Sparse representation of delay differential equation of Pantograph type using multi-wavelets Galerkin method

Behzad Nemati Saray (Department of Mathematics, Institute for Advanced Studies in Basic Sciences, Zanjan, Iran and Research Center for Basic Sciences and Modern Technologies, Institute for Advanced Studies in Basic Sciences (IASBS), Zanjan, Iran)
Jalil Manafian (Young Researchers and Elite Club, Ilkhchi Branch, Islamic Azad University, Ilkhchi, Iran)

Engineering Computations

ISSN: 0264-4401

Publication date: 16 April 2018



The purpose of this study is an application of the multi-wavelets Galerkin method to delay differential equations with vanishing delay known as Pantograph equation.


The method consists of expanding the required approximate solution at the elements of the Alpert multi-wavelets. Using the operational matrices of integration and wavelet transform matrix, the authors reduce the problem to a set of algebraic equations.


Because of the large size of the system, thresholding is used to obtain a new sparse system, and then this new system is solved to reduce the computational effort and related computer run time. The authors demonstrate that the solutions may be efficiently represented in a multi-wavelets basis because of flexible vanishing moments property of this type of multi-wavelets.


The L2 convergence of the scheme for the proposed equation has been investigated. A series of numerical tests is provided to demonstrate the validity and applicability of the technique.



Nemati Saray, B. and Manafian, J. (2018), "Sparse representation of delay differential equation of Pantograph type using multi-wavelets Galerkin method", Engineering Computations, Vol. 35 No. 2, pp. 887-903.

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