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An Explicit Wavelet Method for Solution of Nonlinear Fractional Wave Equations

Author

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  • Jiong Weng

    (Key Laboratory of Mechanics on Disaster and Environment in Western China, The Ministry of Education, College of Civil Engineering and Mechanics, Lanzhou University, Lanzhou 730000, China)

  • Xiaojing Liu

    (Key Laboratory of Mechanics on Disaster and Environment in Western China, The Ministry of Education, College of Civil Engineering and Mechanics, Lanzhou University, Lanzhou 730000, China)

  • Youhe Zhou

    (Key Laboratory of Mechanics on Disaster and Environment in Western China, The Ministry of Education, College of Civil Engineering and Mechanics, Lanzhou University, Lanzhou 730000, China)

  • Jizeng Wang

    (Key Laboratory of Mechanics on Disaster and Environment in Western China, The Ministry of Education, College of Civil Engineering and Mechanics, Lanzhou University, Lanzhou 730000, China)

Abstract

An explicit method for solving time fractional wave equations with various nonlinearity is proposed using techniques of Laplace transform and wavelet approximation of functions and their integrals. To construct this method, a generalized Coiflet with N vanishing moments is adopted as the basis function, where N can be any positive even number. As has been shown, convergence order of these approximations can be N . The original fractional wave equation is transformed into a time Volterra-type integro-differential equation associated with a smooth time kernel and spatial derivatives of unknown function by using the technique of Laplace transform. Then, an explicit solution procedure based on the collocation method and the proposed algorithm on integral approximation is established to solve the transformed nonlinear integro-differential equation. Eventually the nonlinear fractional wave equation can be readily and accurately solved. As examples, this method is applied to solve several fractional wave equations with various nonlinearities. Results show that the proposed method can successfully avoid difficulties in the treatment of singularity associated with fractional derivatives. Compared with other existing methods, this method not only has the advantage of high-order accuracy, but it also does not even need to solve the nonlinear spatial system after time discretization to obtain the numerical solution, which significantly reduces the storage and computation cost.

Suggested Citation

  • Jiong Weng & Xiaojing Liu & Youhe Zhou & Jizeng Wang, 2022. "An Explicit Wavelet Method for Solution of Nonlinear Fractional Wave Equations," Mathematics, MDPI, vol. 10(21), pages 1-14, October.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:21:p:4011-:d:956780
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    References listed on IDEAS

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    1. M. Abu-Shady & Mohammed K. A. Kaabar, 2021. "A Generalized Definition of the Fractional Derivative with Applications," Mathematical Problems in Engineering, Hindawi, vol. 2021, pages 1-9, October.
    2. Abdul Ghafoor & Sirajul Haq & Manzoor Hussain & Poom Kumam & Muhammad Asif Jan, 2019. "Approximate Solutions of Time Fractional Diffusion Wave Models," Mathematics, MDPI, vol. 7(10), pages 1-15, October.
    3. Valentina V. Tarasova & Vasily E. Tarasov, 2017. "Economic interpretation of fractional derivatives," Papers 1712.09575, arXiv.org.
    4. Atangana, Abdon, 2017. "Fractal-fractional differentiation and integration: Connecting fractal calculus and fractional calculus to predict complex system," Chaos, Solitons & Fractals, Elsevier, vol. 102(C), pages 396-406.
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