Mathematical modeling of diffusion problem

Document Type : Research Paper


Department of Mathematics, Karaj Branch, Islamic Azad University, Karaj, Iran


‎This work aims to introduce a numerical approximation procedure based on an operational matrix of block pulse functions‎, ‎which is employed in solving integral-algebraic equations arising from the diffusion model‎. ‎It is known that the integral-algebraic equations belong to the class of singular problems‎. ‎The main advantage of this method is the reduction of these singular systems by using an operational matrix to linear lower triangular systems of algebraic equations‎, ‎which is non-singular‎. ‎An estimation of the error and illustrative instances are discussed to evaluate the validity and applicability of the presented method‎.


[1] M.V. Bulatov and V.F. Chistyakov, The properties of differential-algebraic systems and their integral analogues, Memorial University of Newfoundland, Preprint, 1997.
[2] J.P. Kauthen, The numerical solution of integral-algebraic equations of index-1 by polynomial spline collocation methods, Math. Comp. 236 (2000) 1503—1514.
[3] M. Hadizadeh, F. Ghoreishi and S. Pishbin, Jacobi spectral solution for integral algebraic equations of index-2, Appl. Numer. Math. 61 (2011) 131-–148.
[4] K. Maleknejad, H. Safdari and M. Nouri, Numerical solution of an integral equations system of the first kind by using an operational matrix with block pulse functions, Int. J. Syst. Sci. 42 (2011).
[5] H. Brunner, Collocation Methods for Volterra Integral and Related Functional Equations, University Press, Cambridge, 2004.
[6] A.I. Zenchuk, Combination of inverse spectral transform method and method of characteristics: deformed Pohlmeyer equation, J. Nonlinear Math. Phys. 15 (2008) 437-448.
[7] J.R. Cannon, The One-Dimensional Heat Equation, University Press, Cambridge, 1984.
[8] C.W. Gear, Differential-algebraic equations, indices, and integral-algebraic equations, SIAM. J. Numer. Anal. 27 (1990) 1527—1534.
Volume 13, Issue 1
March 2022
Pages 2065-2073
  • Receive Date: 24 June 2019
  • Accept Date: 03 September 2019