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A new computational approach for the solutions of generalized pantograph-delay differential equations

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Abstract

In this study, a new computational approach is presented to solve the generalized pantograph-delay differential equations (PDDEs). The solutions obtained by our scheme represent by a linear combination of a special kind of basis functions, and can be deduced in a straightforward manner. Firstly, using the least squares approximation method and the Lagrange-multiplier method, the given PDDE is converted to a linear system of algebraic equations, and those unknown coefficients of the solution of the problem are determined by solving this linear system. Secondly, a PDDE related to the error function of the approximate solution is constructed based on the residual error function technique, and error estimation is presented for the suggested method. The convergence of the approximate solution is proved. Several numerical examples are given to demonstrate the accuracy and efficiency. Comparisons are made between the proposed method and other existing methods.

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Notes

  1. We omit here for its complex representation.

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Acknowledgements

The authors are very grateful to the editor and the reviewers for their valuable comments and suggestions to improve the paper. This work was supported by the Scientific Research Fund of Zhejiang Provincial Education Department of China (No. Y201430940) and K.C. Wong Magna Fund in Ningbo University.

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  1. Department of Mathematics, Faculty of Science, Ningbo University, Ningbo, 315211, Zhejiang, People’s Republic of China

    Lie-jun Xie, Cai-lian Zhou & Song Xu

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  1. Lie-jun Xie
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  2. Cai-lian Zhou
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  3. Song Xu
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Correspondence to Lie-jun Xie.

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Xie, Lj., Zhou, Cl. & Xu, S. A new computational approach for the solutions of generalized pantograph-delay differential equations. Comp. Appl. Math. 37, 1756–1783 (2018). https://doi.org/10.1007/s40314-017-0418-0

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  • DOI: https://doi.org/10.1007/s40314-017-0418-0

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