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Legendre wavelets optimization method for variable-order fractional Poisson equation

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  • Heydari, Mohammad Hossein
  • Avazzadeh, Zakieh

Abstract

In this study, the Poisson equation is generalized with the concept of variable-order (V-O) fractional derivatives called variable-order fractional Poisson equation (V-OFPE). In order to find an accurate solution of this system, we establish an optimization method through the Legendre wavelets (LWs). To carry out the method, we firstly derive an operational matrix (OM) of V-O fractional derivative for the LWs to be employed in expanding the unknown solution. Then, the function of residual is applied to reform the V-OFPE to an optimization problem which leads to choose the unknown coefficients optimally. In the final step, we implement the constrained extremum method which adjoins the objective function implied from the two-norm of residual function and the constraints corresponded to the given boundary conditions by a set of Lagrange multipliers. Accordingly, the final optimal conditions are actually the algebraic equations including the expansion coefficients and Lagrange multipliers. Theoretical convergence and error analysis of the approximation procedure using the LWs are investigated. In addition, the applicability and computational efficiency are experimentally examined for some illustrative examples.

Suggested Citation

  • Heydari, Mohammad Hossein & Avazzadeh, Zakieh, 2018. "Legendre wavelets optimization method for variable-order fractional Poisson equation," Chaos, Solitons & Fractals, Elsevier, vol. 112(C), pages 180-190.
    • Handle: RePEc:eee:chsofr:v:112:y:2018:i:c:p:180-190
    DOI: 10.1016/j.chaos.2018.04.028
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    References listed on IDEAS

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    1. Bertram Ross & Stefan Samko, 1995. "Fractional integration operator of variable order in the holder spaces H λ ( x )," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 18, pages 1-12, January.
    2. Sun, HongGuang & Chen, Wen & Chen, YangQuan, 2009. "Variable-order fractional differential operators in anomalous diffusion modeling," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 388(21), pages 4586-4592.
    3. Ali Ahmadian & Mohamed Suleiman & Soheil Salahshour, 2013. "An Operational Matrix Based on Legendre Polynomials for Solving Fuzzy Fractional-Order Differential Equations," Abstract and Applied Analysis, Hindawi, vol. 2013, pages 1-29, June.
    4. D. Baleanu & A. H. Bhrawy & T. M. Taha, 2013. "Two Efficient Generalized Laguerre Spectral Algorithms for Fractional Initial Value Problems," Abstract and Applied Analysis, Hindawi, vol. 2013, pages 1-10, June.
    5. Heydari, M.H. & Hooshmandasl, M.R. & Maalek Ghaini, F.M. & Cattani, C., 2016. "Wavelets method for solving fractional optimal control problems," Applied Mathematics and Computation, Elsevier, vol. 286(C), pages 139-154.
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