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pp. 201-209 | Article Number: ijese.2018.016
Published Online: March 16, 2018
Abstract
We need differential equations for modeling and analyzing a huge amount of issues. Fractional calculus is a branch of mathematical analysis that is used in many fields of mathematical and engineering sciences such as electrical networks, fluid mechanics, control theory, electromagnetism, biology, chemistry, propagation and viscoelasticity. The most important topics in mathematics are differential equations and integral equations which are very practical and have a special place in various sciences, especially engineering sciences. We used approximate methods to obtain the results because we cannot use analytical clustering for this kind of equations. The aim of this paper is to investigate the operational matrix in order to solve the partial differential equation.
Keywords: differential equations, operating matrix, minor derivatives, gamma function, integral equations, special functions, fractional problems
References
Arfken, G. B., & Weber, H. J. (2001). Mathematical Methods for Physics. Amsterdam: Elsevier Academic Press.
Bhrawy, A. H., Zaky, M. A., & Baleanu, D. (2015). New numerical approximations for space–time fractional Burgers’ equations via a Legendre spectral-collocation method. Rom. Rep. Phys, 2, 1221–1451.
Chen, C. S., Liu, F., & Burrage, K. (2014). Numerical simulation of a new two-dimensional variable-order fractional percolation equation in non-homogeneous porous media. Comput. Math. Appl, 67, 1673–1681. https://doi.org/10.1016/j.camwa.2014.03.003
Dehghan, M., Abdi-mazraeh, S., & Lakestani, M., (2014). Numerical solution for a class of fractional convection–diffusion equation using the flatlet oblique multiwavelets. J. Vib. Control, 20, 913–924. https://doi.org/10.1177/1077546312470473
Doha, E. H., Bhrawy, A. H., Abdelkawy, M. A., & Van Gorder, R. A. (2014). Jacobi–Gauss–Lobatto collocation method for the numerical solution of 1 + 1 nonlinear Schrödinger equations. J. Comput. Phys, 261, 244–255. https://doi.org/10.1016/j.jcp.2014.01.003
Garrappa, R. (2009). On some explicit Adams multistep methods for fractional differential equations. J. Comput. Appl. Math., 229, 392-399. https://doi.org/10.1016/j.cam.2008.04.004
Hashemi, B. H., Khodabin, M., & Maleknejad, K. (2017). Numerical method for solving linear stochastic Ito-Volterra integral equations driven by fractional Brownian motion using hat functions. Turkish Journal of Mathematics, 41(3), 611-624. https://doi.org/10.3906/mat-1508-50
Hatamzadeh-Varmazyar, S., & Masouri, Z. (2011). Numerical method for analysis of one- and two-dimensional electromagnetic scattering based on using linear Fredholm integral equation models, Math. Comput. Modell, 54, 2199–2210. https://doi.org/10.1016/j.mcm.2011.05.028
Hatamzadeh-Varmazyar, S., & Masouri, Z. (2013). Numerical expansion-iterative method for analysis of integral equation models arising in one- and twodimensional electromagnetic scattering. Eng. Anal. Bound. Elem, 36, 416–422. https://doi.org/10.1016/j.enganabound.2011.09.008
Jia, J., & Sogabe, T. (2013). On particular solution of ordinary differential equations with constant coefficients. Appl. Math. Comput, 219, 6761–6767. https://doi.org/10.1016/j.amc.2012.12.080
Khan, N. A., Jamil, M., Ara, A., & Das, S. (2011). Explicit solution of time-fractional batch reactor system. Int. J. Chem. React. Eng., 9, 154-161. https://doi.org/10.2202/1542-6580.2602
Maleknejad, K., & Torabi, P. (2012). Application of fixed point method for solving nonlinear Volterra-Hammerstein integral equation. U. P. B. Sci. Bull, 74, 12-23.
Mashayekhi, S., & Razzaghi, M. (2016). Numerical solution of distributed order fractional differential equations by hybrid functions. Journal of Computational Physics, 315, 169-181. https://doi.org/10.1016/j.jcp.2016.01.041
Mirzaee, F., & Hadadiyan, E. (2015). Numerical solution of linear Fredholm integral equations via two-dimensional modification of hat functions. Appl. Math. Comput., 250, 805-816. https://doi.org/10.1016/j.amc.2014.10.128
Neamaty, A. B., & Firozja, M. A. (2014). Numerical solution for boundary value problem of fractional order with approximate integral and derivative. Comput. Methods Differential Equations, 2, 195-204.
Ordokhani, Y., & Dehestani, H. (2014). Numerical solution of the nonlinear Fredholm-Volterra Hammerstein integral equations via Bessel functions. Journal of Information and Computing Science, 9, 123-131.
Parand, K., Dehghan, M., & Pirkhedri, A. (2013). The Sinc-collocation method for solving the Thomas- Fermi equation. J. Comput. Appl. Math., 237, 244-252. https://doi.org/10.1016/j.cam.2012.08.001
Saadatmandi, A. (2014). Bernstein operational matrix of fractional derivatives and its applications. Appl. Math. Modelling, 38, 1365-1372. https://doi.org/10.1016/j.apm.2013.08.007
Sedaghat, S., Ordokhani, Y., & Dehghan, M., (2012). Numerical solution of the delay differential equations of pantograph type via Chebyshev polynomials. Commun. Nonlinear Sci., 17, 4815–4830. https://doi.org/10.1016/j.cnsns.2012.05.009
Stephenson, G. (1974). An introduction to Partial Differential Equations for Science Students. Harlow: Longman.
Yang, C., (2012). Chebyshev polynomials solution of nonlinear integral equations. Journal of Franklin Institute, 349, 947-956. https://doi.org/10.1016/j.jfranklin.2011.10.023
Zlatev, Z., Faragó, I., & Havasi, Á. (2012). Richardson extrapolation combined with the sequential splitting procedure and θ –method. Cent. Eur. J. Math, 10(1), 159–172. https://doi.org/10.2478/s11533-011-0099-7