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Vol3 Issue2 _6

posted Mar 24, 2019, 2:29 AM by Yaseen Raouf Mohammed   [ updated Apr 7, 2019, 5:52 AM ]
 A WAVELET COLLOCATION METHOD FOR THE NUMERICAL SOLUTION OF POPULATION DIFFERENTIAL EQUATION

 Amjad Alipanah

 University of Kurdistan a.alipanah@uok.ac.ir


 ABSTRACT.
In this paper, we discuss about the population balance equations which are described by the following differential equation model [4? ? ? ]
dy (x)
---------- + (1 + κxm) y (x) = 2m+1κxmy (2x) , 0 ≤ x ≤ b, (1)
dx
where κ, b and m are constant. If the binary equal breakage is assumed then the value of m is assumed to be 4 and the initial condition y (x) is y (0) = 1. (2)
As we know, there is no exact solution for this equation [4? ? ], so we have to solve it by approximation methods. Several numerical schemes developed to solve the population balance differential equation, such as Block-pulse method [? ], weighted residual method [? ], the method of orthogonal series expansion and wavelet Galerkin method [4] and rationalized Haar functions [? ]. In this paper, we discuss on the application of auto-correlation functions of Daubechies wavelets to solve the population balance differential equation (1), and compare with some of above methods.


 Keywords:
Population balance differential equation, Daubechies wavelet, auto-correlation functions, Lagrange interpolation


 REFERENCES
[1] Alipanah, A. and Dehghan, M. [2008], ‘Solution of population balance equa- tions via rationalized haar functions’, Kybernetes 37, 1189–1196.
[2] Beylkin, G. [n.d.], ‘On the representation of operators in bases of compactly supported wavelets’, SIAM J. Numer. Anal. 121.
[3] Casazza, P. G. and Kutyniok, G. [2012], Finite Frame: Theory and Applica- tions, Brichauer, Buston.
[4] M.Q. Chen, C. H. and Shih, Y. [n.d.], ‘A wavelet-galerkin method for solving population balance equations’, Computers Chem. Engng. 20.
[5] R. Ansari, C. G. and Kaiser, J. [n.d.], ‘Wavelet construction using lagrange halfband filters’, IEEE Trans. Circuits Syst. 38.

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