Operational matrix approach for solution of integro-differential equations arising in theory of anomalous relaxation processes in vicinity of singular point. (English) Zbl 1426.65220
Summary: In this paper we study the numerical solution of singular Abel-Volterra integro-differential equations, which are typical for the theory of anomalous diffusion and viscoelastic delayed stresses. The proposed method is based on application of the operational and almost operational matrices to derivatives and integrals in a vicinity of the kernel’s singular point. As examples, two orthonormal systems are considered: Bernstein polynomials and Legendre wavelets. The methods convert the singular integro-differential equation in to a system of algebraic equations that implies two advantages: (i) one does not need to introduce artificial smoothing factors into the singular integrand and (ii) the direct estimation of computational error around singular point is possible via the obtained explicit expression. The examples of numerical solution and their discussion are presented.
MSC:
| 65R20 | Numerical methods for integral equations |
| 45E10 | Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type) |
| 65T60 | Numerical methods for wavelets |
| 41A10 | Approximation by polynomials |
| 76M25 | Other numerical methods (fluid mechanics) (MSC2010) |
Keywords:
singular integro-differential equation; Bernstein polynomial; Legendre wavelets; operational matrix of integrationReferences:
| [1] | Sokolov, I. M.; Klafter, J., From diffusion to anomalous diffusion: a century after einstein’s Brownian motion, Chaos, 15, 026103, (2005) · Zbl 1080.82022 |
| [2] | van Aartrijk, M.; Clercx, H. J.H., Vertical dispersion of light inertial particles in stably stratified turbulence: the influence of the Basset force, Phys. Fluids, 22, 013301, (2010) · Zbl 1183.76536 |
| [3] | Grzybowski, B. A.; Bishop, K. J.M.; Campbell, C. J.; Fialkowski, M.; Smoukov, S. K., Micro- and nanotechnology via reaction – diffusion, Soft Matter, 1, 114-128, (2005) |
| [4] | Kamińska, A.; Srokowski, T., Memory effects and diffusion for strongly correlated stochastic systems described by the generalized Langevin equation driven by a jumping process, Acta Phys. Polonica B, 37, 1701-1714, (2006) |
| [5] | Mori, H., Transport, collective motion, and Brownian motion, Prog. Theor. Phys., 33, 423-455, (1965) · Zbl 0127.45002 |
| [6] | Porrà, J. M.; Wang, K.-G.; Masoliver, J., Generalized Langevin equations: anomalous diffusion and probability distributions, Phys. Rev. E, 53, 5872-5881, (1996) |
| [7] | Mainardi, F.; Pironi, P., The fractional Langevin equation: Brownian motion revisited, Extracta Math., 10, 140-154, (1996) |
| [8] | van Hinsberg, M. A.T.; ten Thije Boonkkamp, J. H.M.; Clercx, H. J.H., An efficient, second order method for the approximation of the Basset history force, J. Comput. Phys., 230, 1465-1478, (2011) · Zbl 1391.76593 |
| [9] | Khater, A.; Shamardan, A.; Callebaut, D.; Sakran, M., Numerical solutions of integral and integro-differential equations using Legendre polynomials, Numer. Algor., 46, 195-218, (2007) · Zbl 1131.65111 |
| [10] | Yousefi, S. A., Numerical solution of abel’s integral equation by using Legendre wavelets, Appl. Math. Comput., 175, 574-580, (2006) · Zbl 1088.65124 |
| [11] | Razzaghi, M.; Yousefi, S., The Legendre wavelets operational matrix of integration, Int. J. Syst. Sci., 32, 495-502, (2011) · Zbl 1006.65151 |
| [12] | Singh, A. K.; Singh, V. K.; Singh, O. P., The Bernstein operational matrix of integration, Appl. Math. Sci., 3, 2427-2436, (2009) · Zbl 1185.41006 |
| [13] | Singh, O. P.; Singh, V. K.; Pandey, R. K., A stable numerical inversion of abel’s integral equation using almost Bernstein operational matrix, J. Quant. Spectrosc. Radiat. Transfer, 111, 245-252, (2010) |
| [14] | Hanyga, A., Anisotropic viscoelastic models with singular memory, J. Appl. Geophys., 54, 411-425, (2003) |
| [15] | Hanyga, A.; Seredyńska, M., Power-law attenuation in acoustic and isotropic anelastic media, Geophys. J. Int., 155, 830-838, (2003) |
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.
