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Numerical approach for solving variable-order space–time fractional telegraph equation using transcendental Bernstein series

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Abstract

This paper presents the transcendental Bernstein series (TBS) as a generalization of the classical Bernstein polynomials for solving the variable-order space–time fractional telegraph equation (V-STFTE). An approximation method using optimization techniques and the TBS is introduced. The solution of the problem under consideration is expanded in terms of TBS with unknown free coefficients and control parameters. The new corresponding operational matrices of variable-order fractional derivatives, in the Caputo type, are derived. The proposed approach reduces the V-STFTE to a system of algebraic equations and, subsequently, to find the free coefficients and control parameters using the Lagrange multipliers technique. The convergence analysis of the method is guranteed by means of a new theorem concerning the TBS. The experimental results confirm the high accuracy and computational efficiency of the TBS method.

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References

  1. Agrawal OP (2004) A general formulation and solution scheme for fractional optimal control problems. Nonlinear Dyn 38:323–337

    MathSciNet MATH  Google Scholar 

  2. Engheta N (1996) On fractional calculus and fractional multipoles in electromagnetism. IEEE Trans Antennas Propag 44:554–566

    MathSciNet MATH  Google Scholar 

  3. Magin RL (2010) Fractional calculus models of complex dynamics in biological tissues. Comput Math Appl 59:1586–1593

    MathSciNet MATH  Google Scholar 

  4. Kulish VV, Lage JL (2002) Application of fractional calculus to fluid mechanics. J Fluids Eng 124:803–806

    Google Scholar 

  5. Oldham KB (2010) Fractional differential equations in electrochemistry. Adv Eng Softw 41:9–12

    MATH  Google Scholar 

  6. Gafiychuk V, Datsko B, Meleshko V (2008) Mathematical modeling of time fractional reaction diffusion systems. J Comput Appl Math 220:215–225

    MathSciNet MATH  Google Scholar 

  7. Meerschaert MM, Tadjeran C (2006) Finite difference approximations for two-sided space-fractional partial differential equations. Appl Numer Math 56:80–90

    MathSciNet MATH  Google Scholar 

  8. Moghaddam BP, Machado JAT (2017) Extended algorithms for approximating variable order fractional derivatives with applications. J Sci Comput 71(3):1351–1374

    MathSciNet MATH  Google Scholar 

  9. Odibat Z, Momani S, Xu H (2010) A reliable algorithm of homotopy analysis method for solving nonlinear fractional differential equations. Appl Math Model 34:593–600

    MathSciNet MATH  Google Scholar 

  10. Li X, Xu M, Jiang X (2009) Homotopy perturbation method to time-fractional diffusion equation with a moving boundary. Appl Math Comput 208:434–439

    MathSciNet MATH  Google Scholar 

  11. Heydari MH, Hooshmandasl MR, Ghaini FMM, Cattani C (2016) Wavelets method for solving fractional optimal control problems. Appl Math Comput 286:139–154

    MathSciNet MATH  Google Scholar 

  12. Heydari MH, Hooshmandasl MR, Mohammadi F (2014) Legendre wavelets method for solving fractional partial differential equations with Dirichlet boundary conditions. Appl Math Comput 234:267–276

    MathSciNet MATH  Google Scholar 

  13. El-sayed A, Gaber M (2006) The Adomian decomposition method for solving partial differential equations of fractional order in finite domains. Phys Lett A 359:175–182

    MathSciNet MATH  Google Scholar 

  14. Galeone L, Garrappa R (2006) On multistep methods for differential equations of fractional order. Mediterr J Math 3:565–580

    MathSciNet MATH  Google Scholar 

  15. Diethelm K, Ford NJ, Freed AD (2002) A predictor-corrector approach for the numerical solution of fractional differential equations. Nonlinear Dyn 16:3–22

    MathSciNet MATH  Google Scholar 

  16. Moghaddam BP, Yaghoobi S, Machado JAT (2016) An extended predictor–corrector algorithm for variable-order fractional delay differential equations. J Comput Nonlinear Dyn 11(6):061001 (7 pages)

    Google Scholar 

  17. Bhrawy AH, Doha EH, Baleanu D, Ezz-Eldien SS (2014) A spectral tau algorithm based on jacobi operational matrix for numerical solution of time fractional diffusion-wave equations. J Comput Phys 293:142–156

    MathSciNet MATH  Google Scholar 

  18. Dahaghin MS, Hassani H (2017) A new optimization method for a class of time fractional convection–diffusion-wave equations with variable coefficients. Eur Phys J Plus 132:130

    Google Scholar 

  19. Dahaghin MS, Hassani H (2017) An optimization method based on the generalized polynomials for nonlinear variable-order time fractional diffusion-wave equation. Nonlinear Dyn 88(3):1587–1598

    MathSciNet MATH  Google Scholar 

  20. Jiwari R, Pandit S, Mittal RC (2012) Numerical simulation of two-dimensional sine-Gordon solitons by differential quadrature method. Comput Phys Commun 183:600–616

    MathSciNet MATH  Google Scholar 

  21. Pandit S, Jiwari R, Bedi K, Koksal ME (2017) Haar wavelets operational matrix based algorithm for computational modelling of hyperbolic type wave equations. Eng Comput 34(8):793–2814

    Google Scholar 

  22. Jiwari R (2015) Lagrange interpolation and modified cubic B-spline differential quadrature methods for solving hyperbolic partial differential equations with Dirichelet and Neumann boundary conditions. Comput Phys Commun 193:55–65

    MathSciNet MATH  Google Scholar 

  23. Jiwari R, Pandit S, Mittal RC (2012) A differential quadrature algorithm to solve the two dimensional linear hyperbolic telegraph equation with Dirichlet and Neumann boundary conditions. Appl Math Comput 218:7279–7294

    MathSciNet MATH  Google Scholar 

  24. Pandey RK, Mishra HK (2017) Numerical simulation for solution of space-time fractional telegraphs equations with local fractional derivatives via HAFSTM. New Astron 57:82–93

    Google Scholar 

  25. Mittal RC, Dahiya S (2017) Numerical simulation of three-dimensional telegraphic equation using cubic B-spline differential quadrature method. Appl Math Comput 313:442–452

    MathSciNet MATH  Google Scholar 

  26. Oruç Ö (2018) A numerical procedure based on Hermite wavelets for two-dimensional hyperbolic telegraph equation. Eng Comput 34(4):741–755

    MathSciNet  Google Scholar 

  27. Mollahasani N, Moghadam MM, Afrooz K (2016) A new treatment based on hybrid functions to the solution of telegraph equations of fractional order. Appl Math Model 40(4):2804–2814

    MathSciNet MATH  Google Scholar 

  28. Hashemi MS, Baleanu D (2016) Numerical approximation of higher-order time-fractional telegraph equation by using a combination of a geometric approach and method of line. J Comput Phys 316:10–20

    MathSciNet MATH  Google Scholar 

  29. Jiwari R, Pandit S, Mittal RC (2012) A differential quadrature algorithm for the numerical solution of the second-order one dimensional hyperbolic telegraph equation. Int J Nonlinear Sci 13(3):259–266

    MathSciNet MATH  Google Scholar 

  30. Heydari MH, Hooshmandasl MR, Mohammadi F (2014) Two-dimensional Legendre wavelets for solving time-fractional telegraph equation. Adv Appl Math Mech 6(2):247–260

    MathSciNet MATH  Google Scholar 

  31. Ford NJ, Rodrigues MM, Xiao J, Yan Y (2013) Numerical analysis of a two-parameter fractional telegraph equation. J Comput Appl Math 249:95–106

    MathSciNet MATH  Google Scholar 

  32. Saadatmandi A, Mohabbati M (2015) Numerical solution of fractional telegraph equation via the tau method. Math Rep 17(67):155–166

    MathSciNet MATH  Google Scholar 

  33. Kumar S (2014) A new analytical modelling for fractional telegraph equation via Laplace transform. Appl Math Model 38:3154–3163

    MathSciNet MATH  Google Scholar 

  34. Momani S (2005) Analytic and approximate solutions of the space- and time-fractional telegraph equations. Appl Math Comput 170:1126–34

    MathSciNet MATH  Google Scholar 

  35. Zhao Z, Li C (2012) Fractional difference/finite element approximations for the time-space fractional telegraph equation. Appl Math Comput 219:2975–2988

    MathSciNet MATH  Google Scholar 

  36. Wei L, Dai H, Zhang D, Si Z (2014) Fully discrete local discontinuous Galerkin method for solving the fractional telegraph equation. Calcolo 51:175–192

    MathSciNet MATH  Google Scholar 

  37. Saadatmandi A, Dehghan M (2010) numerical solution of hyperbolic telegraph equation using the Chebyshev tau method. Numer Methods Partial Differ Equations 26:239–252

    MathSciNet MATH  Google Scholar 

  38. Pandit S, Kumar M, Tiwari S (2015) Numerical simulation of second-order hyperbolic telegraph type equations with variable coefficients. Comput Phys Commun 187:83–90

    MathSciNet MATH  Google Scholar 

  39. Chen J, Liu F, Anh V (2008) Analytical solution for the time-fractional telegraph equation by the method of separating variables. J Math Anal Appl 338:1364–1377

    MathSciNet MATH  Google Scholar 

  40. Bhrawy AH, Zaky MA (2017) Highly accurate numerical schemes for multi-dimensional space variable-order fractional Schrödinger equations. Comput Math Appl 73(6):1100–1117

    MathSciNet MATH  Google Scholar 

  41. Kreyszig E (1978) Introductory functional analysis with applications. Wiley, New York

    MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Applied Mathematics, Faculty of Mathematical Sciences, Shahrekord University, Shahrekord, Iran

    H. Hassani

  2. School of Mathematical Sciences, Nanjing Normal University, Nanjing, 210023, China

    Z. Avazzadeh

  3. Department of Electrical Engineering, Institute of Engineering, Polytechnic of Porto, R. Dr. António Bernardino de Almeida, 4249-015, Porto 431, Portugal

    J. A. Tenreiro Machado

Authors
  1. H. Hassani

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  2. Z. Avazzadeh

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  3. J. A. Tenreiro Machado

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Correspondence toH. Hassani.

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Hassani, H., Avazzadeh, Z. & Machado, J.A.T. Numerical approach for solving variable-order space–time fractional telegraph equation using transcendental Bernstein series.Engineering with Computers36, 867–878 (2020). https://doi.org/10.1007/s00366-019-00736-x

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