Laguerre Collocation Method for Solutions of Systems of First Order Linear Differential Equations

Abstract

In this paper, a collocation method based on Laguerre polynomials is presented to solve systems of linear differential equations. The Laguerre polynomials, their derivatives, system of differential equations and conditions are written in the matrix form. Then, by using the constructed matrix forms, collocation points and matrix operations, the system of linear differential equations is transformed into a system of linear algebraic equations. The solution of this system gives the coefficients of the solutions forms. Thus, the solutions based on the Laguerre polynomials is found. Also, error estimation is made by using residual functions. Numerical examples are given to explain the method. The results are compared with results of other methods.   

Keywords

• Laguerre collocation method,Laguerre polynomials,systems of linear differential equations,Collocation method

References

1. Abdel, I.H. and Hassan, H., Application to differential transformation method for solving systems of differential equations, Appl. Math. Modell.,32(2008), 2552–2559.
2. Akyuz-Dascıoglu, A. and Sezer, M., Chebyshev polynomial solutions of systems of higher-order linear Fredholm-Volterra integro-differentialequations, Journal of the Franklin Inst., 342(2005), 688–701.
3. Arikoglu, A. and Ozkol, I., Solutions of integral and integro-differential equation systems by using differential transform method, Comput MathAppl., 56(2008), 2411–2417.
4. Biazar, J., Babolian, E. and Islam, R., Solution of the system of ordinary differential equations by Adomian decomposition method, AppliedMathematics and Computation, 147(2004), 713–719.
5. Gulsu, M. and Sezer, M., Taylor collocation method for solution of systems of high-order linear Fredholm Volterra integro-differential equations,Intern. J. Computer Math., 83(2006), 429–448.
6. Javidi, M., Modified homotopy perturbation method for solving system of linear Fredholm integral equations, Mathematical and ComputerModelling, 50(2009), 159–165. 1Laguerre Collocation Method 20
7. Kurt, N. and Sezer, M., Polynomial solution of high-order linear Fredholm integro-differential equations with constant coefficients, Journal ofthe Frankin Institute., 345(2008), 839–850.
8. Maleknejad, K. and Mirzaee, F., Numerical solution of linear Fredholm integral equations system by rationalized Haar functions method, Int.J. Comput. Math., 80(2003), 1397–1405.

9. Maleknejad, K., Mirzaee, F. and Abbasbandy, S., Solving linear integro-differential equations system by using rationalized Haar functionsmethod, Appl. Math. Comput., 155(2004), 317–328.
10. Pour-Mahmoud, J., Rahimi-Ardabili, M.Y. and Shahmorad, S., Numerical solution of the system of Fredholm integro-differential equations bythe Tau method, Appl. Math. Comput., 168(2005), 465–478.
11. Ramadan, M.A. and Abd El Salam, M.A., Solving systems of ordinary differential equations in unbounded domains by exponential Chebyshevcollocation method, Journal of Abstract and Computational Mathematics, 1(2016), 33–46.
12. Saberi-Nadjafi, J. and Tamamgar, M., The variational iteration method: A highly promising method for solving the system of integro differentialequations, Comput. Math. Appl., 56(2008), 346–351.
13. Sezer, M., Karamete, A. and G¨ulsu, M., Taylor polynomial solutions of systems of linear differential equations with variable coefficients, Int.J. Comput. Math., 82(2005), 755–764.
14. Tatari, M. and Dehghan, M., Improvement of He’s variational iteration method for solving systems of differential equations, Comput. Math.Appl., 58(2009), 2160–2166.
15. Thongmoon, M. and Pusjuso, S., The numerical solutions of differential transform method and the Laplace transform method for a system ofdifferential equations, Nonlinear Analysis: Hybrid Systems, 4(2009), 425–431.
16. Yalcınbas¸, S., Sezer, M. and Sorkun, H.H., Legendre polynomial solutions of high-order linear Fredholm integro-differential equations, Appl.Math. Comput., 210(2009), 334–349.
17. Yusufoglu, E., An efficient algorithm for solving integro-differential equations system, Appl. Math. Comput., 192(2007), 51–55.
18. Yuksel, G., Gulsu, M. and Sezer, M., A Chebyshev polynomial approach for high-order linear Fredholm-Volterra integro-differential equations,Gazi University Journal of Science, 25(2012), 393–401.
19. Yuzbasi, S., Sahin, N. ve Sezer, M., A numerical approach for solving linear differential equation systems, Journal of Advanced Research inDifferential Equations, 3(2011), 8–32.
20. Zarali, B. and Rabbani, M., Solution of Fredholm integro-differential equations system by modified decomposition method, The Journal ofMathematics and Computer Science, 4(2012), 258–264.