Conference Paper
BibTex RIS Cite

A Truncated Bell Series Approach to Solve Systems of Generalized Delay Differential Equations with Variable Coefficients

Year 2019, Volume: 11, 105 - 113, 30.12.2019

Abstract

In this study, a matrix method based on collocation points and Bell polynomials are improved to obtain the approximate solutions of systems of high-order generalized delay differential equations with variable coefficients. The presented technique reduces the solution of the mentioned delay system under the initial conditions to the solution of a matrix equation with the unknown Bell coefficients. Thereby, the approximate solution is obtained in terms of  Bell polynomials. In addition, some examples along with residual error analysis are performed to illustrate the efficiency of the method; the obtained results are scrutinized and interpreted.

References

  • Abdel-Halim, I., \textit{Hassan application to differential transformation method for solving systems of differential equations}, Appl Math Model, \textbf{32(12)}(2008), 2552--2559.
  • Ba\c{s}ar, U., Sezer, M., Numerical Solution Based on Stirling Polynomials for Solving Generalized Linear Integro-Differential Equations with Mixed Functional Arguments, Proceeding of 2. International University Industry Cooperation, R\&D and Innovation Congress, (2018), 141--148.
  • Bell, E.T, \textit{Exponential polynomials}, Ann. Math., \textbf{35(2)}(1934), 258--277.
  • \c{C}am, \c{S}., \textit{Stirling Say\i lar\i }, Matematik D\"{u}nyas\i , (2005), 30--34.
  • \c{C}elik, I., \textit{Collocation method and residual correction using Chebyshev series}, Appl. Math. Comput., \textbf{174(2)}(2006), 910--920.
  • \c{C}etin, M., G\"{u}rb\"{u}z, B., Sezer, M., \textit{Lucas collocation method for system of high order linear functional differential equtions}, Journal of Science and Arts, \textbf{4(45)}(2018), 891--910.
  • G\"{o}kmen, E., I\c{s}\i k, O.R., Sezer, M., \textit{Taylor collocation approach for delayed Lotka-Volterra predator-prey system}, Applied Mathematics and Computation, \textbf{268}(2015), 671--684.
  • G\"{o}kmen, E., Sezer, M., \textit{Taylor collocation method for systems of high-order linear differential-difference equations with variable coefficients}, Ain Shams Engineering Journal, \textbf{4}(2013), 117--125.
  • G\"{u}mg\"{u}m, S., Bayku\c{s} Sava\c{s}aneril, N., K\"{u}rk\c{s}\"{u}, O.K., Sezer, M., \textit{A numerical technique based on Lucas polynomials together with standard and Chebyshev-Lobatto collocation points for solving functional integro-differential equations involving variable delays}, Sakarya University Journal of Science, \textbf{22(6)}(2018), 1659--1668.
  • Maleknejad, K., Mirzae, F., Abbasbandy, S., \textit{Solving linear integro-differential equations system by using rationalized Haar functions method}, Appl. Math. Comput., \textbf{155}(2004), 317--328.
  • Mollao\u{g}lu, T., Sezer, M., \textit{A numerical approach with residual error estimation for solution of high-order linear differential-difference equations by using Gegenbauer polynomials}, CBU J.of Sci., \textbf{13(1)}(2017), 39--49.
  • Oguz, C., Sezer, M., Oguz, A.D., \textit{Chelyshkov collocation approach to solve the systems of linear functional differential equations}, NTMSCI, \textbf{3(4)}(2015), 83--97.
  • Oliveira, F.A., \textit{Collacation and residual correction}, Numer. Math., \textbf{36}(1980), 27--31.
  • Saeed, R.K., Rahman B.M., \textit{Adomian decomposition method for solving system of delay differential equation}, Australian Journal of Basic and Applied Sciences, \textbf{4(8)}(2010), 3613--3621.
  • Sun, Y., Galip Ulsoy, A., Nelson, P.W., \textit{Solution of systems of linear delay differential equations via Laplace transformation}, Proceedings of the 45th IEEE Conference on Decision and Control, (2006), 13--15.
  • Van Gorder, R.A., \textit{Recursive relations for Bell polynomials of arbitrary positive non-integer order}, International Mathematical Forum, \textbf{5(37)}(2010), 1819--1821.
  • Yal\c{c}\i nba\c{s}, S., Akkaya, T., \textit{A Numerical approach for solving linear integro-differential-difference equations with Boubaker polynomial bases}, Ain Shams Engineering Journal, \textbf{3}(2012), 153--161.
  • Zurigat, M., Momani, S., Odibat, Z., Alawneh, A., \textit{The homotopy analysis method for handling systems of fractional differential equations}, Appl Math Modell, \textbf{34}(2010), 24--35.
Year 2019, Volume: 11, 105 - 113, 30.12.2019

Abstract

References

  • Abdel-Halim, I., \textit{Hassan application to differential transformation method for solving systems of differential equations}, Appl Math Model, \textbf{32(12)}(2008), 2552--2559.
  • Ba\c{s}ar, U., Sezer, M., Numerical Solution Based on Stirling Polynomials for Solving Generalized Linear Integro-Differential Equations with Mixed Functional Arguments, Proceeding of 2. International University Industry Cooperation, R\&D and Innovation Congress, (2018), 141--148.
  • Bell, E.T, \textit{Exponential polynomials}, Ann. Math., \textbf{35(2)}(1934), 258--277.
  • \c{C}am, \c{S}., \textit{Stirling Say\i lar\i }, Matematik D\"{u}nyas\i , (2005), 30--34.
  • \c{C}elik, I., \textit{Collocation method and residual correction using Chebyshev series}, Appl. Math. Comput., \textbf{174(2)}(2006), 910--920.
  • \c{C}etin, M., G\"{u}rb\"{u}z, B., Sezer, M., \textit{Lucas collocation method for system of high order linear functional differential equtions}, Journal of Science and Arts, \textbf{4(45)}(2018), 891--910.
  • G\"{o}kmen, E., I\c{s}\i k, O.R., Sezer, M., \textit{Taylor collocation approach for delayed Lotka-Volterra predator-prey system}, Applied Mathematics and Computation, \textbf{268}(2015), 671--684.
  • G\"{o}kmen, E., Sezer, M., \textit{Taylor collocation method for systems of high-order linear differential-difference equations with variable coefficients}, Ain Shams Engineering Journal, \textbf{4}(2013), 117--125.
  • G\"{u}mg\"{u}m, S., Bayku\c{s} Sava\c{s}aneril, N., K\"{u}rk\c{s}\"{u}, O.K., Sezer, M., \textit{A numerical technique based on Lucas polynomials together with standard and Chebyshev-Lobatto collocation points for solving functional integro-differential equations involving variable delays}, Sakarya University Journal of Science, \textbf{22(6)}(2018), 1659--1668.
  • Maleknejad, K., Mirzae, F., Abbasbandy, S., \textit{Solving linear integro-differential equations system by using rationalized Haar functions method}, Appl. Math. Comput., \textbf{155}(2004), 317--328.
  • Mollao\u{g}lu, T., Sezer, M., \textit{A numerical approach with residual error estimation for solution of high-order linear differential-difference equations by using Gegenbauer polynomials}, CBU J.of Sci., \textbf{13(1)}(2017), 39--49.
  • Oguz, C., Sezer, M., Oguz, A.D., \textit{Chelyshkov collocation approach to solve the systems of linear functional differential equations}, NTMSCI, \textbf{3(4)}(2015), 83--97.
  • Oliveira, F.A., \textit{Collacation and residual correction}, Numer. Math., \textbf{36}(1980), 27--31.
  • Saeed, R.K., Rahman B.M., \textit{Adomian decomposition method for solving system of delay differential equation}, Australian Journal of Basic and Applied Sciences, \textbf{4(8)}(2010), 3613--3621.
  • Sun, Y., Galip Ulsoy, A., Nelson, P.W., \textit{Solution of systems of linear delay differential equations via Laplace transformation}, Proceedings of the 45th IEEE Conference on Decision and Control, (2006), 13--15.
  • Van Gorder, R.A., \textit{Recursive relations for Bell polynomials of arbitrary positive non-integer order}, International Mathematical Forum, \textbf{5(37)}(2010), 1819--1821.
  • Yal\c{c}\i nba\c{s}, S., Akkaya, T., \textit{A Numerical approach for solving linear integro-differential-difference equations with Boubaker polynomial bases}, Ain Shams Engineering Journal, \textbf{3}(2012), 153--161.
  • Zurigat, M., Momani, S., Odibat, Z., Alawneh, A., \textit{The homotopy analysis method for handling systems of fractional differential equations}, Appl Math Modell, \textbf{34}(2010), 24--35.
There are 18 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Gökçe Yıldız 0000-0001-9896-6580

Mehmet Sezer This is me 0000-0002-7744-2574

Publication Date December 30, 2019
Published in Issue Year 2019 Volume: 11

Cite

APA Yıldız, G., & Sezer, M. (2019). A Truncated Bell Series Approach to Solve Systems of Generalized Delay Differential Equations with Variable Coefficients. Turkish Journal of Mathematics and Computer Science, 11, 105-113.
AMA Yıldız G, Sezer M. A Truncated Bell Series Approach to Solve Systems of Generalized Delay Differential Equations with Variable Coefficients. TJMCS. December 2019;11:105-113.
Chicago Yıldız, Gökçe, and Mehmet Sezer. “A Truncated Bell Series Approach to Solve Systems of Generalized Delay Differential Equations With Variable Coefficients”. Turkish Journal of Mathematics and Computer Science 11, December (December 2019): 105-13.
EndNote Yıldız G, Sezer M (December 1, 2019) A Truncated Bell Series Approach to Solve Systems of Generalized Delay Differential Equations with Variable Coefficients. Turkish Journal of Mathematics and Computer Science 11 105–113.
IEEE G. Yıldız and M. Sezer, “A Truncated Bell Series Approach to Solve Systems of Generalized Delay Differential Equations with Variable Coefficients”, TJMCS, vol. 11, pp. 105–113, 2019.
ISNAD Yıldız, Gökçe - Sezer, Mehmet. “A Truncated Bell Series Approach to Solve Systems of Generalized Delay Differential Equations With Variable Coefficients”. Turkish Journal of Mathematics and Computer Science 11 (December 2019), 105-113.
JAMA Yıldız G, Sezer M. A Truncated Bell Series Approach to Solve Systems of Generalized Delay Differential Equations with Variable Coefficients. TJMCS. 2019;11:105–113.
MLA Yıldız, Gökçe and Mehmet Sezer. “A Truncated Bell Series Approach to Solve Systems of Generalized Delay Differential Equations With Variable Coefficients”. Turkish Journal of Mathematics and Computer Science, vol. 11, 2019, pp. 105-13.
Vancouver Yıldız G, Sezer M. A Truncated Bell Series Approach to Solve Systems of Generalized Delay Differential Equations with Variable Coefficients. TJMCS. 2019;11:105-13.