A novel numerical implementation for solving time fractional telegraph differential equations having multiple space and time delays via Delannoy polynomial

This paper is concerned with solving numerically the time fractional telegraph equationshaving multiple space and time delays by proposing a novel matrix-collocation methoddependent on the Delannoy polynomial. This method enables easy and fast approximationtool consisting of the matrix expansions of the functions using only the Delannoypolynomial. Thus, the solutions are obtained directly from a unique matrix system. Also, theresidual error computation, which involves the same procedure as the method, provides theimprovement of the solutions. The method is evaluated under some valuable error tests in thenumerical applications. To do this, a unique computer module is devised. The present resultsare compared with those of the existing methods in the literature, in order to oversee theprecision and efficiency of the method. One can express that the proposed method admitsvery consistent approximation for the equations in question.

Keywords:

Delannoy polynomial, matrix-collocation method, multiple delays telegraph equation,

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Caputo, M., \enquote{Elasticit$\grave{a}$e Dissipazione}, Bologna, Zanichelli, 1969.
Moaddy, K., Momani, S., Hashim, I., \enquote{The non-standard finite difference scheme for linear fractional PDEs in fluid mechanics}, Comput. Math. Appl., 61, (2011), 1209--1216.
Hosseini, V.R., Chen, W., Avazzadeh, Z., \enquote{Numerical solution of fractional telegraph equation by using radial basis functions}, Eng. Anal. Bound. Elem., 38, (2014), 31--39.
Faraji, M, Ansari, O.R., \enquote{Linear and nonlinear vibrations of fractional viscoelastic Timoshenko nanobeams considering surface energy effects}, Appl. Math. Model., 43, (2017), 337--350.
Arqub, O.A., \enquote{Numerical solutions for the Robin time-fractional partial differential equations of heat and fluid flows based on the reproducing kernel algorithm}, Int. J. Numer. Method H., 28(4), (2018), 828--856.
Koleva, M.N., Vulkov, L.G., \enquote{Numerical solution of time-fractional Black--Scholes equation}, Comp. Appl. Math., 36, (2017), 1699--1715.
Soori, Z., Aminataei, A., \enquote{A new approximation to Caputo-type fractional diffusion and advection equations on non-uniform meshes}, Appl. Numer. Math., 144, (2019), 21--41.
K\"{u}rk\c{c}\"{u}, \"{O}.K., Aslan, E., Sezer, M., \enquote{An advanced method with convergence analysis for solving space-time fractional partial differential equations with multi delays}, Eur. Phys. J. Plus, 134, (2019), 393.
Dehestani, H., Ordokhani, Y., Razzaghi, M., \enquote{A numerical technique for solving various kinds of fractional partial differential equations via Genocchi hybrid functions}, Rev. R. Acad. Cienc. Exactas, Fis. Nat. Madr., 113, (2019), 3297--3321.
Rihan, F.A., \enquote{Computational methods for delay parabolic and time-fractional partial differential equations}, Numer. Methods Partial Differ. Equ., 26, (2010), 1556--1571.
Usman, M., Hamid, M., Zubair, T., Haq, R.U., Wang, W., Liu, M.B., \enquote{Novel operational matrices-based method for solving fractional-order delay differential equations via shifted Gegenbauer polynomials}, Appl. Math. Comput., 372, (2020), 124985.
Pozar, D.M., \enquote{Microwave Engineering} in Addison-Wesley, 1990.
Mainardi, F., Paradisi, P., \enquote{Fractional diffusive waves}, J. Comput. Acoust., 9(4), (2001), 1417--1436.
Abdou, M.A., \enquote{Adomian decomposition method for solving the telegraph equation in charged particle transport}, J. Quant. Spectrosc. Radiat. Transf., 95, (2005), 407--414.
Orsingher, E., Beghin, L., \enquote{Time-fractional telegraph equations and telegraph processes with brownian time}, Probab. Theory Relat. Fields, 128, (2004), 141--160.
Kumar, A., Bhardwaj, A., Dubey, S., \enquote{A local meshless method to approximate the time-fractional telegraph equation}, Eng. Comput., (2020), https://doi.org/10.1007/s00366-020-01006-x.
Wang, Y.L., Du, M.-J., Temuer, C.-L., Tian, D., \enquote{Using reproducing kernel for solving a class of time-fractional telegraph equation with initial value conditions}, Int. J. Comput. Math., 95(8), (2018), 1609--1621.
Pandey, R.K., Mishra, H.K., \enquote{Numerical simulation for solution of space-time fractional telegraphs equations with local fractional derivatives via HAFSTM}, New Astronomy, 57, (2017), 82--93.
Y\"{u}zba\c{s}\i, \c{S}., Kara\c{c}ay\i r, M., \enquote{A Galerkin-type method to solve one-dimensional telegraph equation using collocation points in initial and boundary conditions}, Int. J. Comput. Methods, 15(1), (2018), 1850031, (16 pages).
Dehghan, M., Shokri, A., \enquote{A numerical method for solving the hyperbolic telegraph equation}, Numer. Methods Partial Differ. Equ., 24, (2008), 1080--1093.
Dehghan, M., Lakestani, M., \enquote{The use of Chebyshev cardinal functions for solution of the second-order one-dimensional telegraph equation}, Numer. Methods Partial Differ. Equ., 25, (2009), 931--938.
Pandit, S., Kumar, M., Tiwari, S., \enquote{Numerical simulation of second-order hyperbolic telegraph type equations with variable coefficients}, Comput. Phys. Commun., 187, (2015), 83--90.
Sharifi, S., Rashidinia, J., \enquote{Numerical solution of hyperbolic telegraph equation by cubic B-spline collocation method}, Appl. Math. Comput., 281, (2016), 28--38.
Delannoy, H., \enquote{Emploi de l'echiquier pour la resolution de certains problµemes de probabilites}, Assoc. Franc. Bordeaux, 24, (1895), 70--90.
Banderier, C., Schwer, S., \enquote{Why Delannoy numbers?}, J. Stat. Plan. Infer., 135(1), (2005), 40--54.
Weisstein, E.W., \enquote{Delannoy Number}, from MathWorld--A Wolfram Web Resource, https://mathworld.wolfram.com/DelannoyNumber.html.
Comtet, L., \enquote{Advanced Combinatorics: The Art of Finite and Infinite Expansions}, Rev. enl. ed. Dordrecht, Netherlands, Reidel, pp. 80-81, 1974.
Sulanke, R.A., \enquote{Objects counted by the central Delannoy numbers}, J. Integer Seq., 6, (2003), Article 03.1.5.
Peart, P., Woan, W.-J., \enquote{A bijective proof of the Delannoy recurrence}, Congressus Numerantium, 158, (2002), 29--33.
Hetyei, G., \enquote{Shifted Jacobi polynomials and Delannoy numbers}, (2009), ArXiv e-prints arXiv:0909.5512v2.
Sun, Z., \enquote{Congruences involving generalized central trinomial coefficients}, Sci. China Math., 57, (2014), 1375--1400.
Sun, Z.-W., \enquote{On Delannoy numbers and Schr\"{o}der numbers}, J. Number Theory, 131, (2011), 2387--2397.
Torres, A., Cabada, A., Nieto, J.J., \enquote{An exact formula for the number of alignments between two DNA sequences}, DNA Seq., 14, (2003), 227--430.
Guo, V.J.W., Zeng, J., \enquote{Proof of some conjectures of Z.-W. Sun on congruences for Apery polynomials}, J. Number Theory, 132, (2012), 1731--1740.
Abramowitz, M., Stegun, I.A., \enquote{Handbook of Mathematical Functions, with Formulas, Graphs, and Mathematical Tables}, Nat. Bureau of Standards, Appl. Math. Series, 55, 1964.

ISSN: 1694-7398
Yayın Aralığı: Yılda 2 Sayı
Başlangıç: 2001
Yayıncı: KIRGIZİSTAN-TÜRKİYE MANAS ÜNİVERSİTESİ

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