Abstract
In this paper we study the numerical solution of initial-boundary problem for parabolic Volterra integro-differential equations in one dimensional. These equations include the partial differentiation of an unknown function and the integral term containing the unknown function which is the memory of problem. We have made an attempt to develop a method for Wavelet Galerkin which provides the approximate solution. Some numerical examples are presented to illustrate the efficiency and accuracy of the proposed method.
Similar content being viewed by others
References
Grasselli, M., Kabanikhin, S.I., Lorenzi, A.: An inverse hyperbolic integrodifferential problem arising in geophysics. Nonlinear Anal. 15, 283–298 (1990)
Bloom, F.: Ill-posed Problems for Integrodifferential Equations in Mechanics and Electromagnetic Theory. SIAM (1981)
Grigoriev, Y.N., Ibragimov, N.H., Kovalev, V.F., Meleshko, S.V.: Symmetries of Integro-Differential Equations: with Applications in Mechanics and Plasma Physics. Springer, New York (2010)
Habetler, G.T., Schiffman, R.L.: A finite difference method for analyzing the compression of poro-viscoelastic media. Computing 6, 342–348 (1970)
Sun, Z., Wu, X.: A fully discrete difference scheme for a diffusion-wave system. Appl. Numer. Math 56, 193–209 (2006)
Zadeh, K.S.: An integro-partial differential equation for modeling biofluids flow in fractured biomaterials. Theor. Biol. 273, 72–79 (2011)
Pachpatte, B.G.: On a nonlinear diffusion system arising in reactor dynamics. Math. Anal. Appl. 94, 501–508 (1983)
Pao, C.V.: Bifurcation analysis of a nonlinear diffusion system in reactor dynamics. Appl. Anal. 9, 107–119 (1979)
Pao, C.V.: Solution of a nonlinear integrodifferential system arising in nuclear reactor dynamics. Math. Anal. Appl. 48, 470–492 (1974)
Sachs, E.W., Strauss, A.K.: Efficient solution of a partial integro-differential equation in finance. Appl. Numer. Math. 58, 1687–1703 (2008)
Abeergel, F., Tachet, R.: A nonlinear partial integro-differential equation from mathematical finance. AIMS 10, 10–20 (2010)
Hepperger, P.: Hedging electricity swaptions using partial integro-differential equations. Stoch. Process. Appl. 122, 600–622 (2012)
Lin, Y., Xu, C.: Finite difference/spectral approximations for the time-fractional diffusion equation. Comput. Phys. 225, 1533–1552 (2007)
Yanik, E.G., Fairweather, G.: Finite element methods for parabolic and hyperbolic partial integro-differential equations. Nonlinear Anal. 12, 785–809 (1988)
Yan, Y., Fairweather, G.: Orthogonal spline collocation methods for some partial integrodifferential equations. SIAM. Numer. Anal. 29, 755–768 (1992)
Larsson, S., Thome, V., Wahlbin, L.: Numerical solution of parabolic integro-differential equations by the discontinuous Galerkin method. Math. Comput. 67, 45–71 (1998)
Sahu, P.K., Ray, S.S.: Legendre wavelets operational method for the numerical solutions of nonlinear Volterra integro-differential equations system. Appl. Math. Comput. 256, 715–723 (2015)
Shamsi, M., Razzaghi, M.: Solution of Hallens integral equation using multiwavelets. Comput. Phys. Commun. 168, 187–197 (2005)
Sahu, P.K., Ray, S.S.: Numerical solutions for the system of Fredholm integral equations of second kind by a new approach involving semiorthogonal B-spline wavelet collocation method. Appl. Math. Comput. 234, 368–379 (2014)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Rostami, Y., Maleknejad, K. Numerical Solution of Partial Integro-Differential Equations by using Projection Method. Mediterr. J. Math. 14, 113 (2017). https://doi.org/10.1007/s00009-017-0904-z
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00009-017-0904-z