IDSOLVER: a general purpose solver for \(n\)th-order integro-differential equations. (English) Zbl 1344.45001

Summary: Many mathematical models of complex processes may be posed as integro-differential equations (IDE). Many numerical methods have been proposed for solving those equations, but most of them are ad hoc thus new equations have to be solved from scratch for translating the IDE into the framework of the specific method chosen. Furthermore, there is a paucity of general-purpose numerical solvers that free the user from additional tasks.
Here we present a general-purpose MATLAB\(^{\circledR}\) solver that has the above features. We have chosen to use a numerical quadrature algorithm combined with an accurate and efficient ODE solver-both within a MATLAB\(^{\circledR}\) environment-to construct a routine (idsolver) capable of solving a wide variety of IDE of arbitrary order, including the Volterra and Fredholm IDE, variable limits on the integral, and non-linear IDE. The solver performs successive relaxation iterations until convergence is achieved. The user has to define a kernel, limits of integration and a forcing function, then launch the routine and get accurate results by tuning in a single tolerance parameter, as described below for several numerical examples. We have found, by solving several numerical examples from the literature, that the method is robust, fast and accurate.


45-04 Software, source code, etc. for problems pertaining to integral equations
65R20 Numerical methods for integral equations
45J05 Integro-ordinary differential equations


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[1] Astarita, G.; Ocone, R., AIChE J., 34, 1299-1309, (1988)
[2] Mantzaris, N. V.; Liou, J. J.; Daoutidis, P.; Srienc, F., J. Biotech., 71, 157-174, (1999)
[3] Jackiewicz, Z.; Rahman, M.; Welfert, B. D., Appl. Math. Comput., 195, 523-536, (2008)
[4] Boyd, J. P., J. Comput. Phys., 189, 98-110, (2003)
[5] Dehghan, M.; Shakeri, F., Prog. Electromagn. Res., 78, 361-376, (2008)
[6] Emmrich, E.; Weckner, O., Math. Mech. Solids, 12, 363-384, (2007)
[7] Lodge, A. S.; Renardy, M.; Nohel, J. A., Viscoelasticity and rheology, (1985), Academic Press Orlando, Fla, University of Wisconsin-Madison, Mathematics Research Center
[8] Medlock, J.; Kot, M., Math. Biosci., 184, 201-222, (2003)
[9] Astarita, G., AIChE J., 35, 529-532, (1989)
[10] Yildirim, A., Comput. Math. Appl., 56, 3175-3180, (2008)
[11] Deeba, E.; Khuri, S. A.; Xie, S. S., Appl. Math. Comput., 115, 123-131, (2000)
[12] Wazwaz, A. M., Appl. Math. Comput., 118, 327-342, (2001)
[13] Akyüz, A.; Sezer, M., Int. J. Comput. Math., 72, 491-507, (1999)
[14] Kajani, M. T.; Ghasemi, M.; Babolian, E., Appl. Math. Comput., 180, 569-574, (2006)
[15] Driscoll, T. A., J. Comput. Phys., 229, 5980-5998, (2010)
[16] Hosseini, S. M.; Shahmorad, S., Appl. Math. Model., 27, 145-154, (2003)
[17] Hosseini, S. M.; Shahmorad, S., Appl. Math. Comput., 136, 559-570, (2003)
[18] Pour-Mahmoud, J.; Rahimi-Ardabili, M. Y.; Shahmorad, S., Appl. Math. Comput., 168, 465-478, (2005)
[19] Arikoglu, A.; Ozkol, I., Appl. Math. Comput., 168, 1145-1158, (2005)
[20] Darania, P.; Ebadian, A., Appl. Math. Comput., 188, 657-668, (2007)
[21] Jorquera, H., Comput. Phys. Comm., 86, 91-96, (1995)
[22] van der Vorst, H. A., SIAM J. Sci. Stat. Comput., 13, 631-644, (1992)
[23] Axelsson, O., Iterative solution methods, (1994), Cambridge University Press Cambridge, England; New York · Zbl 0795.65014
[24] Jaradat, H.; Alsayyed, O.; Al-Shara, S., J. Math. Stat., 4, 250-254, (2008)
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