Efficient spectral-Petrov-Galerkin methods for the integrated forms of third- and fifth-order elliptic differential equations using general parameters generalized Jacobi polynomials. (English) Zbl 1242.65148

Doha, E. H.; Abd-Elhameed, W. M.; Youssri, Y. H.

Appl. Math. Comput. 218, No. 15, 7727-7740 (2012).

Summary: This article analyzes some algorithms for solving numerically the integrated forms of third- and fifth-order differential equations in one variable subject to homogeneous and nonhomogeneous boundary conditions using a dual Petrov-Galerkin method. Two new families of general parameters generalized Jacobi polynomials are introduced and used for this purpose. Numerical results indicating the high accuracy and effectiveness of the proposed algorithms are presented.

MSC:

65L60	Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations
34B05	Linear boundary value problems for ordinary differential equations
65L10	Numerical solution of boundary value problems involving ordinary differential equations

Keywords:

dual-Petrov-Galerkin method; generalized Jacobi polynomials; nonhomogeneous Dirichlet conditions; integrated forms; algorithms; numerical results

References:

[1]	Szego, G., Orthogonal polynomials, Am. math. soc. colloq. pub., 23, (1985) · JFM 65.0278.03
[2]	Bialecki, B.; Fairweather, G.; Karageorghis, A., Matrix decomposition algorithms for elliptic boundary value problems: a survey, Numer. algor., 56, 3467-3478, (2011)
[3]	Boyd, J.P., Chebyshev and Fourier spectral methods, (2001), Dover Publications Mineola · Zbl 0987.65122
[4]	Funaro, D., Polynomial approximation of differential equations, lecturer notes in physics, (1992), Springer-Verlag Heidelberg,Berlin, New York
[5]	Canuto, C.; Hussaini, M.Y.; Quarteroni, A.; Zang, T.A., Spectral methods in fluid dynamics, (1988), Springer-Verlag New York · Zbl 0658.76001
[6]	Fox, L.; Parker, I.B., Chebyshev polynomials in numerical analysis, (1972), Oxford University Press Oxford · Zbl 0153.17502
[7]	Gottlieb, D.; Orszag, S.A., Numerical analysis of spectral methods: theory and applications, (1977), SIAM Philadelphia · Zbl 0412.65058
[8]	Voigt, R.G.; Gottlieb, D.; Hussaini, M.Y., Spectral methods for partial differential equations, (1984), SIAM Philadelphia
[9]	Guo, B.-Y.; Shen, J.; Wang, L.-L., Optimal spectral-Galerkin methods using generalized Jacobi polynomials, J. sci. comput., 27, 305-322, (2006) · Zbl 1102.76047
[10]	Khanal, N.; Sharma, R.; Wu, J., A dual-petrov – galerkin method for extended fifth-order kortweg – de Vries type equations, Disc. contin dynam. syst., supplement, 442-450, (2009) · Zbl 1187.35214
[11]	Swanson, C.A., Comparison and oscillation theory of linear differential equations, (1968), Academic Press New York · Zbl 0191.09904
[12]	Mckelvey, Ed., Lectures on ordinary differential equation, (1970), Academic Press New York · Zbl 0213.10102
[13]	Gregus, M., Third order linear differential equations D, (1987), Reidel Publishing Company Dordrecht
[14]	Doha, E.H.; Abd-Elhameed, W.M., Efficient spectral-Galerkin algorithms for direct solution of second-order equations using ultraspherical polynomials, SIAM J. sci. comput., 24, 548-571, (2002) · Zbl 1020.65088
[15]	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, 3221-3242, (2009) · Zbl 1169.65326
[16]	Doha, E.H.; Bhrawy, A.H., Efficient spectral-Galerkin algorithms for direct solution for second-order differential equations using Jacobi polynomials, Numer. algor., 42, 137-164, (2006) · Zbl 1103.65119
[17]	Doha, E.H.; Bhrawy, A.H., Efficient spectral-Galerkin algorithms for direct solution of the integrated forms for second-order equations using ultraspherical polynomials, Anziam j., 48, 361-386, (2007) · Zbl 1138.65104
[18]	Doha, E.H.; Bhrawy, A.H., Efficient spectral-Galerkin algorithms for direct solution of fourth-order differential equations using Jacobi polynomials, Appl. numer. math., 58, 1224-1244, (2008) · Zbl 1152.65112
[19]	Doha, E.H.; Abd-Elhameed, W.M.; Bhrawy, A.H., Efficient spectral ultraspherical-Galerkin algorithms for the direct solution of 2nth-order linear differential equations, Appl. math. model., 33, 1982-1996, (2009) · Zbl 1205.65224
[20]	Ma, H.; Sun, W., A legendre – petrov – galerkin method and Chebyshev collocation method for third-order differential equations, SIAM J. numer. anal., 38, 1425-1438, (2000) · Zbl 0986.65095
[21]	Ma, H.; Sun, W., Optimal error estimates of the legendre – petrov – galerkin method for the kortweg – de Vries equation, SIAM J. numer. anal., 39, 1380-1394, (2001) · Zbl 1008.65070
[22]	Phillips, T.N.; Karageoghis, A., On the coefficients of integrated expansions of ultraspherical polynomials, SIAM J. numer. anal., 27, 823-830, (1990) · Zbl 0701.33007
[23]	Doha, E.H., On the coefficients of integrated expansions and integrals of ultraspherical polynomials and their applications for solving differential equations, J. comp. appl. math., 139, 275-298, (2002) · Zbl 0991.33003
[24]	Doha, E.H., Explicit formulae for the coefficients of integrated expansions for Jacobi polynomials and their integrals, Integr. transf. spec. F, 14, 69-86, (2003) · Zbl 1051.33004
[25]	()
[26]	Andrews, G.E.; Askey, R.; Roy, R., Special functions, (1999), Cambridge University Press Cambridge
[27]	Doha, E.H., On the construction of recurrence relations for the expansion and connection coefficients in series of Jacobi polynomials, J. phys. A: math. gen., 37, 657-675, (2004) · Zbl 1055.33007
[28]	Shen, J.; Tang, T.; Wang, L.L., Spectral methods: algorithms, analysis and applications, Springer series in computational mathematics, (2011), Springer-Verlag Berline. Heidelberg
[29]	Stewart, G.W., Matrix algorithms, volume I: basic decomposition, (1998), SIAM USA · Zbl 0910.65012