# New Spectral Solutions for High Odd-Order Boundary Value Problems via Generalized Jacobi Polynomials

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## Abstract

In this article, some new efficient and accurate algorithms are developed for solving linear and nonlinear odd-order two-point boundary value problems. The algorithm in linear case is based on the application of Petrov–Galerkin method. For implementing this algorithm, two certain families of generalized Jacobi polynomials are introduced and employed as trial and test functions. The trial functions satisfy the underlying boundary conditions of the differential equations, and the test functions satisfy the dual boundary conditions. The developed algorithm leads to linear systems with band matrices which can be efficiently inverted. These special systems are carefully investigated, especially their complexities. Another algorithm based on the application of the typical collocation method is presented for handling nonlinear odd-order two-point boundary value problems. The use of generalized Jacobi polynomials leads to simplified analysis and very efficient numerical algorithms. Numerical results are presented for the sake of testing the efficiency and applicability of the two proposed algorithms.

### Keywords

Dual-Petrov–Galerkin method Collocation method Generalized Jacobi polynomials Legendre polynomials High odd-order boundary value problems### Mathematics Subject Classification

65M70 65N35 35C10 42C10## Notes

### Acknowledgements

The author is very grateful to the editor and to the referee for their constructive and useful comments which have improved the manuscript in its present form.

### References

- 1.Abd-Elhameed, W.M.: On solving linear and nonlinear sixth-order two point boundary value problems via an elegant harmonic numbers operational matrix of derivatives. CMES Comp. Model. Eng.
**101**(3), 159–185 (2014)MathSciNetMATHGoogle Scholar - 2.Abd-Elhameed, W.M.: New Galerkin operational matrix of derivatives for solving Lane–Emden singular-type equations. Eur. Phys. J. Plus
**130**, 52 (2015)CrossRefGoogle Scholar - 3.Abd-Elhameed, W.M., Doha, E.H., Youssri, Y.H.: Efficient spectral-Petrov–Galerkin methods for third-and fifth-order differential equations using general parameters generalized jacobi polynomials. Quaest. Math.
**36**(1), 15–38 (2013)MathSciNetCrossRefMATHGoogle Scholar - 4.Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables. Courier Dover Publications, New York (2012)MATHGoogle Scholar
- 5.Akram, G.: Quartic spline solution of a third order singularly perturbed boundary value problem. ANZIAM J.
**53**, 44–58 (2012)MathSciNetCrossRefMATHGoogle Scholar - 6.Andrews, G.E., Askey, R., Roy, R.: Special Functions. Cambridge University Press, Cambridge (1999)CrossRefMATHGoogle Scholar
- 7.Boyd, J.P.: Chebyshev and Fourier Spectral Methods. Courier Corporation, North Chelmsford (2001)MATHGoogle Scholar
- 8.Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral Methods in Fluid Dynamics. Springer, Berlin (1988)CrossRefMATHGoogle Scholar
- 9.Davies, A.R., Karageorghis, A., Phillips, T.N.: Spectral Galerkin methods for the primary two-point boundary value problem in modelling viscoelastic flows. Int. J. Numer. Methods Eng.
**26**(3), 647–662 (1988)CrossRefMATHGoogle Scholar - 10.Doha, E.H., Abd-Elhameed, W.M.: Efficient spectral ultraspherical-dual-Petrov–Galerkin algorithms for the direct solution of (2n \(+\) 1)th-order linear differential equations. Math. Comput. Simul.
**79**(11), 3221–3242 (2009)MathSciNetCrossRefMATHGoogle Scholar - 11.Doha, E.H., Abd-Elhameed, W.M.: Efficient solutions of multidimensional sixth-order boundary value problems using symmetric generalized Jacobi–Galerkin method, vol 2012, Article ID 749370 (2012)Google Scholar
- 12.Doha, E.H., Abd-Elhameed, W.M., Bassuony, M.A.: On the coefficients of differentiated expansions and derivatives of Chebyshev polynomials of the third and fourth kinds. Acta Math. Sin.
**35**(2), 326–338 (2015)MathSciNetMATHGoogle Scholar - 13.Doha, E.H., Abd-Elhameed, W.M., Bhrawy, A.H.: New spectral-Galerkin algorithms for direct solution of high even-order differential equations using symmetric generalized Jacobi polynomials. Collect. Math.
**64**(3), 373–394 (2013)MathSciNetCrossRefMATHGoogle Scholar - 14.Doha, E.H., Abd-Elhameed, W.M., Youssri, Y.H.: Efficient spectral-Petrov–Galerkin methods for the integrated forms of third-and fifth-order elliptic differential equations using general parameters generalized Jacobi polynomials. Appl. Math. Comput.
**218**(15), 7727–7740 (2012)MathSciNetMATHGoogle Scholar - 15.Guo, B.-Y., Shen, J., Wang, Li-Lian: Optimal spectral-Galerkin methods using generalized Jacobi polynomials. J. Sci. Comput.
**27**(1–3), 305–322 (2006)MathSciNetCrossRefMATHGoogle Scholar - 16.Howes, F.A.: Singular Perturbations and Differential Inequalities, vol. 168. American Mathematical Society, Providence (1976)MATHGoogle Scholar
- 17.Huang, W., Sloan, D.M.: The pseudospectral method for third-order differential equations. SIAM J. Numer. Anal.
**29**(6), 1626–1647 (1992)MathSciNetCrossRefMATHGoogle Scholar - 18.Kelevedjiev, P.: Existence of positive solutions to a singular second order boundary value problem. Nonlinear Anal. Theory Methods Appl.
**50**(8), 1107–1118 (2002)MathSciNetCrossRefMATHGoogle Scholar - 19.Lang, F.-G., Xu, X.-P.: Quartic B-spline collocation method for fifth order boundary value problems. Computing
**92**(4), 365–378 (2011)MathSciNetCrossRefMATHGoogle Scholar - 20.Mason, J.C., Handscomb, D.C.: Chebyshev Polynomials. CRC Press, Boca Raton (2002)CrossRefMATHGoogle Scholar
- 21.Merryfield, W.J., Shizgal, B.: Properties of collocation third-derivative operators. J. Comput. Phys.
**105**(1), 182–185 (1993)MathSciNetCrossRefMATHGoogle Scholar - 22.Rainville, E.D.: Special Functions. Macmillan, New York (1960)MATHGoogle Scholar
- 23.Rashidinia, J., Ghasemi, M., Jalilian, R.: An \( {O}(h^6)\) numerical solution of general nonlinear fifth-order two point boundary value problems. Numer. Algoritm.
**55**(4), 403–428 (2010)MathSciNetCrossRefMATHGoogle Scholar - 24.Schatzman, M., Taylor, J.: Numerical Analysis: A Mathematical Introduction. Oxford University Press, Oxford (2002)Google Scholar
- 25.Shizgal, B.: Spectral Methods in Chemistry and Physics. Springer, Berlin (2014)MATHGoogle Scholar
- 26.Wazwaz, A.M.: Approximate solutions to boundary value problems of higher order by the modified decomposition method. Comput. Math. Appl.
**40**(6), 679–691 (2000)MathSciNetCrossRefMATHGoogle Scholar - 27.Wazwaz, A.M.: The numerical solution of fifth-order boundary value problems by the decomposition method. J. Comput. Appl. Math.
**136**(1), 259–270 (2001)MathSciNetCrossRefMATHGoogle Scholar - 28.Yüzbaşi, Ş., Şahin, N.: On the solutions of a class of nonlinear ordinary differential equations by the Bessel polynomials. J. Numer. Math.
**20**(1), 55–80 (2012)MathSciNetMATHGoogle Scholar - 29.Zhang, J.: The numerical solution of fifth-order boundary value problems by the variational iteration method. Comput. Math. Appl.
**58**(11), 2347–2350 (2009)MathSciNetCrossRefMATHGoogle Scholar