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A numerical method for fractional variable order pantograph differential equations based on Haar wavelet

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Abstract

In this article, Haar wavelet collocation technique is developed for the solution of variable order fractional pantograph differential equations. The Haar technique reduces the given equations into a system of linear algebraic equations. The derived system is then solved by Gaussian elimination method. Through fixed point theory, we develop some requisite conditions for the existence of at most one solution of the underlying problem endowed with initial conditions. Some numerical examples are also given for checking the validation and convergence of the concerned method. The mean square root and maximum absolute errors for different number of collocation points have been calculated. The results indicate that Haar method is efficient for solving such equations. Fractional derivative is described in the Caputo sense.

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Acknowledgements

We are thankful to the reviewers for their careful reading and suggestions.

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Authors and Affiliations

  1. College of Engineering, Al Ain University, Al Ain, United Arab Emirates

    Hussam Alrabaiah

  2. Department of Mathematics, Tafila Technical University, Tafila, Jordan

    Hussam Alrabaiah

  3. Department of Mathematics, University of Malakand, Chakdara Dir (Lower), Khyber Pakhtunkhawa, Pakistan

    Israr Ahmad & Kamal Shah

  4. Department of Mathematics, University of Peshawar, Peshawar, Khyber Pakhtunkhwa, Pakistan

    Rohul Amin

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  1. Hussam Alrabaiah
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  2. Israr Ahmad
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  4. Kamal Shah
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Correspondence to Hussam Alrabaiah.

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Alrabaiah, H., Ahmad, I., Amin, R. et al. A numerical method for fractional variable order pantograph differential equations based on Haar wavelet. Engineering with Computers 38, 2655–2668 (2022). https://doi.org/10.1007/s00366-020-01227-0

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  • DOI: https://doi.org/10.1007/s00366-020-01227-0

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