Abstract
In recent times, operational matrix methods become overmuch popular. Actually, we have many more operational matrix methods. In this study, a new remodeled method is offered to solve linear Fredholm–Volterra integro-differential equations (FVIDEs) with piecewise intervals using Chebyshev operational matrix method. Using the properties of the Chebyshev polynomials, the Chebyshev operational matrix method is used to reduce FVIDEs into a linear algebraic equations. Some numerical examples are solved to show the accuracy and validity of the proposed method. Moreover, the numerical results are compared with some numerical algorithm.
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References
Delves, L.M., Mohamed, J.L.: Computational Methods for Integral Equations. Cambridge University Press, Cambridge (1985)
Agarwal, R.P. (ed.): Contributions in Numerical Mathematics. World Scientific Publishing, Singapore (1993)
Agarwal, R.P. (ed.): Dynamical Systems and Applications. World Scientific Publishing, Singapore (1995)
Wazwaz, A.M.: A First Course in Integral Equations. World Scientifics, Singapore (1997)
Bhrawy, A.H., Tohidi, E., Soleymani, F.: A new Bernoulli matrix method for solving high-order linear and nonlinear Fredholm integro-differential equations with piecewise intervals. Appl. Math. Comp. 219, 482–497 (2012)
Biçer, G.G., Öztürk, Y., Gülsu, M.: Numerical approach for solving linear Fredholm integro-differential equation with piecewise intervals by Bernoulli polynomials. Inter. J. Comp. Math. 95, 2100–2111 (2018)
Acar, N.İ, Daşçıoğlu, A.: A projection method for linear Fredholm-Volterraintegro differential equations. J. Taibah Univ. Sci. 13(1), 644–650 (2019)
Kürkçü, Ö.K., Aslan, E., Sezer, M.: A novel collocation method based on residual error analysis for solving integro differential equations using hybrid Dickson and Taylor polynomials. Sains Malaysiana 46, 335–347 (2017)
Yüksel, G., Gülsu, M., Sezer, M.: A Chebyshev polynomial approach for high-order linear Fredholm-Volterra integro-differential equations. GU J Sci. 25, 393–401 (2012)
Ebrahimi, N., Rashidinia, J.: Spline collocation for Fredholm and Volterra integro-differential equations. Int. J. Math. Model. Comput. 4(3), 289–298 (2014)
Kürkçü, Ö.K., Ersin, A., Sezer, M.: A numerical approach with error estimation to solve general integro-differential difference equations using Dickson polynomials. Appl. Math. Comput. 276, 324–339 (2016)
Gümgüm, S., Baykuş, N., Savaşaneril, N., Kürkçü, Ö., Sezer, M.: A numerical technique based on Lucas polynomials together with standard and Chebyshev-Lobatto collocation points for solving functional integro-differential equations involving variable delays. Sakarya Univ. J. Sci. 22(6), 1659–1668 (2018)
Kurt, N., Sezer, M.: Polynomial solution of high-order linear Fredholm integro-differential equations with constant coefficients. J. Franklin Inst. 345, 839–850 (2008)
Reutskiy, S.Y.: The backward substitution method for multipoint problems with linear Volterra-Fredholm integro-differential equations of the neutral type. J. Comput. Appl. Math. 296, 724–738 (2016)
Dehghan, M., Shakeri, F.: Solution of an integro-differential equation arising in oscillating magnetic field using He’s homotopy perturbation method. Prog. Electromagnet. Res. PIER 78, 361–376 (2008)
Kürkçü, Ö.K.: A numerical method with a control parameter for integro differential delay equations with state dependent bounds via generalized Mott polynomials. Math. Sci. 14, 43–52 (2020)
Gürbüz, B., Sezer, M., Güler, C.: Laguerre collocation method for solving Fredholm integro-differential equations with functional arguments. J. Appl. Math. 682398 (2014)
Gülsu, M., Sezer, M.: Taylor collocation for the solution of systems of high-order linear Fredholm-Volterraintegro-differential equations. Int. J. Comput. Math. 83, 429–448 (2006)
Sahu, P.K., Saha Ray, S.: Numerical solutions for the system of Fredholm integral equations of second kind by a new approach involving semiorthogonal B-spline wavelet collocation method. Appl. Math. Comput. 234, 368–379 (2014)
Shahmorad, S.: Numerical solution of the general form linear Fredholm-Volterra integro differential equations by the Tau method with an error estimation. App. Math. Comput. 167, 1418–1429 (2005)
Turkyilmazoglu, M.: An effective approach for numerical solutions of high-order Fredholm integro-differential equations. Appl. Math. Comput. 227, 384–398 (2014)
Mason, J.C., Handscomb, D.C.: Chebyshev Polynomials. Chapman and Hall/CRC, New York (2003)
Vanani, S.K., Aminataei, A.: Operational Tau approximation for a general class fractional integro-differential equations. Comp. Appl. Math. 30(3), 655–674 (2011)
Öztürk, Y., Gülsu, M.: An operational matrix method for solving Lane-Emden equations arising in astrophysics. Math. Methods App. Sci. 37, 2227–2235 (2014)
Babolian, E., Fattahzadeh, F.: Numerical solution of differential equations by using Chebyshev wavelet operational matrix of integration. Appl. Math. Comp. 188, 417–425 (2007)
Saadatmandi, A.: Mehdi Dehghan, A new operational matrix for solving fractional-order differential equations. Comp. Math. Appl. 59, 1326–1336 (2010)
Öztürk, Y.: Numerical solution of systems of differential equations using operational matrix method with Chebyshev polynomials. J. Taibah Uni. Scie. 12(2), 155–162 (2018)
Öztürk, Y., Gülsu, M.: Numerical solution of Abel equation using operational matrix method with Chebyshev polynomials. Asian-Eur. J. Math. 10(3), 175053 (2017)
Öztürk, Y., Gülsu, M.: An operational matrix method for solving a class of nonlinear Volterra integro-differential equations by operational matrix method. Inter. J. Appl. Comput. Math. 3, 3279–3294 (2017)
Rivlin, T.J.: Introduction to the Approximation of Functions. London (1969)
Body, J.P.: Chebyshev and fourier spectral methods. University of Michigan, New York (2000)
Epperson, J.F.: An Introduction to Numerical Methods and Analysis. Wiley & Sons, Inc., Hoboken, New Jersey (2013)
Gülsu, M., Öztürk, Y.: On the numerical solution of linear Fredholm-Volterra integro differential difference equations with piecewise intervals. Appl. Appl. Math. Int. J. 7(2), 556–570 (2012)
Shang, X., Han, D.: Application of the variational iteration method for solving n-th order integro differential equations. J. Comput. Appl. Math. 234, 1442–1447 (2010)
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Öztürk, Y., Gülsu, M. An operational matrix method to solve linear Fredholm–Volterra integro-differential equations with piecewise intervals. Math Sci 15, 189–197 (2021). https://doi.org/10.1007/s40096-021-00401-9
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DOI: https://doi.org/10.1007/s40096-021-00401-9