×

Numerical solution of nonlinear Volterra integro-differential equations of arbitrary order by CAS wavelets. (English) Zbl 1221.65140

Summary: A computational method for numerical solution of a nonlinear Volterra integro-differential equation of fractional (arbitrary) order which is based on CAS wavelets and BPFs is introduced. The CAS wavelet operational matrix of fractional integration is derived and used to transform the main equation to a system of algebraic equations. Some examples are included to demonstrate the validity and applicability of the technique.

MSC:

65J20 Numerical solutions of ill-posed problems in abstract spaces; regularization
65R20 Numerical methods for integral equations
26A33 Fractional derivatives and integrals
45J05 Integro-ordinary differential equations
65T60 Numerical methods for wavelets
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Caputo, M., Linear models of dissipation whose Q is almost frequency independent - II, Geophys J royal astron soc, 13, 529539, (1967)
[2] Mainardi, F., Fractional calculus: some basic problems in continuum and statistical mechanics, (), 291348
[3] Olmstead, W.; Handelsman, R., Diffusion in a semi-infinite region with nonlinear surface dissipation, SIAM rev, 18, 275291, (1976)
[4] Momani, S., Local and global existence theorems on fractional integro-differential equations, J fract calculus, 18, 8186, (2000)
[5] Rawashdeh, E.A., Numerical solution of fractional integro-differential equations by collocation method, Appl math comput, 176, 1-6, (2006) · Zbl 1100.65126
[6] Boyadjiev, L.; Dobner, H.J.; Kalla, S.L., A fractional integro-differential equation of Volterra type, Math comput modell, 28, 103-113, (1998) · Zbl 0993.65153
[7] Nazari D, Shahmorad S. Application of the fractional differential transform method to fractional-order integro-differential equations with nonlocal boundary conditions. J Comput Appl Math. doi:10.1016/j.cam.2010.01.053. · Zbl 1188.65174
[8] Arikoglu, A.; Ozkol, I., Solution of fractional integro-differential equations by using fractional differential transform method, Chaos solitons fract, 40, 521-529, (2009) · Zbl 1197.45001
[9] Momani, S.; Qaralleh, R., An efficient method for solving systems of fractional integro-differential equations, Comput math appl, 52, 459-470, (2006) · Zbl 1137.65072
[10] Boyadjiev, L.; Kalla, S.L.; Khajah, H.G., Analytical and numerical treatment of a fractional integro-differential equation of Volterra type, Math comput modell, 25, 1-9, (1997) · Zbl 0932.45012
[11] Saberi-Nadjafi, J.; Ghorbani, A., He’s homotopy perturbation method: an effective tool for solving nonlinear integral and integro-differential equations, Comput math appl, 58, 23792390, (2009) · Zbl 1189.65173
[12] Kilicman, A.; Al Zhour, Z.A.A., Kronecker operational matrices for fractional calculus and some applications, Appl math comput, 187, 1, 25065, (2007)
[13] Momani, S.; Noor, M., Numerical methods for fourth order fractional integro-differential equations, Appl math comput, 182, 754-760, (2006) · Zbl 1107.65120
[14] Podlubny, I., Fractional differential equations, (1999), Academic Press New York · Zbl 0918.34010
[15] Rao, C.R., Piecewise orthogonal functions and their applications on system and control, (1983), Springer Berlin
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.