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New Analytical Solutions for Time-Fractional Kolmogorov-Petrovsky-Piskunov Equation with Variety of Initial Boundary Conditions

Author

Listed:
  • Thanon Korkiatsakul

    (Department of Mathematics, Faculty of Applied Science, King Mongkut’s University of Technology North Bangkok, Bangkok 10800, Thailand)

  • Sanoe Koonprasert

    (Department of Mathematics, Faculty of Applied Science, King Mongkut’s University of Technology North Bangkok, Bangkok 10800, Thailand
    Centre of Excellence in Mathematics, CHE, Si Ayutthaya Rd., Bangkok 10400, Thailand)

  • Khomsan Neamprem

    (Department of Mathematics, Faculty of Applied Science, King Mongkut’s University of Technology North Bangkok, Bangkok 10800, Thailand
    Centre of Excellence in Mathematics, CHE, Si Ayutthaya Rd., Bangkok 10400, Thailand)

Abstract

The generalized time fractional Kolmogorov-Petrovsky-Piskunov equation (FKPP), D t α ω ( x , t ) = a ( x , t ) D x x ω ( x , t ) + F ( ω ( x , t ) ) , which plays an important role in engineering, chemical reaction problem is proposed by Caputo fractional order derivative sense. In this paper, we develop a framework wavelet, including shift Chebyshev polynomial of the first kind as a mother wavelet, and also construct some operational matrices that represent Caputo fractional derivative to obtain analytical solutions for FKPP equation with three different types of Initial Boundary conditions (Dirichlet, Dirichlet-Neumann, and Neumann-Robin). Our results shown that the Chebyshev wavelet is a powerful method, due to its simplicity, efficiency in analytical approximations, and its fast convergence. The comparison of the Chebyshev wavelet results indicates that the proposed method not only gives satisfactory results but also do not need large amount of CPU times.

Suggested Citation

  • Thanon Korkiatsakul & Sanoe Koonprasert & Khomsan Neamprem, 2019. "New Analytical Solutions for Time-Fractional Kolmogorov-Petrovsky-Piskunov Equation with Variety of Initial Boundary Conditions," Mathematics, MDPI, vol. 7(9), pages 1-20, September.
    • Handle: RePEc:gam:jmathe:v:7:y:2019:i:9:p:813-:d:263668
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    References listed on IDEAS

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    1. Pundikala Veeresha & Doddabhadrappla Gowda Prakasha & Dumitru Baleanu, 2019. "An Efficient Numerical Technique for the Nonlinear Fractional Kolmogorov–Petrovskii–Piskunov Equation," Mathematics, MDPI, vol. 7(3), pages 1-18, March.
    2. Pakdaman, M. & Ahmadian, A. & Effati, S. & Salahshour, S. & Baleanu, D., 2017. "Solving differential equations of fractional order using an optimization technique based on training artificial neural network," Applied Mathematics and Computation, Elsevier, vol. 293(C), pages 81-95.
    3. Allahviranloo, T. & Gouyandeh, Z. & Armand, A., 2015. "Numerical solutions for fractional differential equations by Tau-Collocation method," Applied Mathematics and Computation, Elsevier, vol. 271(C), pages 979-990.
    4. Hojatollah Adibi & Pouria Assari, 2010. "Chebyshev Wavelet Method for Numerical Solution of Fredholm Integral Equations of the First Kind," Mathematical Problems in Engineering, Hindawi, vol. 2010, pages 1-17, June.
    Full references (including those not matched with items on IDEAS)

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