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DOI：http://dx.doi.org/10.26855/jamc.2022.03.014

This article uses residual correction procedure for improving the Galerkin approximate solutions to higher order boundary value problem (BVP). The residual function of a differential equation is found from the approximate solution of a BVP and setting it as nonhomogeneous term we get the error differential equation. We exploit Bernstein and Bernoulli polynomials as basis functions to solve the two differential equations, namely, original and its error equations, by Galerkin technique subject to the corresponding boundary conditions. Linear and nonlinear problems of fourth order BVPs are considered to verify the proposed method. The resulting numerical solutions are compared with the analytic solutions as well as the results of other approaches those have been reported in the literature. This method is also applied to sixth order BVPs. The comparison reveals that the current procedure is more accurate.

[1] Lima, P. M. and Morgado, L. (2010). Numerical modeling of oxygen diffusion in cells with Michaelis-Menten uptake kinetics. J. Math. Chem., 48(1), 145-158.

[2] Rach, R., Duan, J. S., and Wazwaz, A. M. (2014). Solving coupled Lane-Emden boundary value problems in catalytic diffusion reactions by the Adomian decomposition method. J. Math. Chem., 52(1), 255-267.

[3] Chandrasekhar, S. (2013). Hydrodynamic and hydromagnetic stability. Courier Corporation.

[4] Davies, A. R., Karageorghis, A., and Phillips, T. N. (1988). Spectral Galerkin methods for the primary two-point boundary value problem in modelling viscoelastic ﬂows. Int. J. Num. Eng., 26, 647-662.

[5] Ma, T. F. and Da Silva, J. (2004). Iterative solutions for a beam equation with nonlinear boundary conditions of third order. Appl. Math. Comput., 159(1), 11-18.

[6] Chawla, M. M. and Katti, C. P. (1979). Finite difference methods for two-point boundary value problems involving high order differential equations. BIT Numer. Math., 19(1), 27-33.

[7] Graef, J. R., Kong, L., Kong, Q., and Yang, B. (2011). Positive solutions to a fourth Order boundary value problem. Results Math., 59(1-2), 141-155.

[8] Singh, R., Kumar, J., and Nelakanti, G. (2014). Approximate series solution of fourth-order boundary value problems using decomposition method with Green’s function. J. Math. Chem., 52(4), 1099-1118.

[9] Sun, T. and Yi, L. (2016). A new Galerkin spectral element method for fourth-order boundary value problems. Int. J. Comput. Math., 93(6), 915-928.

[10] Kürkçü, Ö. K., Aslan, E., and Sezer, M. (2017). A numerical method for solving some model problems arising in science and convergence analysis based on residual function. Appl. Numer. Math., 121,134-148.

[11] Abd-Elhameed, W. M. and Youssri, Y. H. (2020). Connection formulae between generalized Lucas polynomials and some Jacobi polynomials: Application to certain types of fourth-order BVPs. Int. J. Appl. Comput. Math., 6(2), 1-19.

[12] Wazwaz, A. (2002). The numerical solution of special fourth- order boundary value problems by the modified decomposition method. Int. J. Comput. Math., 79(3), 345-356.

[13] Geng, F. (2009). A new reproducing kernel Hilbert space method for solving nonlinear fourth-order boundary value problems. Appl. Math. Comput., 213(1), 163-169.

[14] Mohyud-Din, S. T. and Noor, M. A. (2006). Homotopy perturbation method for solving fourth-order boundary value problems. Math. Probl. Eng., 2007, 1-15.

[15] Cherruault, Y. (2005). A comparison of numerical solutions of fourth-order boundary value problems. Kybernetes, 34(7-8), 960-968.

[16] Islam, M. S. and Hossain, M. B. (2014). On the use of piecewise standard polynomials in the numerical solutions of fourth order boundary value problems. GANIT J. Bangladesh Math. Soc., 33, 53-64.

[17] Hossain, M. B. and Islam, M. S. (2014). Numerical Solutions of Sixth Order Linear and Nonlinear Boundary Value Problems. Journal of Advances in Mathematics, 7(2), 1180-1190.

[18] Wazwaz, A. M. (2001). The numerical solution of sixth-order boundary value problems by the modified decomposition method. Appl. Math. Comput., 118(2-3), 311-325.

[19] Momani, S. and Moadi, K. (2006). A reliable algorithm for solving fourth-order boundary value problems. J. Appl. Math. Comput., 22(3), 185-197.

[20] Duan, J. S. and Rach, R. (2011). A new modification of the Adomian decomposition method for solving boundary value problems for higher order nonlinear differential equations. Appl. Math. Comput., 218(8), 4090-4118.

[21] He, J.-H. (2006). New interpretation of homotopy perturbation method. Int. J. Modern Phys. B., 20(18), 2561-2568.

[22] Liang, S. and Jeffrey, D. J. (2009). An efficient analytical approach for solving fourth order boundary value problems. Comput. Phys. Commun., 180(11), 2034-2040.

[23] Ertürk, V. S. and Momani, S. (2007). Comparing numerical methods for solving fourth-order boundary value problems. Appl. Math. Comput., 188(2), 1963-1968.

[24] Momani, S. and Noor, M. A. (2007). Numerical comparison of methods for solving a special fourth-order boundary value problem. Appl. Math. Comput., 191(1), 218-224.

[25] Adak, M. and Mandal, A. (2021). Numerical solution of fourth-order boundary value problems for Euler-Bernoulli beam equation using FDM. J. Phys. Conf. Ser., 2070, 1-6.

[26] Toomore, J., Zahn, J. P., Latour, J., and Spiegel, E. A. (1976). Stellar convection theory II: single-mode study of the second convection zone in A-type stars. Astrophys. J., 207, 545-563.

[27] Twizell, E. H. and Boutayeb, A. (1990). Numerical methods for the solution of special and eneral sixth-order boun-dary-value problems, with applications to Bénard layer eigenvalue problems. Proc. R. Soc. London. Ser. A Math. Phys. Sci., 431, 433-450.

[28] Boutayeb, A. and Twizell, E. H. (1992). Numerical methods for the solution of special sixth-order boundary-value problems. Int. J. Comput. Math., 45(3-4), 207-223.

[29] Agarwal, R. P. (1986). Boundary value problems for higher order differential equations. Singapore: World Scientific.

[30] Oliveira, F. A. (1980). Collocation and residual correction. Numer. Math., 36, 27-31.

[31] Çelik, I. (2006). Collocation method and residual correction using Chebyshev series. Appl. Math. Comput., 174(2), 910-920.

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

[33] El-Gamel, M., Cannon, J. R., and Zayed, A. I. (2003). Sinc-Galerkin method for solving linear sixth-order boundary-value problems. 73(247), 1325-1343.

[34] Siddiqi, S. S., Akram, G., and Nazeer, S. (2007). Quintic spline solution of linear sixth-order boundary value problems. Appl. Math. Comput., 189(1), 887-892.

[35] Siddiqi, S. S. and Akram, G. (2008). Septic spline solutions of sixth-order boundary value problems. J. Comput. Appl. Math., 215(1), 288-301.

[36] Khan, A. and Sultana, T. (2012). Parametric quintic spline solution for sixth order two point boundary value problems. Filomat, 26(6), 1233-1245.

[37] Haq, F., Ali, A., and Hussain, I. (2012). Solution of sixth-order boundary-value problems by collocation method using Haar wavelets. Int. J. Phys. Sci., 7(43), 5729-5735.

[38] Reinkenhof, J. (1977). Differentiation and integration using Bernstein’s polynomials. Int. J. Numer. Methods Eng., 11(10), 1627-1630.

[39] Bhatta, D. D. and Bhatti, M. I. (2006). Numerical solution of KdV equation using modified Bernstein polynomials. Appl. Math. Comput., 174(2), 1255-1268.

[40] Mandal, B. N. and Bhattacharya, S. (2007). Numerical solution of some classes of integral equations using Bernstein polynomials. Appl. Math. Comput., 190(2), 1707-1716.

[41] Bhatti, M. I. and Bracken, P. (2007). Solutions of differential equations in a Bernstein polynomial basis. J. Comput. Appl. Math., 205(1), 272-280.

[42] Lu, D. Q. (2011). Some properties of Bernoulli polynomials and their generalizations. Appl. Math. Lett., 24(5), 746-751.

[43] Lehmer, D. H. (1988). A new approach to Bernoulli polynomials. Am. Math. Mon., 9(10), 905-911.

[44] Dil, A., Kurt, V., and Cenkci, M. (2007). Algorithms for Bernoulli and related polynomials. J. Integer Seq., 10(5), 1-14.

[45] Zhang, Z. and Yang, H. (2008). Several identities for the generalized Apostol-Bernoulli polynomials. Comput. Math. with Appl., 56(12), 2993-2999.

[46] Qi, F., and Chapman, R. J. (2016). Two closed forms for the Bernoulli polynomials. J. Number Theory, 159, 89-100.

[47] Lewis, P. E. and Ward, J. P. (1991). The Finite Element Method, Principles and Applications. Wokingham, England: Addison-Wesley.

[48] Loghmani, G. B. and Alavizadeh, S. R. (2007). Numerical solution of fourth-order problems with separated boundary conditions. Appl. Math. Comput., 191(2), 571-581.

[49] Ramadan, M. A., Lashien, I. F., and Zahra, W. K. (2009). Quintic nonpolynomial spline solutions for fourth order two-point boundary value problem. Commun. Nonlinear Sci. Numer. Simul., 14(4), 1105-1114.

[50] Usmani, R. A. (1992). The use of quartic splines in the numerical solution of a fourth-order boundary value problem. 44, 187-99.

[51] Rashidinia, J. and Golbabaee, A. (2005). Convergence of numerical solution of a fourth-order boundary value problem. Appl. Math. Comput., 171(2), 1296-1305.

[52] Noor, M. A. and Mohyud-Din, S. T. (2007). An efﬁcient method for fourth-order boundary value problems. Comput. Math. Appl., 54(2007), 1101-1111.

[53] Akram, G. and Siddiqi, S. S. (2006). Solution of sixth order boundary value problems using non-polynomial spline technique. Appl. Math. Comput., 181(1), 708-720.

Galerkin Residual Correction for Fourth Order BVP

**How to cite this paper:** Md. Nurunnabi Sohel, Md. Shariful Islam, Md. Shafiqul Islam. (2022) Galerkin Residual Correction for Fourth Order BVP. *Journal of Applied Mathematics and Computation*, **6**(**1**), 127-138.

DOI: http://dx.doi.org/10.26855/jamc.2022.03.014

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