Shifted Gegenbauer-Gauss Collocation Method for Solving Fractional Neutral Functional-Differential Equations with Proportional Delays

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DOI: 10.46793/KgJMat2206.981H


In this paper, the shifted Gegenbauer-Gauss collocation (SGGC) method is applied to fractional neutral functional-differential equations with proportional delays. The technique we have used is based on shifted Gegenbauer polynomials and Gauss quadrature integration. The shifted Gegenbauer-Gauss method reduces solving the generalized fractional pantograph equation fractional neutral functional-differential equations to a system of algebraic equations. Reasonable numerical results are obtained by selecting few shifted Gegenbauer-Gauss collocation points. Numerical results demonstrate its accuracy, and versatility of the proposed techniques.


Neutral fractional functional-differential equations, proportional delay, collocation method, shifted Gegenbauer-Gauss quadrature, shifted Gegenbauer polynomials.


[1]   W. M. Abd-Elhameed and Y. H. Youssri, New ultraspherical wavelets spectral solutions for fractional Riccati differential equations, Abstr. Appl. Anal. 2014 (2014), 8 pages.

[2]   W. M. Abd-Elhameed and Y. H. Youssri, New spectral solutions of multi-term fractional-order initial value problems with error analysis, CMES-Comp. Model. Eng. Sci. 105 (2015), 375–398.

[3]   A. H. Bhrawy, A. A. AL-Zahrani, Y. A. Alhamed and D. Baleanu, A new generalized Laguerre-Gauss collocation scheme for numerical solution of generalized fractional pantograph equations, Rom. Journ. Phys. 59 (2014), 646–657.

[4]   R. L. Bagley and P. J. Torvik, A theoritical asis for the appliation of fractional calculus to viscoelastiity, J. Rheology 27 (1983), 201–210.

[5]   R. L. Bagley and P. J. Torvik, Fractional calculus in the transient analysis of viscoelastially damped struturesy, AIAA J. 23 (1985), 918–925.

[6]   R. Baillie, Long memory processes and fractional integration in econometrices, J. Econometrices 73 (1996), 5–59.

[7]   A. H. Bhrawy, A. Al Zahrani, D. Baleanu and Y. Alhamed, A modified generalized laguerre-gauss collocation method for fractional neutral functional-differential equations on the half-line, Abstr. Appl. Anal. 2014 (2014), 7 pages.

[8]   E. H. Doha, A. H. Bhrawy, D. Baleanu and R. M. Hafez, Efficient Jacobi-Gauss collocation method for solving initial value problems of Bratu type, Comp. Math. Math. Phys. 53 (2013), 1292–1302.

[9]   E. H. Doha, A. H. Bhrawy, D. Baleanu and R. M. Hafez, A new Jacobi rational-Gauss collocation method for numerical solution of generalized pantograph equations, Appl. Numer. Math. 77 (2014), 43–54.

[10]   T. S. Chow, Fractional dynamics of interfaces etween soft-nanoparticls and rough sustrates, Phys. Lett. A 342 (2005), 148–155.

[11]   S. Das, Functional Fractional Calculus for System Identification and Controls, Springer, New York, 2008.

[12]   E. H. Doha, The coefficients of differentiated expansions and derivatives of ultraspherical polynomials, J. Comp. Math. Appl. 19 (1991), 115–122.

[13]   M. Ghasemi, M. Fardi and R. K. Ghaziani, Numerical solution of nonlinear delay differential equations of fractional order in reproducing kernel Hilbert space, Appl. Math. Comput. 268 (2015), 815–831.

[14]   B. Fornberg, A practical guide to pseudospectral methods, Cambridge University Press, Cambridge, 1998.

[15]   J. H. He, Nonlinear oscillation with fractional derivatives and its applications in: International Conference on Vibrating Engineering’98, Dalia, Chinan, 1998.

[16]   J. H. He, Some applications of nonlinear fractional differential equations and thier approximations bull, Sci. Technol. 15 (1999), 86–90.

[17]   M. Ishteva, Properties and applications of the Caputo fractional operator, PhD thesis, Department of Mathematics, Universitat Karlsruhe (TH), Sola, Bulgaria, 2005.

[18]   C. Li and W. Deng, Remarks on fractional derivatives, Appl. Math. Comput. 187 (2007), 777–784.

[19]   C. Li and Z. Zhao, Introduction to fractional integrability and differentiability, The European Physical Journal Special Topics 193 (2011), 5–26.

[20]   R. Magin, Frational calculus in bioengineering, Crit. Rev. Biomed. Eng. 32 (2004), 1–104.

[21]   R. Magin, Frational calculus in bioengineering-part 2, Crit. Rev. Biomed. Eng. 32 (2004), 105–193.

[22]   R. Magin, Frational calculus in bioengineering-part 3, Crit. Rev. Biomed. Eng. 32 (2004), 194–377.

[23]   F. Mainardi, Fractional calculus: some basic prolems in continuum and statical mechanis, in: A. Carpinteri and F. Mainardi (Eds.), Fratals and Fractional Calculus in Continuum Mechanics, Springer, Verlag, New York, 1997, 291–348.

[24]   B. Mandelbort, Some noises with 1∕f spectrum, a bridge between direct current and white noise, IEEE Trans. Inform. Theory 13 (1967), 289–298.

[25]   R. Metzler and J. Klafter, The restaurant at the end of the random walk: Recent developments in the description of anomalos transport by fractional dyuamics, J. Phys. A 37 (2004), 161–208.

[26]   K. S. Miller and B. Ross, An Introduaction to the Frational Calculus and Fractional Differential Equations, Wiley, New York, 1993.

[27]   K. B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, New York, 1974.

[28]   P. Rahimkhani, Y. Ordokhani and E. Babolian, Numerical solution of fractional pantograph differential equations by using generalized fractional-order Bernoulli wavelet, J. Comput. Appl. Math. 309 (2017), 493–510.

[29]   A. Anapali, Y. Öztürk and M. Gülsu, Numerical approach for solving fractional pantograph equation, IJCA 113 (2015), 45–52.

[30]   R. Peyret, Spectral Methods for Incompressible Viscous Flow, New York, Springer, 2002.

[31]   I. Podluny, Fractional Differential Equations, Academic Press, San Diego, 1999.

[32]   C. Li D. Qian and Y. Chen, On Riemann-Liouville and Caputo derivatives, Discrete Dynamics in Nature and Society 2011 (2011), 15 pages.

[33]   C. Canuto, M. Y. Hussaini, A. Quarteroni and T. A. Zang, Spectral Methods in Fluid Dynamics, New York, Springer, 1988.

[34]   E. D. Rainville, Special functions, Chelsea Pub Co., New York, 1971.

[35]   Y. A. Rossikhin and M. V. Shitikova, Applications of fractional calculus to dynamic prolems of linear and nonlinear hereditary mechanics of solids, Appl. Mech. Rev. 50 (1997), 15–67.

[36]   A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, San Diego, 2006.

[37]   G. Szegö, Orthogonal Polynomials, American Mathematical Society Colloquium Publications, Providence, Rhode Island, 1975.

[38]   L. N. Trefethen, Spectral Methods in MATLAB, Philadelphia, PA: SIAM, 2000.

[39]   Y. Yang and Y. Huang, Spectral-collocation methods for fractional pantograph delay-integrodifferential equations, Adv. Math. Phys. 2013 (2013), 14 pages.