwood Cliffs, NJ : Prentice Hall . chapter 7
   Google Scholar
• Nagle , R. K. and Saff , E. B. 1986 . Fundamentals of Differential Equations , 284 – 305 . The Benjamin/Cummings Publishing Company . chapter 6
   Google Scholar
• Davies , A. R. , Karageorghis , A. and Philips , T. N. 1988 . Spectral Galerkin methods for the primary two-point boundary-value problem in modelling visco-elastic flows . International Journal for Numerical Methods in Engineering , 26 : 647 – 662 .
   Web of Science ®Google Scholar
• Khan , M. S. 1994 . “ Finite-difference solutions of fifth-order boundary-value problems ” . 19 – 20 . Brunel University . chapter 1
   Google Scholar
• Chandrasekhar , S. 1981 . Hydrodynamic and Hydromagnatic Stability , Oxford : Clarendon Press . (reprinted: Dover Books, New York)
   Google Scholar
• Boutayeb , A. 1990 . “ Numerical methods for high-order ordinary differential equations with apllication to eigenvalue problems ” . 1 – 12 . Brunel University . PhD thesis, chapter 1
   Google Scholar
• Toomre , J. , Zahn , J. R. , Labour , J. and Spiegel , E. A. 1976 . Stellar convection theory II: single-mode study of the second convection zone in A-type stars . Astrophysics Journal , 207 : 545 – 563 .
   Web of Science ®Google Scholar
• Glatzmaier , G. A. 1985 . Numerical simulations of stellar convective dynamics III: at the base of the convection zone . Geophysics and Astrophysics Fluid Dynamics , 31 : 137 – 150 .
   Web of Science ®Google Scholar
• Whithead , J. A. and Chen , M. M. 1970 . Thermal instability and convection of a thin fluid layer bounded by a stably stratified region . Journal of Fluid Mechanics , 40 : 549 – 570 .
   Web of Science ®Google Scholar
• Baldwin , P. 1987 . A localized instability in Bénard layer . Applicable Analysis , 24 : 117 – 156 .
   Google Scholar
• Baldwin , P. 1987 . Asymptotic estimates of the eigenvalues of a sixth-order boundary-value problem obtained by using global phase-integral methods . Philosophical Transactions of the Royal Society of London A , 322 : 281 – 305 .
   Web of Science ®Google Scholar
• Agarwal , R. P. and Krishnamoorthy , P. R. 1979 . Boudary-value problems for nth-order ordinary differential equations . Bulletin of the Institute of Mathematics, Academia Sinica , 7 : 211 – 230 .
   Google Scholar
• Agarwal , R. P. 1981 . Boudary-value problems for higher-order differential equations . Bulletin of the Institute of Mathematics, Academia Sinica , 9 : 47 – 61 .
   Google Scholar
• Agarwal , R. P. and Akrivis , G. 1982 . Boundary value problems occurring in plate deflection theory . Journal of Computational and Applied Mathematics , 8 : 145 – 154 .
   Google Scholar
• Agarwal , R. P. 1986 . Boudary-value Problems for Higher-order Differential Equations , Singapore : World Scientific .
   Google Scholar
• Davis , J. M. and Henderson , J. 2000 . Monotone methods applied to some higher order boundary-value problems . Journal of Inequalities in Pure and Applied Mathematics , 2 : 1 – 9 .
   Google Scholar
• Granhoff , B. and Bleistein , N. 1972 . Asymptotic solutions of a 6th order differential equation with two turning points. Part 1: Derivation by method of steepest descent . SIAM Journal of Mathematical Analysis , 3 : 45 – 57 .
   Google Scholar
• Sloan , D. M. and Huang , W. 1991 . The pseudospectral method for solving Bénard-type differential eigenproblems . Strathclyde Mathematics Research Report
   Google Scholar
• Tirmizi , S. I.A. 1991 . On numerical solution of third-order boundary-value problems . Communications in Applied Numerical Methods , 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 Applied Numerical Methods in Engineering , 10 : 943 – 988 .
   Web of Science ®Google Scholar
• Tirmizi , S. I.A. and Twizell , E. H. 2001 . A fourth-order extrapolation method for special non-linear third-order boundary-value problems . Communications in Applied Numerical Methods in Engineering , in press
   Google Scholar
• Khan , A. and Aziz , T. 2002 . The numerical solution of third-order boundary-value problems using quintic splines . International Journal of Computer Mathematics , 79 : 1025 – 1036 .
   Web of Science ®Google Scholar
• Awoyemi , D. O. and Idowu , O. M. 2005 . A class of hybrid collocation methods for third-order ordinary differential equations . International Journal of Computer Mathematics , 82 : 1287 – 1293 .
   Web of Science ®Google Scholar
• Tirmizi , I. A. , Twizell , E. H. and Siraj-Ul-Islam . 2005 . A numerical method for third-order non-linear boundary-value problems in engineering . International Journal of Computer Mathematics , 82 : 103 – 109 .
   Web of Science ®Google Scholar
• Siraj-Ul-Islam , Khan , M. A. , Tirmizi , I. A. and Twizell , E. H. 2005 . Non polynomial spline approach to the solution of a system of third-order boundary-value problems . Applied Mathematics and Computation , 168 : 152 – 163 .
   Web of Science ®Google Scholar
• Çaglar , H. N. , Çaglar , S. H. and Twizell , E. H. 1999 . The numerical solution of third-order boundary-value problems with fourth-degree B-spline functions . International Journal of Computer Mathematics , 71 : 373 – 381 .
   Web of Science ®Google Scholar
• Twizell , E. H. 1986 . A fourth-order extrapolation method for special non linear fourth-order boundary-value problems . Communications in Applied Numerical Methods , 2 : 593 – 602 .
   Web of Science ®Google Scholar
• Twizell , E. H. 1987 . A sixth-order extrapolation method for special non linear fourth-order boundary-value problems . Computer Methods in Applied Mechanics and Engineering , 62 : 593 – 602 .
   Web of Science ®Google Scholar
• Usmani , R. A. and Sakai , M. 1987 . Two finite difference methods for computing eigenvalues of a fourth order linear boundary-value problem . International Journal of Mathematics and Mathematical Sciences , 10 : 525 – 530 .
   Google Scholar
• Usmani , R. A. and Taylor , P. J. 1983 . Finite difference methods for solving p(4)(x)+q(x)y=r(x) . International Journal of Computer Mathematics , 14 : 277 – 293 .
   Web of Science ®Google Scholar
• Taylor , P. J. and Usmani , R. A. 1980 . Convergence of numerical approximation to a fourth-order boundary-value problem . International Journal of Computer Mathematics , 8 : 265 – 270 .
   Web of Science ®Google Scholar
• Usmani , R. A. and Marsden , M. J. Convergence of a numerical procedure for the solution of a fourth-order boundary-value problem . Proceedings of the Indian Academy of Sciences , Vol. 88 , pp. 21 – 30 .
   Google Scholar
• Usmani , R. A. 1979 . Discrete methods for boundary-value problems with application in plate deflection theory . ZAMP , 30 : 87 – 99 .
   Web of Science ®Google Scholar
• Usmani , R. A. 1979 . On the numerical integration of a boundary-value problem involving a fourth order linear differential equation . BIT , 17 : 227 – 234 .
   Google Scholar
• Usmani , R. A. A uniqueness theorem for a boundary-value problem . Proceedings of the American Mathematical Society , Vol. 77 , pp. 329 – 335 .
   Google Scholar
• Al-Said , E. A. , Noor , M. A. , Kaya , D. and Al-Khaled , K. 2004 . Finite difference method for solving fourth-order obstacle problems . International Journal of Computer Mathematics , 81 : 741 – 748 .
   Web of Science ®Google Scholar
• Al-Said , E. A. and Noor , M. A. 1995 . Computational methods for fourth-order obstacle boundary-value problems . Communications in Applied Non Linear Analysis , 2 : 73 – 83 .
   Google Scholar
• Usmani , R. A. 1986 . Finite difference methods for computing eigenvalues of fourth order boundary-value problem . International Journal of Mathematics and Mathematical Sciences , 9 : 137 – 143 .
   Google Scholar
• Jain , M. K. , Lyenger , S. R.K. and Saldanha , J. S.V. 1977 . Numerical integration of a fourth-order differential equation . Journal Engineering Mathematics , 11 : 373 – 380 .
   Web of Science ®Google Scholar
• Chawla , M. M. 1983 . A new fourth-order finite difference method for computing eigenvalues of a fourth order two-point . IMA Journal of Numerical Analysis , 3 : 291 – 293 .
   Web of Science ®Google Scholar
• Chawla , M. M. and Katti , C. P. 1982 . A new symmetric five-diagonal finite difference method for computing eigenvalues of fourth order two-point . Journal of Computational and Applied Mathematics , 8 : 135 – 136 .
   Google Scholar
• Li , J. 2005 . High-order finite difference schemes for differential equations containing higher derivatives . Applied Mathematics and Computation , 171 : 1157 – 1176 .
   Web of Science ®Google Scholar
• Twizell , E. H. 1988 . “ Numerical methods for sixth-order boundary-value problems ” . In Numerical Mathematics , Edited by: Agarwal , R. P. , Chow , Y. M. and Xilson , S. J. 495 – 506 . Basel : Birkhausezr .
   Google Scholar
• Boutayeb , A. and Twizell , E. H. 1992 . Numerical methods for the solution of special sixth-order boundary-value problems . International Journal of Computer Mathematics , 45 : 207 – 223 .
   Web of Science ®Google Scholar
• Twizell , E. H. and Boutayeb , A. Numerical methods for the solution of special and general sixth-order boundary-value problems with application to Bénard eigenvalue problems . Proceedings of the Royal Society of London , Vol. 431 , pp. 433 – 450 .
   Google Scholar
• Boutayeb , A. and Twizell , E. H. 1993 . Finite difference methods for the solution of special eighth-order boundary-value problems . International Journal of Computer Mathematics , 48 : 63 – 75 .
   Web of Science ®Google Scholar
• Twizell , E. H. , Boutayeb , A. and Djidjeli , K. 1994 . Numerical methods for eighth-, tenth- and twelfth-order eigenvalue problems arising in thermal instability . Advances in Computational Mathematics , 2 : 407 – 436 .
   Google Scholar
• Boutayeb , A. and Twizell , E. H. 1993 . Finite-difference methods for twelfth-order boundary-value problems . Journal of Computational and Applied Mathematics , 35 : 133 – 138 .
   Web of Science ®Google Scholar
• Inc , M. and Evans , D. J. 2004 . An efficient approach to approximate solutions of eighth-order boundary-value problems . International Journal of Computer Mathematics , 81 : 685 – 692 .
   Web of Science ®Google Scholar
• Djidjeli , K. , Twizell , E. H. and Boutayeb , A. 1993 . Numerical methods for special non linear boundary-value problems of order 2m . Journal of Computational and Applied Mathematics , 47 : 35 – 45 .
   Web of Science ®Google Scholar
• Chawla , M. M. and Katti , C. P. 1979 . Finite difference methods for two-point boundary-value problems involving higher-order differential equations . BIT , 19 : 27 – 33 .
   Google Scholar
• Zhu , S. and Phan , H. T. 2003 . A new numerical approach for solving high-order non-linear ordinary differential equations . Communications in Numerical Methods in Engineering , 19 : 601 – 614 .
   Web of Science ®Google Scholar
• Subasi , M. 2004 . Approximations to the solution of boundary-value problems of high order (2N) . Applied Numerical Mathematics , 48 : 103 – 111 .
   Web of Science ®Google Scholar
• Mai-Duy , N. 2005 . Solving high order ordinary differential equations with radial basis function networks . International Journal for Numerical Methods in Engineering , 62 : 824 – 852 .
   Web of Science ®Google Scholar
• Wazwaz , A. M. 2000 . Approximate solutions to boundary-value problems of higher-order by the modified decomposition method . Computer Mathematics with Applications , 40 : 679 – 91 .
   Web of Science ®Google Scholar
• Khaliq , A. Q.M. and Twizell , E. H. 1989 . Global extrapolation on three and four grids for special initial value problems . Applied Mathematics Letters , 2 : 35 – 37 .
   Google Scholar
• Twizell , E. H. 1988 . Family of numerical methods for the solution of high-order general initial value problems . Computer Methods in Applied Mechanics and Engineering , 67 : 15 – 25 .
   Web of Science ®Google Scholar
• Wazwaz , A. M. 2001 . The numerical solution of fifth-order boundary-value problems by the decomposition method . Journal of Computational and Applied Mathematics , 136 : 259 – 270 .
   Web of Science ®Google Scholar
• Wazwaz , A. M. 2002 . The numerical solution of fifth-order boundary-value problems by the decomposition method . International Journal of Computer Mathematics , 79 : 345 – 356 .
   Web of Science ®Google Scholar
• Adomian , G. 1988 . A review of the decompostion method in applied mathematics . Journal of Mathematical Analysis and Applications , 135 : 501 – 544 .
   Web of Science ®Google Scholar
• Adomian , G. 1994 . Solving Frontier Problems of Physics. The Decomposition Method , Boston : Kluwer .
   Google Scholar
• Wazwaz , A. M. 1998 . A comparison between Adomian decomposition and Taylor series method in series solutions . Applied Mathematics and Computation , 79 : 37 – 44 .
   Web of Science ®Google Scholar
• Wazwaz , A. M. 1999 . A reliable modification of Adomian's decomposition method . Applied Mathematics and Computation , 102 : 77 – 86 .
   Web of Science ®Google Scholar
• Cherrault , Y. 1990 . Convergence of Adomian's method . Mathematical Computer Modelling , 14 : 83 – 86 .
   Google Scholar
• Cherrault , Y. and Adomian , G. 1993 . Decomposition method: a new proof of convergence . Mathematical Computer Modelling , 18 : 103 – 106 .
   Google Scholar
• Cherrault , Y. and Saccomandi , B. 1992 . Some new results for convergence of Adomian's method applied to integral equations . Mathematical Computer Modelling , 16 : 85 – 93 .
   Google Scholar
• Cherrault , Y. 2005 . A comparison of numerical solutions of fourth-order boundary-value problems . International Journal of Systems & Cybernetics , 34 : 960 – 968 .
   Web of Science ®Google Scholar
• Wazwaz , A. M. 2001 . The numerical solution of sixth-order boundary-value problems by the modified decomposition method . Applied Mathematics and Computation , 118 : 311 – 325 .
   Web of Science ®Google Scholar
• Wazwaz , A. M. 2002 . The numerical solution of special fourth-order boundary-value problems by the modified decomposition method . International Journal of Computer Mathematics , 79 : 345 – 356 .
   Web of Science ®Google Scholar
• Wazwaz , A. M. 2000 . The modified Adomian decomposition method for solving linear and non linear boundary-value problems of 10th-order and 12th-order . International Journal of Nonlinear Sciences and Numerical Simulation , 1 : 17 – 24 .
   Google Scholar
• Al-Said , E. A. , Noor , M. A. and Rassias , T. M. 2006 . Cubic splines method for solving fourth-order obstacle problems . Applied Mathematics and Computation , 174 : 180 – 187 .
   Web of Science ®Google Scholar
• Papamichael , N. and Worsey , A. J. 1981 . A cubic spline method for the solution of linear fourth-order two-point boundary-value problem . Journal of Computational and Applied Mathematics , 7 : 187 – 189 .
   Google Scholar
• Al-Said , E. A. and Noor , M. A. 2002 . Quartic spline method for solving fourth-order obstacle boundary-value problems . Journal of Computational and applied Mathematics , 143 : 107 – 116 .
   Web of Science ®Google Scholar
• Khalifa , A. K. and Noor , M. A. 1990 . Quintic splines solutions of a class of contact problems . Mathematical Computer Modelling , 13 : 51 – 58 .
   Google Scholar
• Usmani , R. A. 1992 . The use of quatric splines in the numerical solution of fourth-order boundary-value problems . Journal of Computational and Applied Mathematics , 44 : 187 – 199 .
   Web of Science ®Google Scholar
• Siddiqi , S. S. and Akram , G. 2006 . Solution of fifth order boundary-value problems using non polynomial spline technique . Applied Mathematics and Computation , 175 : 1574 – 1581 .
   Web of Science ®Google Scholar
• Khan , M. A. , Siraj-ul-Islam , Tirmizi , I. A. , Twizell , E. H. and Ashraf , S. 2006 . A class of methods based on non-polynomial sextic spline functions for the solution of a special fifth-order boundary-value problems . Journal of Mathematical Analysis and Applications , 321 : 651 – 660 .
   Web of Science ®Google Scholar
• Çaglar , H. N. , Çaglar , S. H. and Twizell , E. H. 1999 . The numerical solution of fifth-order boundary-value problems with sixth-degree B-spline functions . Applied Mathematics Letters , 12 : 25 – 30 .
   Web of Science ®Google Scholar
• Fyfe , D. J. 1971 . Linear dependence relations connecting equal interval Nth degree splines and their derivatives . Journal of Applied Mathematics , 7 : 398 – 406 .
   Google Scholar
• Akram , G. and Siddiqi , S. S. 2006 . Solution of sixth-order boundary-value problems using non polynomial spline technique . Applied Mathematics and Computation , in press
   PubMed Web of Science ®Google Scholar
• Siddiqi , S. S. and Twizell , E. H. 1996 . Spline solutions of linear sixth-order boundary-value problems . International Journal of Computer Mathematics , 60 : 295 – 304 .
   Web of Science ®Google Scholar
• Siddiqi , S. S. and Twizell , E. H. 1996 . Spline solutions of linear eighth-order boundary-value problems . Computational Methods in Applied Mechanical Engineering , 131 : 309 – 325 .
   Web of Science ®Google Scholar
• Siddiqi , S. S. and Twizell , E. H. 1997 . Spline solutions of linear twelfth-order boundary-value problems . Journal of Computational and Applied Mathematics , 78 : 371 – 390 .
   Web of Science ®Google Scholar
• Siddiqi , S. S. and Akram , G. 1998 . Spline solutions of linear tenth-order boundary-value problems . International Journal of Computer Mathematics , 68 : 345 – 362 .
   Web of Science ®Google Scholar
• Akram , G. and Siddiqi , S. S. 2006 . Nonic spline solution of eighth order boundary-value problems . Applied Mathematics and Computation , in press
   PubMed Web of Science ®Google Scholar
• Wang , Y. , Zhao , Y. B. and Wei , G. W. 2003 . A note on the numerical solution of high-order differential equations . Journal of Computaional and Applied Mathematics , 159 : 387 – 398 .
   Web of Science ®Google Scholar
• Wu , T. Y. and Liu , G. R. 2000 . Application of generalized differential quadrature rule to sixth-order differential equations . Communications in Numerical Methods in Engineering , 16 : 777 – 784 .
   Web of Science ®Google Scholar
• Liu , G. R. and Wu , T. Y. 2002 . Differential quadrature solutions of eighth-order boundary-value differential equations . Journal of Computaional and Applied Mathematics , 145 : 223 – 235 .
   Web of Science ®Google Scholar
• Wu , T. Y. and Liu , G. R. 2001 . The generalized differential quadrature rule for fourth-order differential equations . International Journal of Numerical Methods in Engineering , 50 : 1907 – 1929 .
   Web of Science ®Google Scholar
• He , J. H. 2003 . Variational approach to the sixth-order boundary-value problem . Applied Mathematics and Computation , 143 : 537 – 538 .
   Web of Science ®Google Scholar
• Noor , M. A. and Al-Said , E. A. 2000 . Numerical solutions of fourth-order variational inequalities . International Journal of Computer Mathematics , 75 : 387 – 395 .
   Web of Science ®Google Scholar
• Noor , M. A. 1997 . Some recent advances in variational inequalities, Part I. Basic Concepts . New Zeland Journal of Mathematics , 26 : 53 – 80 .
   Google Scholar
• Noor , M. A. 1997 . Some recent advances in variational inequalities, Part II. Other concepts . New Zeland Journal of Mathematics , 26 : 229 – 255 .
   Google Scholar
• El-Gamel , M. , Behiry , S. H. and Hashish , H. 2003 . Numerical method for the solution of special non linear fourth-order boundary-value problems . Applied Mathematics and Computation , 145 : 717 – 734 .
   Web of Science ®Google Scholar
• Barrett , K. E. 1977 . Finite element analysis for flow between rotating discs using exponential weighted basis functions . International Journal of Numerical Methods in Engineering , 12 : 1809 – 1817 .
   Web of Science ®Google Scholar
• Banerjee , J. R. and Su , H. 2006 . Free transverse and lateral vibration of beams with torsional coupling . Journal of Aerospace Engineering , 19 : 13 – 20 .
   Web of Science ®Google Scholar
• Lee , S. Y. and Chao , J. C. 2001 . Exact solutions for out-of-plane vibration of curved nonuniform beams . Journal of Applied Mechanics , 68 : 186 – 191 .
   Web of Science ®Google Scholar
• Shanthi , V. and Ramanujam , N. 2003 . Asymptotic numerical methods for singularly perturbed fourth-order ordinary differential equations of reaction-diffusion type . Computers and Mathematics with Applications , 46 : 463 – 478 .
   Web of Science ®Google Scholar
• Shanthi , V. and Ramanujam , N. 2004 . A boundary-value technique for boundary-value problems for singularly perturbed fourth-order ordinary differential equations . Computers and Mathematics with Applications , 47 : 1673 – 1688 .
   Web of Science ®Google Scholar
• Shanthi , V. and Ramanujam , N. 2004 . Computational methods for reaction-diffusion problems for fourth order ordinary differential equations with a small parameter at the highest derivative . Applied Mathematics and Computation , 147 : 97 – 113 .
   Web of Science ®Google Scholar
• Shanthi , V. and Ramanujam , N. 2006 . Asymptotic numerical method for boundary-value problems for singularly perturbed fourth-order ordinary differential equations with a weak interior layer . Applied Mathematics and Computation , 172 : 252 – 266 .
   Web of Science ®Google Scholar
• Smith , R. C. 1991 . The Sinc–Galerkin method for fourth-order differential equations . SIAM Journal of Numerical Analysis , 28 : 760 – 788 .
   Web of Science ®Google Scholar
• El-Gamel , M. , Cannon , J. R. and Zayed , A. I. 2004 . Sinc–Galerkin method for solving linear sixth-order boundary-value problems . Mathematcs of Computation , 7 : 1325 – 1343 .
   Google Scholar
• El-Gamel , M. and Zayed , A. I. 2004 . Sinc–Galerkin method for solving non linear boundary-value problems . Computer and Mathematics with Applications , 48 : 1285 – 1298 .
   Web of Science ®Google Scholar
• Ralph , C. and Bowers , K. 1991 . The Sinc–Galerkin method for fourth-order differential equations . SIAM Journal of Numererical Analysis , 28 : 760 – 788 .
   Web of Science ®Google Scholar
• Davis , A. R. , Karageorphis , A. and Philips , T. N. 1988 . Spectral Galerkin methods for the primary two-point boundary-value problem in modeling viscelasticitic flows . International Journal of Numerical Methods in Engineering , 26 : 647 – 662 .
   Web of Science ®Google Scholar
• Davis , A. R. , Karageorphis , A. and Philips , T. N. 1988 . Spectral collocation methods for the primary two-point boundary-value problem in modeling viscelasticitic flows . International Journal of Numerical Methods in Engineering , 26 : 805 – 813 .
   Web of Science ®Google Scholar
• Taiwo , O. A. and Evans , D. J. 1997 . Collocation approximation for fourth-order boundary-value problems . International Journal of Computer Mathematics , 63 : 57 – 66 .
   Web of Science ®Google Scholar
• Nurmuhammad , A. , Muhammad , M. , Mori , M. and Sugihara , M. 2005 . Double exponential transformation in the Sinc-collocation method for a boundary-value problem with fourth-order ordinary differential equation . Journal of Computational and Applied Mathematics , 182 : 32 – 50 .
   Web of Science ®Google Scholar