A block-by-block method for nonlinear variable-order fractional quadratic integral equations. (English) Zbl 07657510
Summary: This paper is dedicated to solving a class of nonlinear fractional quadratic integral equations of variable order. The block-by-block method based on the Gauss-Lobatto quadrature technique has been developed to solve such integral equations with smooth and weakly singular kernels. In this approach, several values of the unknown functions are obtained through numerical integration without need to any starting value for beginning. The analysis of convergence of the presented method is proved and a rate of convergence is found. Some examples are solved to demonstrate the accuracy of the established approach.
MSC:
| 65R20 | Numerical methods for integral equations |
| 45G10 | Other nonlinear integral equations |
| 26A33 | Fractional derivatives and integrals |
Keywords:
variable-order; quadratic integral equation; weakly singular kernels; block-by-block methodReferences:
| [1] | Asgari M (2014) Block pulse approximation of fractional stochastic integro-differential equation. Commun Numer Anal 1-7:2014 |
| [2] | Atanackovic, TM; Stankovic, B., On a system of differential equations with fractional derivatives arising in rod theory, J Phys A Math Gen, 37, 4, 1241-1250 (2004) · Zbl 1059.35011 · doi:10.1088/0305-4470/37/4/012 |
| [3] | Chen, Y.; Liu, L.; Li, B.; Sun, Y., Numerical solution for the variable order linear cable equation with Bernstein polynomials, Appl Math Comput, 238, 329-341 (2014) · Zbl 1334.65167 |
| [4] | Cioica, PA; Dahlke, S., Spatial Besov regularity for semilinear stochastic partial differential equations on bounded Lipschitz domains, Int J Comput Math, 89, 18, 2443-2459 (2012) · Zbl 1411.60093 · doi:10.1080/00207160.2011.631530 |
| [5] | Deimling, K., Nonlinear functional analysis (1985), Berlin: Springer, Berlin · Zbl 0559.47040 · doi:10.1007/978-3-662-00547-7 |
| [6] | El-Sayed, AMA; Saleh, MM; Ziada, EAA, Numerical and analytic solution for nonlinear quadratic integral equations, Math Sci Res J, 12, 8, 183-191 (2008) · Zbl 1173.26304 |
| [7] | El-Sayed, AMA; Hashem, HHG; Ziada, EAA, Picard and Adomian methods for quadratic integral equation, J Comput Appl Math, 29, 3, 447-463 (2010) · Zbl 1208.45007 |
| [8] | El-Sayed, AMA; Mohamed, MSh; Mohamed, FFS, Existence of positive continuous solution of a quadratic integral equation of fractional orders, Math Sci Lett, 1, 9, 1-7 (2011) · Zbl 1488.45007 |
| [9] | El-Sayed, AMA; Hashem, HHG; Omar, YMY, Positive continuous solution of a quadratic integral equation of fractional orders, Math Sci Lett, 2, 1, 19-27 (2013) · doi:10.12785/msl/020103 |
| [10] | El-Sayed, AMA; Hashem, HHG; Ziada, EAA, Picard and Adomian decomposition methods for a quadratic integral equation of fractional order, J Comput Appl Math, 33, 1, 95-109 (2014) · Zbl 1308.26010 |
| [11] | Evans, RM; Katugampola, UN; Edwards, DA, Applications of fractional calculus in solving Abel-type integral equations: surface-volume reaction problem, J Comput Appl Math, 73, 1-17 (2017) · Zbl 1409.65114 · doi:10.1016/j.camwa.2016.12.005 |
| [12] | He, JH, Some applications of nonlinear fractional differential equations and their approximations, Bull Sci Technol Soc, 15, 2, 86-90 (1999) |
| [13] | Heydari, MH, Chebyshev cardinal wavelets for nonlinear variable-order fractional quadratic integral equations, Appl Numer Math, 144, 190-203 (2019) · Zbl 1433.65352 · doi:10.1016/j.apnum.2019.04.019 |
| [14] | Heydari, MH; Hooshmandasl, MR; Maalek Ghaini, FM; Li, M., Chebyshev wavelets method for solution of nonlinear fractional integrodifferential equations in a large interval, Adv Math Phys (2013) · Zbl 1291.65245 · doi:10.1155/2013/482083 |
| [15] | Heydari, MH; Hooshmandasl, MR; Mohammadi, F.; Cattani, C., Wavelets method for solving systems of nonlinear singular fractional Volterra integro-differential equations, Commun Nonlinear Sci Numer Simul, 19, 37-48 (2014) · Zbl 1344.65126 · doi:10.1016/j.cnsns.2013.04.026 |
| [16] | Hilfer, R., Applications of fractional calculus in physics (2000), Singapore: World Scientific, Singapore · Zbl 0998.26002 · doi:10.1142/3779 |
| [17] | Katani, R.; Shahmorad, S., Block by block method for the systems of nonlinear Volterra integral equations, Appl Math Model, 34, 2, 400-406 (2010) · Zbl 1185.65237 · doi:10.1016/j.apm.2009.04.013 |
| [18] | Katani, R.; Shahmorad, S., The block-by-block method with Romberg quadrature for the solution of nonlinear Volterra integral equations on large intervals, Ukr Math J, 64, 7, 1050-1063 (2012) · Zbl 1259.65215 · doi:10.1007/s11253-012-0698-x |
| [19] | Katani, R.; Shahmorad, S., A block by block method with Romberg quadrature for the system of Urysohn type Volterra integral equations, Comput Appl Math, 31, 1, 191-203 (2012) · Zbl 1252.65211 · doi:10.1590/S1807-03022012000100010 |
| [20] | Khane Keshi, F.; ParsaMoghaddam, B.; Aghili, A., A numerical approach for solving a class of variable-order fractional functional integral equations, J Comput Appl Math, 37, 4821-4834 (2018) · Zbl 1432.65106 |
| [21] | Linz, P., A method for solving nonlinear Volterra integral equations of the second kind, Math Comput, 23, 107, 595-599 (1969) · Zbl 0177.20601 · doi:10.1090/S0025-5718-1969-0247794-7 |
| [22] | Linz, P., Analytical and numerical methods for Volterra equations (1985), Philadelphia: Siam, Philadelphia · Zbl 0566.65094 · doi:10.1137/1.9781611970852 |
| [23] | Ma, X.; Huang, C., Numerical solution of fractional integro-differential equations by a hybrid collocation method, Appl Math Comput, 219, 12, 6750-6760 (2013) · Zbl 1290.65130 |
| [24] | Mirzaee, F.; Alipour, S., Approximate solution of nonlinear quadratic integral equations of fractional order via piecewise linear functions, J Comput Appl Math, 331, 217-227 (2018) · Zbl 1377.65171 · doi:10.1016/j.cam.2017.09.038 |
| [25] | Mirzaee, F.; Hadadian, E., Application of modified hat functions for solving nonlinear quadratic integral equations, Iran J Numer Anal Optim, 6, 2, 65-84 (2016) · Zbl 1357.65323 |
| [26] | Panda, R.; Dash, M., Fractional generalized splines and signal processing, Signal Process, 86, 2340-2350 (2006) · Zbl 1172.65315 · doi:10.1016/j.sigpro.2005.10.017 |
| [27] | Podlubny, I., Fractional differential equations (1999), San Diego: Academic Press, San Diego · Zbl 0924.34008 |
| [28] | Saify SAA (2005) Numerical methods for a system of linear Volterra integral equations. University of Technology Baghdad, Iraq |
| [29] | Sweilam, NH; Nagy, AM; El-Sayed, AA, Numerical approach for solving space fractional order diffusion equations using shifted Chebyshev polynomials of the fourth kind, Turk J Math, 40, 1283-1297 (2016) · Zbl 1438.35442 · doi:10.3906/mat-1503-20 |
| [30] | Sweilam, NH; Nagy, AM; El-Sayed, AA, Non-standard finite difference and Chebyshev collocation methods for solving fractional diffusion equation, Phys A, 500, 40-49 (2018) · Zbl 1514.65139 · doi:10.1016/j.physa.2018.02.014 |
| [31] | Young, A., The application of approximate product-integration to the numerical solution of integral equations, Proc R Soc Lond Ser A, 224, 561-873 (1954) · Zbl 0055.35803 · doi:10.1098/rspa.1954.0180 |
| [32] | Ziada, EAA, Numerical solution for nonlinear quadratic integral equations, Fract Calc Appl Anal, 7, 1-11 (2013) |
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.
