Numerical solution of partial integrodifferential equations of diffusion type. (English) Zbl 1427.65280
Summary: A collocation method based on linear Legendre multiwavelets is developed for numerical solution of one-dimensional parabolic partial integrodifferential equations of diffusion type. Such equations have numerous applications in many problems in the applied sciences to model dynamical systems. The proposed numerical method is validated by applying it to various benchmark problems from the existing literature. The numerical results confirm the accuracy, efficiency, and robustness of the proposed method.
MSC:
| 65M70 | Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs |
| 65R20 | Numerical methods for integral equations |
| 65T60 | Numerical methods for wavelets |
| 45K05 | Integro-partial differential equations |
References:
| [1] | Shakeri, F.; Dehghan, M., A high order finite volume element method for solving elliptic partial integro-differential equations, Applied Numerical Mathematics, 65, 105-118, (2013) · Zbl 1263.65137 · doi:10.1016/j.apnum.2012.10.002 |
| [2] | Dehghan, M.; Salehi, R., The numerical solution of the non-linear integro-differential equations based on the meshless method, Journal of Computational and Applied Mathematics, 236, 9, 2367-2377, (2012) · Zbl 1243.65154 · doi:10.1016/j.cam.2011.11.022 |
| [3] | Dehghan, M., Solution of a partial integro-differential equation arising from viscoelasticity, International Journal of Computer Mathematics, 83, 1, 123-129, (2006) · Zbl 1087.65119 · doi:10.1080/00207160500069847 |
| [4] | Yanik, E. G.; Fairweather, G., Finite element methods for parabolic and hyperbolic partial integro-differential equations, Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal, 12, 8, 785-809, (1988) · Zbl 0657.65142 · doi:10.1016/0362-546X(88)90039-9 |
| [5] | Meurer, T., Flatness of a class of linear Volterra partial integro-differential equations, IFAC-PapersOnLine, 49, 8, 174-179, (2016) · doi:10.1016/j.ifacol.2016.07.432 |
| [6] | Chen, Y. Z.; Lin, X. Y.; Wang, Z. X.; Nik Long, N. M. A., Solution of contact problem for an arc crack using hypersingular integral equation, International Journal of Computational Methods, 5, 1, 119-133, (2008) · Zbl 1257.74112 · doi:10.1142/S0219876208001418 |
| [7] | Hosseini, S. M.; Shahmorad, S., Numerical solution of a class of integro-differential equations by the tau method with an error estimation, Applied Mathematics and Computation, 136, 2-3, 559-570, (2003) · Zbl 1027.65182 · doi:10.1016/S0096-3003(02)00081-4 |
| [8] | Raza, N.; Sial, S.; Neuberger, J. W.; Ahmad, M. O., Numerical solutions of integro-differential equations using Sobolev gradient methods, International Journal of Computational Methods, 9, 4, (2012) · Zbl 1360.65253 · doi:10.1142/S0219876212500466 |
| [9] | Kumar, S., A new analytical modelling for fractional telegraph equation via Laplace transform, Applied Mathematical Modelling, 38, 13, 3154-3163, (2014) · Zbl 1427.35327 · doi:10.1016/j.apm.2013.11.035 |
| [10] | Jha, N.; Mohanty, R. K.; Mishra, B. K., Alternating group explicit iterative method for nonlinear singular Fredholm integro-differential boundary value problems, International Journal of Computer Mathematics, 86, 9, 1645-1656, (2009) · Zbl 1172.65070 · doi:10.1080/00207160801965214 |
| [11] | Mohanty, R. K.; Dhall, D., Third order accurate variable mesh discretization and application of {TAGE} iterative method for the non-linear two-point boundary value problems with homogeneous functions in integral form, Applied Mathematics and Computation, 215, 6, 2024-2034, (2009) · Zbl 1178.65086 · doi:10.1016/j.amc.2009.07.046 |
| [12] | Mohanty, R. K.; Jain, M. K.; Dhall, D., A cubic spline approximation and application of TAGE iterative method for the solution of two point boundary value problems with forcing function in integral form, Applied Mathematical Modelling, 35, 6, 3036-3047, (2011) · Zbl 1219.65072 · doi:10.1016/j.apm.2010.12.013 |
| [13] | Mohanty, R. K., A combined arithmetic average discretization and {TAGE} iterative method for non-linear two point boundary value problems with a source function in integral form, Differential Equations and Dynamical Systems, 20, 4, 423-440, (2012) · Zbl 1256.65076 · doi:10.1007/s12591-012-0140-8 |
| [14] | Shahmorad, S.; Ostadzad, M. H., An Operational Matrix Method for Solving Delay Fredholm and Volterra Integro-Differential Equations, International Journal of Computational Methods, 13, 6, (2016) · Zbl 1359.65137 · doi:10.1142/S0219876216500407 |
| [15] | Shidfar, A.; Molabahrami, A.; Babaei, A.; Yazdanian, A., A series solution of the nonlinear Volterra and Fredholm integro-differential equations, Communications in Nonlinear Science and Numerical Simulation, 15, 2, 205-215, (2010) · Zbl 1221.65343 · doi:10.1016/j.cnsns.2009.03.015 |
| [16] | Zarebnia, M., Sinc numerical solution for the Volterra integro-differential equation, Communications in Nonlinear Science and Numerical Simulation, 15, 3, 700-706, (2010) · Zbl 1221.65346 · doi:10.1016/j.cnsns.2009.04.021 |
| [17] | Li, B.; Chen, X.; He, Z., A wavelet-based error estimator and an adaptive scheme for plate bending problems, International Journal of Computational Methods, 7, 2, 241-259, (2010) · Zbl 1267.74111 · doi:10.1142/S0219876210002155 |
| [18] | Dahmen, W.; Kurdila, A.; Oswald, P., Multiscale Wavelet Methods for Partial Differential Equations, (1997), Academic Press · Zbl 1528.65002 |
| [19] | Aziz, I.; Islam, S., New algorithms for the numerical solution of nonlinear Fredholm and Volterra integral equations using Haar wavelets, Journal of Computational and Applied Mathematics, 239, 333-345, (2013) · Zbl 1255.65235 |
| [20] | Heydari, M. H.; Hooshmandasl, M. R.; Mohammadi, F.; Cattani, C., Wavelets method for solving systems of nonlinear singular fractional Volterra integro-differential equations, Communications in Nonlinear Science and Numerical Simulation, 19, 1, 37-48, (2014) · Zbl 1344.65126 · doi:10.1016/j.cnsns.2013.04.026 |
| [21] | Heydari, M. H.; Hooshmandasl, M. R.; Mohammadi, F.; Cattani, C., Wavelets method for the time fractional diffusion-wave equation, Physics Letters A, 379, 71-76, (2015) · Zbl 1304.35748 |
| [22] | Islam, S.; Aziz, I.; Sarler, B., The numerical solution of second-order boundary-value problems by collocation method with the Haar wavelets, Mathematical and Computer Modelling, 5, 1577-1590, (2010) · Zbl 1205.74187 |
| [23] | Comincioli, V.; Naldi, G.; Scapolla, T., Wavelet-based method for numerical solution of nonlinear evolution equations, Applied Numerical Mathematics, 33, 1, 291-297, (2000) · Zbl 0964.65112 · doi:10.1016/S0168-9274(99)00095-1 |
| [24] | Díaz, L. A.; Martín, M. T.; Vampa, V., Daubechies wavelet beam and plate finite elements, Finite Elements in Analysis and Design, 45, 3, 200-209, (2009) · doi:10.1016/j.finel.2008.09.006 |
| [25] | Zhu, X.; Lei, G.; Pan, G., On application of fast and adaptive periodic Battle-Lemarie wavelets to modeling of multiple lossy transmission lines, Journal of Computational Physics, 132, 2, 299-311, (1997) · Zbl 0882.65123 · doi:10.1006/jcph.1996.5637 |
| [26] | Majak, J.; Shvartsman, B. S.; Kirs, M.; Pohlak, M.; Herranen, H., Convergence theorem for the haar wavelet based discretization method, Composite Structures, 126, 227-232, (2015) · doi:10.1016/j.compstruct.2015.02.050 |
| [27] | Majak, J.; Shvartsman, B.; Karjust, K.; Mikola, M.; Haavajõe, A.; Pohlak, M., On the accuracy of the Haar wavelet discretization method, Composites Part B: Engineering, 80, 321-327, (2015) · doi:10.1016/j.compositesb.2015.06.008 |
| [28] | Lepik, U.; Hein, H., Haar Wavelets with Applications, (2014), Springer · Zbl 1287.65146 |
| [29] | Singh, V. K.; Singh, O. P.; Pandey, R. K., Numerical evaluation of the Hankel transform by using linear Legendre multi-wavelets, Computer Physics Communications, 179, 6, 424-429, (2008) · Zbl 1197.65237 · doi:10.1016/j.cpc.2008.04.006 |
| [30] | Fakhar-Izadi, F.; Dehghan, M., The spectral methods for parabolic Volterra integro-differential equations, Journal of Computational and Applied Mathematics, 235, 14, 4032-4046, (2011) · Zbl 1219.65161 · doi:10.1016/j.cam.2011.02.030 |
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.
