Polynomial collocation method for initial value problem of mixed integro-differential equations

Document Type : Original Article

Authors

1 Department of Mathematics, Faculty of Science, University of Lagos, Lagos, Nigeria

2 Department of Mathematics, Caleb University, Imota, Lagos, Nigeria

3 Department of Mathematics, Covenant University, Canaan Land, Ota, Ogun State, Nigeria

4 Department of Mathematics, Nigerian Army University, Biu, Borno State, Nigeria

Abstract

This paper presents the development and implementation of a numerical method for
the solution of one dimensional Mixed Fredholm Volterra Intergro-Differential Equations
(MFVIDEs). The new technique transformed MFVIDEs into an integral equation which
is then approximated using a polynomial collocation method. Standard collocation points
are then used to convert the problem into a system of algebraic equations. Some numerical
examples are used to test the efficiency and accuracy of the method. The results show
that the new method is efficient, accurate and easy to implement.

Keywords

Main Subjects

References

[1] O.A. Agbolade, T.A. Anake, Solution of First-Order Volterra Type Linear Integro Differential Equations by Collocation Method, J. Appl. Math., 2017.
[2] S. Almezel, Q.H. Ansari, M.A. Khamsi, Topics in fixed point theory, Springer International Publishing, Switzerland, 2014.
[3] Z.P. Atabakan, A.K Nasab, A. Kilicman, Z.K. Eshkuvatov, Numerical solution of nonlinear Fredhlom integro differential equations using Homotopy Analysis method, Math. Probl. Eng, 2013.
[4] V. Berinde, Iterative approximation of fixed points, Editura Efemeride, Baia Mare, Romania
[5] A. Borhanifar, Kh. Sadri, A new operational approach for numerical solution of generalized functional integro differential equations, J. Comput. Appl. Math., 279 2015, 80-96.
[6] P. Darania, K. Ivan, Numerical solution of nonlinear Volterra Fredholm integro differential equations, Comput. Math. Appl., 58 2008, 2197-2209.
[7] H.L. Dastjerdi, H.M.M. Ghaini, Numerical solution of Volterra fredholm integral equations by moving least square method and Chebyshev polynomial, Appl. Math. Model., 36 2012, 3281-3288.
[8] A.H. Hamoud, K.P. Ghadle, S.M.Atshan, The approximate solution of fractional integro differential equations by using modified Adomian decomposition method, Khayyam J. Maths, 5(1) 2019, 21-40.
[9] A.A. Hamoud, N.M. Mohammed, K.P. Ghadle, S.L. Dhondge, Solving Integro-Differential Equations by Using Numerical Techniques, International Journal of Applied Engineering Research, 14 2019, 3219-3225.
[10] A. Khani, M.M. Moghadam, S. Shahmorad, Approximate solution of system of nonlinear Volterra integro differential equations, Comput. Method Appl. Math., 8(1) 2008, 77-85.
[11] W.A. Kirk, B. Sims, Hand book of matrix fixed point theory, Kluwert Academic Publisher, 1st edition, 2001.
[12] D.A. Maturi, E.A.M. Simbawa, Tme modified decomposition method for solving Volterra Fredholm integro-differential equations using Maple, International Journal of GEOMATE, 18(67) 2020, 84-89.
[13] H.D.B. Mohammed, S.N. Jafar, G.A. Asghar, A novel method for solving nonlinear Volterra integro differential equations system, Abstr. Appl. Anal., 2018.
[14] R. Mohammed, R. Kamal, H. Adel, N. Mohmoud, Numerical solution of high order linear integro differential equations with variable coefficient using two proposed schemes for rational Chebyshev functions, NTMSCI, 4(3) 2016, 22-35.
[15] N.A. Mohamed, Z.A. Majid, Multistep Block Method for Solving Volterra Integro Differential Equations, Malays. J. Math. Sci., 10(8) 2016, 33-48.
[16] G. Mustapha, O. Yalcin, S. Mehmet, A new collocation method for solution of mixed linear integro differential difference equation, Appl. Math. Comput., 216 2010, 2183-2198.
[17] K. Parand, M. Nikarya, New numerical method based on generalized Bessel function to solve nonlinear Abel fractional differential equations of the first kind, Nonlinear Engineering, 8 2017, 438-448.
[18] J. Rahidinia, A. Tahmasebi, Taylor series method for the system of linear Volterra integro differential equations, TJMCS, 4(3) 2012, 331-343.
[19] M. Rahman, Integral Equations and their Applications, Southampton, Boston: WIT Press. 2007.
[20] S. Tate, V.V. Kharat, H.T. Dinde, On nonlinear mixed fractional integro differential equations with positive constant coefficients, Filomat, 33(19) 2019.
[21] W.F. Trench, Introduction To Real Analysis, Pearson Education, USA.
[22] S.G. Venkatesh, S.K. Ayyaswamy, S.R. Balachandar, K. Kannan, Wavelet solution for class of nonlinear integro differential equations, Indian Journal of Science and Technology, 6(6) 2013, 4670-4677.
[23] A. M. Wazwaz, Linear and nonlinear integral equations-methods and applications, Springer-Verlag Berlin Heidelberg, 2011.
Mathematics and Computational Sciences
Volume 4, Issue 2
June 2023
Pages 1-12

Files

History

  • Receive Date: 15 November 2022
  • Revise Date: 31 March 2023
  • Accept Date: 02 April 2023

Share

How to cite

Statistics