A collocation method based on the Bernoulli operational matrix for solving high-order linear complex differential equations in a rectangular domain. (English) Zbl 1275.65041

Summary: This paper contributes a new matrix method for the solution of high-order linear complex differential equations with variable coefficients in rectangular domains under the considered initial conditions. On the basis of the presented approach, the matrix forms of the Bernoulli polynomials and their derivatives are constructed, and then by substituting the collocation points into the matrix forms, the fundamental matrix equation is formed. This matrix equation corresponds to a system of linear algebraic equations. By solving this system, the unknown Bernoulli coefficients are determined and thus the approximate solutions are obtained. Also, an error analysis based on the use of the Bernoulli polynomials is provided under several mild conditions. To illustrate the efficiency of our method, some numerical examples are given.


65L05 Numerical methods for initial value problems involving ordinary differential equations
34M03 Linear ordinary differential equations and systems in the complex domain
65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations
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[1] Cveticanin, L., Free vibration of a strong non-linear system described with complex functions, Journal of Sound and Vibration, 277, 4-5, 815-824, (2004) · Zbl 1236.70022
[2] Cveticanin, L., Approximate solution of a strongly nonlinear complex differential equation, Journal of Sound and Vibration, 284, 1-2, 503-512, (2005) · Zbl 1237.34058
[3] Barsegian, G. A., Gamma Lines: On the Geometry of Real and Complex Functions. Gamma Lines: On the Geometry of Real and Complex Functions, Asian Mathematics Series, 5, x+176, (2002), New York, NY, USA: Taylor & Francis, New York, NY, USA · Zbl 1059.30023
[4] Barsegian, G.; Lê, D. T., On a topological description of solutions of complex differential equations, Complex Variables, 50, 5, 307-318, (2005) · Zbl 1082.34076
[5] Ishizaki, K.; Tohge, K., On the complex oscillation of some linear differential equations, Journal of Mathematical Analysis and Applications, 206, 2, 503-517, (1997) · Zbl 0877.34009
[6] Heittokangas, J.; Korhonen, R.; Rättyä, J., Growth estimates for solutions of linear complex differential equations, Annales Academiæ Scientiarum Fennicæ, 29, 1, 233-246, (2004) · Zbl 1057.34111
[7] Andrievskii, V., Polynomial approximation of analytic functions on a finite number of continua in the complex plane, Journal of Approximation Theory, 133, 2, 238-244, (2005) · Zbl 1085.30034
[8] Prokhorov, V. A., On best rational approximation of analytic functions, Journal of Approximation Theory, 133, 2, 284-296, (2005) · Zbl 1073.30029
[9] Gülsu, M.; Sezer, M., Approximate solution to linear complex differential equation by a new approximate approach, Applied Mathematics and Computation, 185, 1, 636-645, (2007) · Zbl 1107.65328
[10] 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
[11] Öztürk, Y.; Gülsu, M., Approximate solution of linear generalized pantograph equations with variable coefficients on Chebyshev-Gauss grid, Journal of Advanced Research in Scientific Computing, 4, 1, 36-51, (2012)
[12] Sezer, M.; Akyuz-Das-cioglu, A., A Taylor method for numerical solution of generalized pantograph equations with linear functional argument, Journal of Computational and Applied Mathematics, 200, 1, 217-225, (2007) · Zbl 1112.34063
[13] Sezer, M.; Gülsu, M.; Tanay, B., A Taylor collocation method for the numerical solution of complex differential equations with mixed conditions in elliptic domains, Applied Mathematics and Computation, 182, 1, 498-508, (2006) · Zbl 1106.65061
[14] Sezer, M.; Yuzbasi, S., A collocation method to solve higher order linear complex differential equations in rectangular domains, Numerical Methods for Partial Differential Equations, 26, 3, 596-611, (2010) · Zbl 1189.65152
[15] Tohidi, E., Legendre approximation for solving linear HPDEs and comparison with Taylor and Bernoulli matrix methods, Applied Mathematics, 3, 5, 410-416, (2012)
[16] Yuzbasi, S.; Aynigul, 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
[17] Yuzbasi, S., A numerical approximation based on the Bessel functions of first kind for solutions of Riccati type differential difference equations, Computers & Mathematics with Applications, 64, 6, 1691-1705, (2012) · Zbl 1268.65090
[18] Yuzbasi, S.; Sahin, N.; Sezer, M., A collocation approach for solving linear complex differential equations in rectangular domains, Mathematical Methods in the Applied Sciences, 35, 10, 1126-1139, (2012) · Zbl 1250.65108
[19] Yuzbasi, S.; Sahin, N.; Gulso, M., A collocation approach for solving a class of complex differential equations in elliptic domains, Journal of Numerical Mathematics, 19, 3, 225-246, (2011) · Zbl 1229.65141
[20] 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
[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
[22] Tohidi, E., Bernoulli matrix approach for solving two dimensional linear hyperbolic partial differential equations with constant coefficients, Journal of Computational and Applied Mathematics, 2, 4, 136-139, (2012)
[23] 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
[24] 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
[25] Mashayekhi, S.; Ordokhani, Y.; Razzaghi, M., Hybrid functions approach for nonlinear constrained optimal control problems, Communications in Nonlinear Science and Numerical Simulation, 17, 4, 1831-1843, (2012) · Zbl 1239.49043
[26] Sezer, M.; Gülsu, M., Approximate solution of complex differential equations for a rectangular domain with Taylor collocation method, Applied Mathematics and Computation, 177, 2, 844-851, (2006) · Zbl 1096.65075
[27] Krylov, V. I., Approximate Calculation of Integrals, x+357, (1962), Mineola, NY, USA: Dover, Mineola, NY, USA · Zbl 1152.65005
[28] Costabile, F. A.; Dell’Accio, F., Expansion over a rectangle of real functions in Bernoulli polynomials and applications, BIT Numerical Mathematics, 41, 3, 451-464, (2001) · Zbl 0989.65014
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