×

Chebyshev polynomial solutions of systems of higher-order linear Fredholm-Volterra integro-differential equations. (English) Zbl 1086.65121

The authors consider systems of \(k\) linear integro-differential equations of Fredholm-Volterra type in the form \[ \!\sum_{n=0}^m\sum_{j=1}^kp_{ij}^n(x)y_j^{(n)}(x)\!=\!g_i(x)\!+\! \int_{-1}^1\sum_{j=1}^kF_{ij}(x,t)y_j(t)\,dt\!+\! \int_{-1}^x\sum_{j=1}^kK_{ij}(x,t)y_j(t)\,dt,\tag{1} \]
\[ i=1,2,\dots,k,\quad -1\leqslant x\leqslant1, \]
under the mixed conditions
\[ \sum_{n=0}^{m-1}\mathbf a_j^ny_j^{(n)}(-1) + \mathbf b_j^ny_j^{(n)}(1) +\mathbf c_j^ny_j^{(n)}(c) =\mathbf\lambda_j,\quad j=1,2,\dots,k,\quad -1<c< 1, \tag{2} \]
where \(\mathbf\lambda_j\), \(\mathbf a_j^n\), \(\mathbf b_j^n\) and \(\mathbf c_j^n\) are real-valued column matrices with \(m\times 1\) dimension and \(y_j^{(n)}\) indicates the \(n\)th-order derivative and \(y_j^{(0)}(x) = y_j(x)\). The aim of this study is to get a solution as truncated Chebyshev series defined by
\[ y_j(x)=\sum_{r=0}^Na_{jr}T_r(x),\quad j = 1,2,\dots,k,\quad -1\leqslant x \leqslant 1, \]
where \(T_r(x)\) denotes the Chebyshev polynomials of the first kind, \(a_{jr}\) are unknown Chebyshev coefficients, and \(N\) is chosen any positive integer such that \(N\geqslant m\).
The authors transform the system (1) and the given conditions (2) into matrix equations via Chebyshev collocation points. By merging these results, a new system which corresponds to a system of linear algebraic equations is obtained. The solution of this system yields the Chebyshev coefficients of the solution function. An interesting feature of this method is that when system of integro-differential equations (1) has linearly independent polynomial solution of degree \(N\) or less than \(N\), the method can be used for finding the analytical solution. Besides, when the truncation limit \(N\) is increased, there exists a solution, which is closer to the exact solution. Some numerical results are also given to illustrate the efficiency of the method.

MSC:

65R20 Numerical methods for integral equations
45J05 Integro-ordinary differential equations
45F05 Systems of nonsingular linear integral equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Baker, C. T.H., A perspective on the numerical treatment of Volterra equations, J. Comput. Appl. Math., 125, 217-249 (2000) · Zbl 0976.65121
[2] Linz, P., Analytical and Numerical Methods for Volterra Equations (1985), SIAM: SIAM Philadelphia, PA · Zbl 0566.65094
[3] Delves, L. M.; Mohamed, J. L., Computational Methods for Integral Equations (1985), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0592.65093
[4] Bloom, F., Asymptotic bounds for solutions to a system of damped integrodifferential equations of electromagnetic theory, J. Math. Anal. Appl., 73, 524-542 (1980) · Zbl 0434.45018
[5] Holmaker, K., Global asymptotic stability for a stationary solution of a system of integro-differential equations describing the formation of liver zones, SIAM J. Math. Anal., 24, 1, 116-128 (1993) · Zbl 0767.45005
[6] Abdou, M. A., Fredholm-Volterra integral equation of the first kind and contact problem, Appl. Math. Comput., 125, 177-193 (2002) · Zbl 1028.45003
[7] Forbes, L. K.; Crozier, S.; Doddrell, D. M., Calculating current densities and fields produced by shielded magnetic resonance imaging probes, SIAM J. Appl. Math., 57, 2, 401-425 (1997) · Zbl 0871.65116
[8] van der Houwen, P. J.; Sommeijer, B. P., Euler-Chebyshev methods for integro-differential equations, Appl. Numer. Math., 24, 203-218 (1997) · Zbl 0881.65141
[9] Enright, W. H.; Hu, M., Continuous Runge-Kutta methods for neutral Volterra integro-differential equations with delay, Appl. Numer. Math., 24, 175-190 (1997) · Zbl 0876.65089
[10] Maleknejad, K.; Mirzae, F.; Abbasbandy, S., Solving linear integro-differential equations system by using rationalized Haar functions method, Appl. Math. Comput., 155, 317-328 (2004) · Zbl 1056.65144
[11] Maleknejad, K.; Tavassoli Kajani, M., Solving linear integro-differential equation system by Galerkin methods with hybrid functions, Appl. Math. Comput., 159, 603-612 (2004) · Zbl 1063.65145
[12] Akyüz, A.; Sezer, M., A Chebyshev collocation method for the solution of linear integro-differential equations, Int. J. Comput. Math., 72, 4, 491-507 (1999) · Zbl 0947.65142
[13] Akyüz, A.; Sezer, M., Chebyshev polynomial solutions of systems of high-order linear differential equations with variable coefficients, Appl. Math. Comput., 144, 237-247 (2003) · Zbl 1024.65059
[14] Akyüz-Daşcıoğlu, A., Chebyshev polynomial solutions of systems of linear integral equations, Appl. Math. Comput., 151, 221-232 (2004) · Zbl 1049.65149
[15] Sezer, M.; Kaynak, M., Chebyshev polynomial solutions of linear differential equations, Int. J. Math. Educ. Sci. Technol., 27, 4, 607-618 (1996) · Zbl 0887.34012
[16] Clenshaw, C. W.; Curtis, A. R., A method for numerical integration on an automatic computer, Numer. Math., 2, 197-205 (1960) · Zbl 0093.14006
[17] Fox, L.; Parker, I. B., Chebyshev Polynomials in Numerical Analysis (1968), Oxford University Press: Oxford University Press London · Zbl 0153.17502
[18] A. Akyüz, Chebyshev Collocation method for solution of linear integro-differential equations, M.Sc. Thesis, Dokuz Eylül University, Graduate School of Natural and Applied Sciences, 1997.; A. Akyüz, Chebyshev Collocation method for solution of linear integro-differential equations, M.Sc. Thesis, Dokuz Eylül University, Graduate School of Natural and Applied Sciences, 1997.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.