AN IMPROVED APPROACH FOR SOLUTIONS OF SYSTEMS OF LINEAR FREDHOLM INTEGRO DIFFERENTIAL EQUATIONS

In this paper, a numerical matrix method is used to solve the systems of high-order linear Fredholm integro-differential equations with variable coefficients under mixed conditions. The technique consists of collocation points and the Morgan-Voyce polynomials. The residual error functions of numerical solutions of the method are also presented. Firstly, the approximate solutions are formed and secondly, an error problem is constituted in favor of the residual error function. The numerical solutions are computed for this error problem by using the present method. The approximate solutions of the original problem and the error problem are the corrected Morgan-Voyce polynomial solutions. When the exact solutions of the problem are not known, the absolute errors can be approximately constructed through the approximate solutions of the error problem. Numerical examples are included to demonstrate the validity and the applicability of the technique, and also the results are compared with the different methods. All numerical computations have been performed using MATLAB v7.11.0 (R2010b).

Keywords:

Morgan-Voyce Polynomials, Systems of Linear Fredholm Integro-Differential Equations, Collocation Points Residual Error,

PDF

___

Wang, H., Fu, H.M., Zhang, H.F., "A practical thermodynamic method to calculate the best glassforming composition for bulk metallic glasses", Int J Nonlinear Sci Numer Simul, 8.2., 171-8, 2007.
Sun, F.Z., Gao, M., Lei, S.H., "The fractal dimension of the fractal model of dropwise condensation and its experimental study", Int J Nonlinear Sci Numer Simul, 8. 2., 211-22, 2007.
Bo, T.L., Xie, L., Zheng, X.L., "Numerical approach to wind ripple in desert", Int J Nonlinear Sci Numer Simul, 8. 2., 223-8, 2007.
Bellomo, N., Firmani, B., Guerri, L., "Bifurcation analysis for a nonlinear system of integro-differential equations modelling tumor-immune cells competition", App. Math. Lett, 12., 39-44, 1999.
Wazwaz, A.M., "The existence of noise terms for systems of inhomogeneous differential and integral equations", Appl. Math. Comput., 16., 81-92, 2003.
Yusufoğlu, E., "A homotopy perturbation algorithm to solve a system of Fredholm-Volterra type integral equations", Math. Comput. Modelling, 47., 1099-1107, 2008.
Yusufoğlu, E., (Agadjanov), "Numerical expansion methods for solving systems of linear integral equations using interpolation and quadrature rules", Int. J. Comput. Math., 84. 1., 133-149, 2007.
Javidi, M., "Modified homotopy perturbation method for solving system of linear Fredholm integral equations", Math. Comput. Modelling, 50., 159-165, 2009.
Aziz, I., Al-Fhaid, A. S. "An improved method based on Haar wavelets for numerical solution of nonlinear integral and integro-differential equations of first and higher orders.", Journal of Computational and Applied Mathematics, 260: 449-469, 2014.
Mirzaee, F., Hoseini, S. F., "Solving systems of linear Fredholm integro-differential equations with Fibonacci polynomials.", Ain Shams Engineering Journal, 5(1): 271- 283, 2014.
Gülsu, M., Sezer, M., "Taylor collocation method for solution of systems of high-order linear Fredholm- Volterra integro-differential equations", Int. J. Comput. Math. 83., 429-448, 2006.
İlhan, Ö., "An Improved Morgan-Voyce Collocation Method for Numerical Solution of Generalized Pantograph Equations", Journal of Scientific and Engineering Research, 4., 10., 320-332, 2017.
İlhan, Ö., Şahin, N., "On Morgan-Voyce Polynomials Approximation For Linear Differential Equations", Turkish Journal of Mathematics and Computer Science, 1., 1., 2014.
İlhan, Ö., "An improved Morgan-voyce collocation method for numerical solution of multi-pantograph equations", New Trends in Mathematical Science, 4., 5., 248-260, 2017.
Özel, M., Kürkçü, Ö.K., Sezer M., "Morgan-Voyce Matrix Method For Generalized Functional Integro-Differential Equations Of Volterra-Type", Journal of Science and Arts, 2 (47): 295-310, 2019 .
Kürkçü, K. Ö., Aslan, E., Sezer, M., İlhan, Ö., "A Numerical Approach Technique for Solving Generalized Delay Integro-Differential Equations with Functional Bounds by Means of Dickson Polynomials", International Journal of Computational Methods, 15., 5., 2018.
Gökmen, E., Sezer, M., "Taylor collocation approach for delayed Lotka–Volterra predator–prey system.", Journal of the Frankin Institute., 345(8): 839-850, 2008.
Karamete, A., Işık, O. R., Sezer, M., "A Taylor collocation method for the solution of linear integro-differential equations.", Appl. Math. and Comput., 268: 671-684, (2015).
Kürkçü, Ö.K., Sezer M., "A directly convergent numerical method based on orthoexponential polynomials for solving integro-differential-delay equations with variable coefficients and infinite boundary on half-line", Journal of Computational and Applied Mathematics, 386: 113250, 2021.
Gülsu, M., Öztürk, Y., Anapalı, A., "Numerical solution the fractional Bagley–Torvik equation arising in fluid mechanics", Int. J. Comput. Math., 94., 1., 173-184, 2017.
Biçer, G. G., Öztürk, Y., Gülsu, M., "Numerical approach for solving linear Fredholm integro-differential equation with piecewise intervals by Bernoulli polynomials",", Int. J. Comput. Math., 95., 10., 2100-2111, 2018.
Yüzbaşı, Ş., Şahin, N., Sezer, M., "Numerical solutions of systems of Linear Fredholm integro-differential equations with Bessel polynomial bases". Computers and Mathematics with Applications. 61., 10., 3079-96, 2011.
Demir, D. D., Lukonde, A. P., Kürkçü, Ö.K., Sezer M., "Pell–Lucas series approach for a class of Fredholm-type delay integro-differential equations with variable delays.", Math Sci., doi:10.1007/s40096-020-00370-5, 2021.
Kürkçü, K. Ö., Aslan, E., Sezer, M., "A numerical approach with error estimation to solve general integro-differential–difference equations using Dickson polynomials", Applied Mathematics and Computation, 276: 324-339, 2016.
Kürkçü, K. Ö., Aslan, E., Sezer, M., "A numerical method for solving some model problems arising in science and convergence analysis based on residual function", Applied Numerical Mathematics, 121: 134-148, 2017.
Türkyılmaz, B., Gürbüz, B., Sezer, M., "Morgan-Voyce Polynomial Approach for Solution of High-Order Linear Differential-Difference Equations with Residual Error Estimation." Düzce Üniversitesi Bilim ve Teknoloji Dergisi, 4(1), 2016.
Gürbüz, B., Sezer, M., "An hybrid numerical algorithm with error estimation for a class of functional integro-differential equations." Gazi University Journal of Science, 29(2): 419-434, 2016.
Gürbüz, B., "Hybrid approximation for solutions of high-order integro-differential equations including variable delay.", In Journal of Physics: Conference Series, 1641(1): p. 012062. IOP Publishing, 2020.
Swamy, M. N. S., Bhattacharryya, B. B., "A Study of Recurrent Ladders Using the Polynomials Defined by Morgan-Voyce", İEEE Transactions on Circuit Theory, ct-14., No.3, 260-264, 1967.
Çelik, İ., "Collocation method and residual correction using Chebyshev series", Appl. Math. Comput., 174., 910-920, 2006.