Bhrawy, Ali H.; Alghamdi, Mohammed A. A shifted Jacobi-Gauss-Lobatto collocation method for solving nonlinear fractional Langevin equation involving two fractional orders in different intervals. (English) Zbl 1280.65079 Bound. Value Probl. 2012, Paper No. 62, 13 p. (2012). Summary: We develop a Jacobi-Gauss-Lobatto collocation method for solving the nonlinear fractional Langevin equation with three-point boundary conditions. The fractional derivative is described in the Caputo sense. The shifted Jacobi-Gauss-Lobatto points are used as collocation nodes. The main characteristic behind the Jacobi-Gauss-Lobatto collocation approach is that it reduces such a problem to those of solving a system of algebraic equations. This system is written in a compact matrix form. Through several numerical examples, we evaluate the accuracy and performance of the proposed method. The method is easy to implement and yields very accurate results. Cited in 37 Documents MSC: 65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations 34A08 Fractional ordinary differential equations and fractional differential inclusions 34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations 34B15 Nonlinear boundary value problems for ordinary differential equations Keywords:nonlinear fractional Langevin equation; three-point boundary conditions; collocation method; Jacobi-Gauss-Lobatto quadrature; shifted Jacobi polynomials; numerical examples PDF BibTeX XML Cite \textit{A. H. Bhrawy} and \textit{M. A. Alghamdi}, Bound. Value Probl. 2012, Paper No. 62, 13 p. (2012; Zbl 1280.65079) Full Text: DOI References: [1] doi:10.1016/j.camwa.2011.04.024 · Zbl 1228.65253 · doi:10.1016/j.camwa.2011.04.024 [2] doi:10.1016/j.camwa.2011.03.044 · Zbl 1228.65109 · doi:10.1016/j.camwa.2011.03.044 [3] doi:10.1016/j.cnsns.2011.07.018 · Zbl 1276.65015 · doi:10.1016/j.cnsns.2011.07.018 [4] doi:10.1016/j.aml.2011.06.016 · Zbl 1269.65068 · doi:10.1016/j.aml.2011.06.016 [5] doi:10.1016/j.apm.2011.05.011 · Zbl 1228.65126 · doi:10.1016/j.apm.2011.05.011 [6] doi:10.1016/j.camwa.2011.07.024 · Zbl 1231.65126 · doi:10.1016/j.camwa.2011.07.024 [7] doi:10.1016/j.nonrwa.2011.07.052 · Zbl 1238.34008 · doi:10.1016/j.nonrwa.2011.07.052 [8] doi:10.1140/epje/i2007-10224-2 · doi:10.1140/epje/i2007-10224-2 [9] doi:10.1016/j.physleta.2008.08.045 · Zbl 1225.82049 · doi:10.1016/j.physleta.2008.08.045 [10] doi:10.1016/j.physa.2010.02.041 · doi:10.1016/j.physa.2010.02.041 [11] doi:10.1016/j.cnsns.2011.04.025 · Zbl 1244.65099 · doi:10.1016/j.cnsns.2011.04.025 [12] doi:10.1016/j.apnum.2008.08.007 · Zbl 1162.65374 · doi:10.1016/j.apnum.2008.08.007 [13] doi:10.1016/j.cnsns.2011.06.018 · Zbl 1244.65114 · doi:10.1016/j.cnsns.2011.06.018 [14] doi:10.1016/j.cnsns.2012.02.027 · Zbl 1251.65112 · doi:10.1016/j.cnsns.2012.02.027 [15] doi:10.1016/j.apnum.2007.07.001 · Zbl 1152.65112 · doi:10.1016/j.apnum.2007.07.001 [16] doi:10.1002/num.20369 · Zbl 1170.65099 · doi:10.1002/num.20369 [17] doi:10.1016/j.amc.2012.01.031 · Zbl 1242.65148 · doi:10.1016/j.amc.2012.01.031 [18] doi:10.1016/j.cnsns.2011.10.014 · Zbl 1335.45002 · doi:10.1016/j.cnsns.2011.10.014 [19] doi:10.1088/0305-4470/35/15/308 · Zbl 0997.33004 · doi:10.1088/0305-4470/35/15/308 [20] doi:10.1088/0305-4470/37/3/010 · Zbl 1055.33007 · doi:10.1088/0305-4470/37/3/010 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.