Bernoulli polynomial and the numerical solution of high-order boundary value problems

Volume 4, Issue 1, pp 45--59 http://dx.doi.org/10.22436/mns.04.01.05

Publication Date: August 20, 2019 Submission Date: November 28, 2018 Revision Date: March 03, 2019 Accteptance Date: March 05, 2019

Download PDF

Download XML

• Export citations
  • CITATIONS & REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • ARTICLE CITATION
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)

• 1807 Downloads
• 3664 Views

Authors

Mohamed El-Gamel - Department of Mathematics and Engineering Physics, Faculty of Engineering, Mansoura University, Egypt. Waleed Adel - Department of Mathematics and Engineering Physics, Faculty of Engineering, Mansoura University, Egypt. M. S. El-Azab - Department of Mathematics and Engineering Physics, Faculty of Engineering, Mansoura University, Egypt.

Abstract

In this work we present a fast and accurate numerical approach for the higher-order boundary value problems via Bernoulli collocation method. Properties of Bernoulli polynomial along with their operational matrices are presented which is used to reduce the problems to systems of either linear or nonlinear algebraic equations. Error analysis is included. Numerical examples illustrate the pertinent characteristic of the method and its applications to a wide variety of model problems. The results are compared to other methods.

Share and Cite

Keywords

• Bernoulli
• collocation
• higher-order
• astrophysics
• error analysis

MSC

•  35G15
•  74H15

References

• [1] S. Abbasbandy, A. Shirzadi, The variational iteration method for a class of eighth-order boundary-value differential equations, Z. Naturforsch. A, 63 (2008), 745--751
  • View Article
  • Google Scholar


• [2] M. Abramowitz, I. A. Stegun, Handbook of mathematical functions with formulas, Graphs, and Mathematical Tables, John Wiley & Sons, New York (1972)
  • View Article
  • MathSciNet
  • Google Scholar


• [3] R. P. Agarwal, Boundary Value Problems for Higher Order Differential Equations, World Scientific Publishing Co., Teaneck (1986)
  • View Article
  • MathSciNet
  • Google Scholar


• [4] G. Akram, H. U. Rehman, Numerical solution of eighth order boundary value problems in reproducing kernel space, Numer. Algorithms, 62 (2013), 527--540
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [5] U. M. Asher, R. M. M. Mattheij, R. D. Russel, Numerical solution of boundary value problems for ordinary differential equations, Siam, Philadelphia (1995)
  • View Article
  • Google Scholar


• [6] E. Babolian, Z. Masouri, Direct method to solve Volterra integral equation of the first kind using operational matrix with block-pulse functions, J. Comput. Appl. Math., 220 (2008), 51--57
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [7] P. Baldwin, Asymptotic estimates of the eigenvalues of a sixth-order boundary value problem obtained by using global phase-integral methods, Philos. Trans. Roy. Soc. London Ser. A, 322 (1987), 281--305
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [8] A. H. Bhrawy, E. Tohidi, F. Soleymani, A new Bernoulli matrix method for solving high-order linear and nonlinear Fredholm integro-differential equations with piecewise intervals, Appl. Math. Comput., 219 (2012), 482--497
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [9] K. E. Biçer, S. Yalçinbaş, Numerical solution of telegraph equation using Bernoulli collocation method, Proc. Natl. Acad. Sci. India. Sect. A: Phys. Sci., 2018 (2018), 1--7
  • View Article
  • Google Scholar


• [10] R. E. D. Bishop, S. M. Cannon, S. Miao, On coupled bending and torsional vibration of uniform beams, J. Sound Vib., 131 (1989), 457--464
  • View Article
  • Google Scholar
  • MATH


• [11] A. Boutayeb, E. H. Twizell, Numerical methods for the solution of special sixth-order boundary value problems, Int. J. Comput. Math., 45 (1992), 207--233
  • View Article
  • Google Scholar
  • MATH


• [12] A. Boutayeb, E. H. Twizell, Finite difference methods for the solution of eighth-order boundary value problems, Int. J. Comput. Math., 48 (1993), 63--75
  • View Article
  • Google Scholar
  • MATH


• [13] S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability, Clarendon Press, Oxford (1961)
  • View Article
  • MathSciNet
  • Google Scholar


• [14] M. M. Chawla, C. P. Katti, Finite difference methods for two-point boundary value problems involving higher order differential equations, BIT Numer. Math., 19 (1979), 27--33
  • View Article
  • Google Scholar


• [15] A. R. Davies, A. Karageoghis, T. N. Phillips, Spectral Galerkin methods for the primary two-point boundary value problem in modelling viscoelastic flows, Int. J. Numer. Math. Eng., 26 (1988), 647--652
  • View Article
  • Google Scholar
  • MATH


• [16] M. El-Gamel, A. Abdrabou, Sinc-Galerkin solution to eighth-order boundary value problems, SeMA J., 76 (2019), 249--270
  • View Article
  • MathSciNet
  • Google Scholar


• [17] M. El-Gamel, W. Adel, Numerical investigation of the solution of higher-order boundary value problems via Euler matrix method, SeMA J., 75 (2018), 349--364
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [18] M. El-Gamel, W. Adel, M. S. El-Azab, Collocation method based on Bernoulli polynomial and shifted Chebychev for solving the Bratu equation, J. Appl. Computat. Math., 7 (2018), 7 pages
  • View Article


• [19] M. El-Gamel, M. S. El-Azab, M. Fathy, The numerical solution of sixth order boundary value problems by the Legendre Galerkin method, Acta Univ. Apulensis Math. Inform., 40 (2014), 145--165
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [20] M. El-Gamel, A. I. Zayed, Sinc-Galerkin method for solving non-linear boundary value problems, Comput. Math. Appl., 48 (2004), 1285--1298
  • View Article
  • Google Scholar


• [21] K. Erdem, S. Yalçinbaş, M. Sezer, A Bernoulli polynomial approach with residual correction for solving mixed linear Fredholm integro-differential-difference equations, J. Difference Equ. Appl., 19 (2013), 1619--1631
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [22] V. S. Ertürk, S. Momani, Comparing numerical methods for solving fourth-order boundary value problems, Appl. Math. Comput., 188 (2007), 1963--1968
  • View Article
  • Google Scholar
  • MATH


• [23] F. Z. Geng, A new reproducing kernel Hilbert space method for solving nonlinear fourth-order boundary value problems, Appl. Math. Comput., 213 (2009), 163--169
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [24] A. Golbabai, S. P. A. Beik, An efficient method based on operational matrices of Bernoulli polynomials for solving matrix differential equations, Comput. Appl. Math., 34 (2015), 159--175
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [25] A. Golbabai, M. Javidi, Application of homotopy perturbation method for solving eighth-order boundary value problems, Appl. Math. Comput., 191 (2007), 334--346
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [26] F. Haq, A. Ali, Numerical solution of fourth-order boundary value problems using Haar Wavelets, Appl. Math. Sci. (Ruse), 5 (2011), 3131--3146
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [27] F. Haq, A. Ali, I. Hussain, Solution of sixth-order boundary-value problems by collocation method using Haar wavelets, Int. J. Phys. Sci., 7 (2012), 5729--5735
  • View Article
  • Google Scholar


• [28] E. Hesameddini, M. Riahi, Bernoulli Galerkin matrix method and its convergence analysis for solving system of volterra–fredholm integro-differential Equations, Iran. J. Sci. Technol. Trans. Sci., 43 (2019), 1203--1214
  • View Article
  • MathSciNet
  • Google Scholar


• [29] M. Inc, D. J. Evans, An efficient approach to approximate solutions of eighth-order boundary value problems, Int. J. Comput. Math., 81 (2004), 685--692
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [30] M. S. Islam, M. B. Hossain, Numerical solutions of eighth-order bvp by the Galerkin residual technique with Bernstein and Legendre polynomials, Appl. Math. Comput., 261 (2015), 48--59
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [31] M. S. Islam, I. A. Tirmizi, F. Haq, M. A. Khan, Non-polynomial splines approach to the solution of sixth-order boundary-value problems, Appl. Math. Comput., 195 (2008), 270--284
  • View Article
  • MathSciNet
  • Google Scholar


• [32] V. I. Krylov, Approximate calculations of integrals, The Macmillan Co., New York-London (1962)
  • View Article
  • MathSciNet
  • Google Scholar


• [33] F.-G. Lang, X.-P. Xu, An effective method for numerical solution and numerical derivatives for sixth order two-Point boundary value problems, Comput. Math. Math. Phys., 55 (2015), 811--822
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [34] S. X. Liang, D. J. Jeffrey, An efficient analytical approach for solving fourth-order boundary value problems, Comput. Phys. Comm., 180 (2009), 2034--2040
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [35] G. R. Liu, T. Y. Wu, Differential quadrature solutions of eighth-order boundary value differential equations, J. Comput. Appl. Math., 145 (2002), 223--235
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [36] K. Maleknejad, R. Mollapourasl, M. Alizadeh, Numerical solution of the Volterra type integral equation of the first kind with wavelet basis, Appl. Math. Comput., 194 (2007), 400--405
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [37] K. Maleknejad, B. Rahimi, Modification of block pulse functions and their application to solve numerically Volterra integral equation of the first kind, Commun. Nonlinear Sci. Numer. Simul., 16 (2011), 2469--2477
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [38] M. Matinfar, H. A. Lashaki, M. Akbari, Numerical approximation based on the Bernoulli polynomials for solving volterra integro-differential equations of high order, Math. Rese., 3 (2017), 19--32
  • View Article


• [39] F. Mirzaee, S. Bimesl, An efficient numerical approach for solving systems of high-order linear Volterra integral equations, Scientia Iranica, 21 (2014), 2250--2263
  • View Article
  • Google Scholar


• [40] S. T. Mohyud-Din, M. A. Noor, Homotopy perturbation method for solving fourth-order boundary value problems, Math. Probl. Eng., 2007 (2007), 15 pages
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [41] S. Momani, M. A. Noor, Numerical comparison of methods for solving a special fourth-order boundary value problem, Appl. Math. Comput., 191 (2007), 218--224
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [42] A. Napoli, Solutions of linear second order initial value problems by using Bernoulli polynomials, Appl. Numer. Math., 99 (2016), 109--120
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [43] M. A. Noor, S. T. Mohyud-Din, Homotopy perturbation method for solving sixth-order boundary value problems, Comput. Math. Appl., 55 (2008), 2953--2972
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [44] M. A. Noor, K. I. Noor, S. T. Mohyud-Din, Variational iteration method for solving sixth-order boundary value problems, Commun. Nonlinear Sci. Numer. Simul., 14 (2009), 2571--2580
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [45] M. G. Porshokouhi, B. Ghanbari, M. Gholami, M. Rashidi, Numerical solution of eighth-order boundary value problems with variational iteration method, Gen. Math. Notes, 2 (2011), 128--133
  • View Article
  • Google Scholar
  • MATH


• [46] S. S. Siddiqi, G. Akram, Solution of eighth-order boundary value problems using the non-polynomial spline technique, Int. J. Comput. Math., 84 (2007), 347--368
  • View Article
  • Google Scholar
  • MATH


• [47] S. S. Siddiqi, E. H. Twizell, Spline solutions of linear eighth-order boundary value problems, Comput. Methods Appl. Mech. Eng., 131 (1996), 309--325
  • View Article
  • MATH


• [48] S. S. Siddiqi, E. H. Twizell, Spline solutions of linear sixth-order boundary value problems, Int. J. Comput. Math., 60 (1996), 295--304
  • View Article
  • MATH


• [49] E. Tohidi, A. H. Bhrawy, K. Erfani, A collocation method based on Bernoulli operational matrix for numerical solution of generalized pantograph equation, Appl. Math. Model., 37 (2013), 4283--4294
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [50] E. Tohidi, K. Erfani, M. Gachpazan, S. Shateyi, A new Tau method for solving nonlinear Lane-Emden type equations via Bernoulli operational matrix of differentiation, J. Appl. Math., 2013 (2013), 9 pages
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [51] E. Tohidi, A. Kiliçman, A Collocation method Based on the Bernoulli operational matrix for solving nonlinear BVPs which arise from the problems in calculus of variation, Math. Probl. Eng., 2013 (2013), 9 pages
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [52] F. Toutounian, E. Tohidi, A new Bernoulli matrix method for solving second order linear partial differential equations with the convergence analysis, Appl. Math. Comput., 223 (2013), 298--310
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [53] F. Toutounian, E. Tohidi, S. Shateyi, A collocation method based on Bernoulli operational matrix for solving high-order linear complex differential equations in a rectangular domain, Abstr. Appl. Anal., 2013 (2013), 12 pages
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [54] K. N. S. K. Viswanadham, S. Ballem, Numerical solution of eighth-order boundary value problems by Galerkin method with quintic B-splines, Int. J. Comput. Appl., 89 (2014), 8880--8887
  • View Article
  • Google Scholar


• [55] A.-M. Wazwaz, The numerical solutions of special eighth-order boundary value problems by the modified decomposition method, Neural Parallel Sci. Comput., 8 (2000), 133--146
  • View Article
  • Google Scholar


• [56] A.-M. Wazwaz, The numerical solution of special fourth-order boundary value problems by the modified decomposition method, Int. J. Comput. Math., 79 (2002), 345--356
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [57] https://www.mathworks.com/help/symbolic/dsolve.html, , , (),