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Fourier operational matrices of differentiation and transmission: introduction and applications. (English) Zbl 1275.65036

Summary: This paper introduces Fourier operational matrices of differentiation and transmission for solving high-order linear differential and difference equations with constant coefficients. Moreover, we extend our methods for generalized pantograph equations with variable coefficients by using Legendre Gauss collocation nodes. In the case of numerical solutions of pantograph equation, an error problem is constructed by means of the residual function and this error problem is solved by using the mentioned collocation scheme. When the exact solution of the problem is not known, the absolute errors can be computed approximately by the numerical solution of the error problem. The reliability and efficiency of the presented approaches are demonstrated by several numerical examples, and also the results are compared with different methods.

MSC:

65L03 Numerical methods for functional-differential equations
34K28 Numerical approximation of solutions of functional-differential equations (MSC2010)
65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations
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[1] Rao, G. P.; Palaisamy, K. R., Walsh stretch matrices and functional differential equations, IEEE Transactions on Automatic Control, 27, 1, 272-276 (1982) · Zbl 0488.34060
[2] Hwang, C.; Shih, Y. P., Laguerre series solution of a functional differential equation, International Journal of Systems Science, 13, 7, 783-788 (1982) · Zbl 0483.93056
[3] Sezer, M.; Yalçinbaş, S.; Gülsu, M., A Taylor polynomial approach for solving generalized pantograph equations with nonhomogenous term, International Journal of Computer Mathematics, 85, 7, 1055-1063 (2008) · Zbl 1145.65048 · doi:10.1080/00207160701466784
[4] Yalçinbaç, S.; Aynigül, M.; Sezer, M., A collocation method using Hermite polynomials for approximate solution of pantograph equations, Journal of the Franklin Institute, 348, 6, 1128-1139 (2011) · Zbl 1221.65187 · doi:10.1016/j.jfranklin.2011.05.003
[5] Corrington, M. S., Solution of differential and integral equations with walsh functions, IEEE Transactions on Circuit Theory, 20, 5, 470-476 (1973)
[6] Chen, C. F.; Hsiao, C. H., Time-domain synthesis via walsh functions, Proceedings of the Institution of Electrical Engineers, 122, 5, 565-570 (1975)
[7] Chen, C. F.; Hsiao, C. H., Walsh series analysis in optimal control, International Journal of Control, 21, 6, 881-897 (1975) · Zbl 0308.49035
[8] Hsu, N. S.; Cheng, B., Analysis and optimal control of time-varying linear systems via block-pulse functions, International Journal of Control, 33, 6, 1107-1122 (1981) · Zbl 0464.93027
[9] Hwang, C.; Shih, Y. P., Parameter identification via laguerre polynomials, International Journal of Systems Science, 13, 2, 209-217 (1982) · Zbl 0475.93033
[10] Horng, I. R.; Chou, J. H., Shifted chebyshev direct method for solving variational problems, International Journal of Systems Science, 16, 7, 855-861 (1985) · Zbl 0568.49019
[11] Chang, R. Y.; Wang, M. L., Shifted Legendre direct method for variational problems, Journal of Optimization Theory and Applications, 39, 2, 299-307 (1983) · Zbl 0481.49004 · doi:10.1007/BF00934535
[12] Kekkeris, G. T.; Paraskevopoulos, P. N., Hermite series approach to optimal control, International Journal of Control, 47, 2, 557-567 (1988) · Zbl 0636.93036
[13] Razzaghi, M.; Razzaghi, M., Fourier series direct method for variational problems, International Journal of Control, 48, 3, 887-895 (1988) · Zbl 0651.49012
[14] Doha, E. H.; Bhrawy, A. H.; Saker, M. A., Integrals of Bernstein polynomials: an application for the solution of high even-order differential equations, Applied Mathematics Letters, 24, 4, 559-565 (2011) · Zbl 1236.65091 · doi:10.1016/j.aml.2010.11.013
[15] Paraskevopoulos, P. N.; Sklavounos, P. G.; Georgiou, G. C., The operational matrix of integration for Bessel functions, Journal of the Franklin Institute, 327, 2, 329-341 (1990) · Zbl 0717.93012
[16] 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, Applied Mathematics and Computation, 219, 2, 482-497 (2012) · Zbl 1302.65274 · doi:10.1016/j.amc.2012.06.020
[17] Akyüz-Dascioglu, A., Chebyshev polynomial approximation for high-order partial differential equations with complicated conditions, Numerical Methods for Partial Differential Equations, 25, 3, 610-621 (2009) · Zbl 1167.65067 · doi:10.1002/num.20362
[18] Gülsu, M.; Gürbüz, B.; Öztürk, Y.; Sezer, M., Laguerre polynomial approach for solving linear delay difference equations, Applied Mathematics and Computation, 217, 15, 6765-6776 (2011) · Zbl 1211.65166 · doi:10.1016/j.amc.2011.01.112
[19] Sezer, M.; Yalçinbaş, S.; Şahin, N., Approximate solution of multi-pantograph equation with variable coefficients, Journal of Computational and Applied Mathematics, 214, 2, 406-416 (2008) · Zbl 1135.65345 · doi:10.1016/j.cam.2007.03.024
[20] Tohidi, E., Legendre approximation for solving linear HPDEs and comparison with taylor and bernoulli matrix methods, Applied Mathematics, 3, 410-416 (2012)
[21] Tohidi, E.; Bhrawy, A. H.; Erfani, Kh., A collocation method based on Bernoulli operational matrix for numerical solution of generalized pantograph equation, Applied Mathematical Modelling, 37, 6, 4283-4294 (2013) · Zbl 1273.34082 · doi:10.1016/j.apm.2012.09.032
[22] Tohidi, E., Bernoulli matrix approach for solving two dimensional linear hyperbolic partial differential equations with constant coefficients, American Journal of Computational and Applied Mathematics, 2, 4, 136-139 (2012) · doi:10.5923/j.ajcam.20120204.01
[23] Toutounian, F.; Tohidi, E.; Shateyi, S., A collocation method based on Bernoulli operational matrix for solving high order linear complex differential equations in a rectangular domain, Abstract, Applied, Analysis (2013) · Zbl 1275.65041 · doi:10.1155/2013/823098
[24] Yousefi, S. A.; Behroozifar, M., Operational matrices of Bernstein polynomials and their applications, International Journal of Systems Science, 41, 6, 709-716 (2010) · Zbl 1195.65061 · doi:10.1080/00207720903154783
[25] Yuzbasi, S., Bessel polynomial solutions of linear differential, integral and integro-differential equations [M.S. thesis] (2009), Graduate School of Natural and Applied Sciences, Mugla University
[26] Yuzbasi, S., A numerical approximation based on the Bessel functions of first kind for solutions of Riccati type differen- tialdifference equations, Computers & Mathematics with Applications, 64, 6, 1691-1705 (2012) · Zbl 1268.65090 · doi:10.1016/j.camwa.2012.01.026
[27] Yuzbasi, S.; Sezer, M., An improved Bessel collocation method with a residual error function to solve a class of LaneEmden differential equations, Computer Modeling, 57, 1298-1311 (2013)
[28] Yuzbasi, S.; Sezer, M.; Kemanci, B., Numerical solutions of integro-differential equations and application of a population model with an improved Legendre method, Applied Mathematical Modelling, 37, 2086-2101 (2013) · Zbl 1349.65728
[29] Saad, Y., Iterative Methods for Sparse Linear Systems (2003), Philadelphia, Pa, USA: SIAM, Philadelphia, Pa, USA · Zbl 1002.65042
[30] Samadi, O. R. N.; Tohidi, E., The spectral method for solving systems of Volterra integral equations, Journal of Applied Mathematics and Computing, 40, 1-2, 477-497 (2012) · Zbl 1295.65128 · doi:10.1007/s12190-012-0582-8
[31] Tohidi, E.; Samadi, O. R. N., Optimal control of nonlinear Volterra integral equations via Legendre polynomials, IMA Journal of Mathematical Control and Information, 30, 1, 67-83 (2013) · Zbl 1275.49056 · doi:10.1093/imamci/dns014
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