Li, Changpin; Zeng, Fanhai The finite difference methods for fractional ordinary differential equations. (English) Zbl 1267.65094 Numer. Funct. Anal. Optim. 34, No. 2, 149-179 (2013). The authors prove the stability and convergence of the fractional Euler method, the fractional Adams method, and the high-order methods based on the convolution formula by using the generalized discrete Gronwall inequality. Numerical examples are also presented. Reviewer: Răzvan Răducanu (Iaşi) Cited in 88 Documents MSC: 65L12 Finite difference and finite volume methods for ordinary differential equations 34A08 Fractional ordinary differential equations 65L05 Numerical methods for initial value problems involving ordinary differential equations 65L20 Stability and convergence of numerical methods for ordinary differential equations Keywords:Caputo derivative; convergence; fractional Adams method; fractional differential equations; fractional Euler method; high-order methods; Riemann-Liouville derivative; stability; numerical examples Software:Eigtool; Seigtool PDFBibTeX XMLCite \textit{C. Li} and \textit{F. Zeng}, Numer. Funct. Anal. Optim. 34, No. 2, 149--179 (2013; Zbl 1267.65094) Full Text: DOI References: [1] Miller K., An Introduction to the Fractional Calculus and Fractional Differential Equations (1993) · Zbl 0789.26002 [2] Podlubny I., Fractional Differential Equations (1999) · Zbl 0924.34008 [3] DOI: 10.1016/S0370-1573(02)00331-9 · Zbl 0999.82053 · doi:10.1016/S0370-1573(02)00331-9 [4] Oldham K. B., The Fractional Calculus (2006) [5] DOI: 10.1006/jmaa.2000.7194 · doi:10.1006/jmaa.2000.7194 [6] DOI: 10.1023/B:NUMA.0000027736.85078.be · Zbl 1055.65098 · doi:10.1023/B:NUMA.0000027736.85078.be [7] DOI: 10.1016/j.cma.2004.06.006 · Zbl 1119.65352 · doi:10.1016/j.cma.2004.06.006 [8] Li C. P., Comput. Math. Appl. 58 pp 1573– (2009) · Zbl 1189.65142 · doi:10.1016/j.camwa.2009.07.050 [9] Li C. P., J. Comput. Phys. 230 pp 3352– (2011) · Zbl 1218.65070 · doi:10.1016/j.jcp.2011.01.030 [10] Odibat Z., J. Appl. Math. Informatics 26 pp 15– (2008) [11] Yang C., ANZIAM J. 47 pp C137– (2006) · doi:10.21914/anziamj.v47i0.1037 [12] Yin C., J. Algorithm Comput. Tech. 1 pp 427– (2007) · doi:10.1260/174830107783133888 [13] Deng W. H., J. Comput. Phys. 227 pp 1510– (2007) · Zbl 1388.35095 · doi:10.1016/j.jcp.2007.09.015 [14] DOI: 10.1016/j.camwa.2011.03.002 · Zbl 1228.93114 · doi:10.1016/j.camwa.2011.03.002 [15] DOI: 10.1016/j.jcp.2009.01.014 · Zbl 1160.65308 · doi:10.1016/j.jcp.2009.01.014 [16] Lin R., Nonlinear Analysis 66 pp 856– (2007) · Zbl 1118.65079 · doi:10.1016/j.na.2005.12.027 [17] DOI: 10.1137/0517050 · Zbl 0624.65015 · doi:10.1137/0517050 [18] DOI: 10.1016/0168-9274(93)90012-G · Zbl 0768.65093 · doi:10.1016/0168-9274(93)90012-G [19] Sheng Q., Math. Comput. Model. 21 pp 1– (1995) · Zbl 0828.65148 · doi:10.1016/0895-7177(95)00066-B [20] Baker C. T. H., J. Comput. Appl. Math. 125 pp 217– (2002) · Zbl 0976.65121 · doi:10.1016/S0377-0427(00)00470-2 [21] Diethelm K., J. Comput. Appl. Math. 186 pp 482– (2006) · Zbl 1078.65550 · doi:10.1016/j.cam.2005.03.023 [22] Galeone L., J. Comput. Appl. Math. 228 pp 548– (2009) · Zbl 1169.65121 · doi:10.1016/j.cam.2008.03.025 [23] DOI: 10.1016/j.cam.2008.04.004 · Zbl 1171.65098 · doi:10.1016/j.cam.2008.04.004 [24] DOI: 10.1093/imanum/6.1.87 · Zbl 0587.65090 · doi:10.1093/imanum/6.1.87 [25] DOI: 10.1080/00207160802624331 · Zbl 1206.65197 · doi:10.1080/00207160802624331 [26] Huang Y., J. Math. Anal. Appl. 282 pp 56– (2003) · Zbl 1030.65140 · doi:10.1016/S0022-247X(02)00369-4 [27] Dixon J., BIT 25 pp 623– (1985) · Zbl 0584.65091 · doi:10.1007/BF01936141 [28] Ma H. P., SIAM J. Numer. Anal. 39 pp 1380– (2001) · Zbl 1008.65070 · doi:10.1137/S0036142900378327 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.