A new numerical method to solve nonlinear Volterra-Fredholm integro-differential equations
Abstract
In this paper, a new method combining the simplified reproducing kernel method (SRKM) and the homotopy perturbation method (HPM) to solve the nonlinear Volterra-Fredholm integro-differential equations (V-FIDE) is proposed. Firstly the HPM can convert nonlinear problems into linear problems. After that we use the SRKM to solve the linear problems. Secondly, we prove the uniform convergence of the approximate solution. Finally, some numerical calculations are proposed to verify the effectiveness of the approach.
Keyword : nonlinear Volterra-Fredholm integro-differential equations, simplified reproducing kernel method, homotopy perturbation method
How to Cite
Hou, J., Niu, J., Xu, M., & Ngolo, W. (2021). A new numerical method to solve nonlinear Volterra-Fredholm integro-differential equations. Mathematical Modelling and Analysis, 26(3), 469-478. https://doi.org/10.3846/mma.2021.12923
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
A. Abdi and S.A. Hosseini. The barycentric rational differencequadrature scheme for systems of Volterra integro-differential equations. SIAM Journal on Scientific Computing, 40(3):A1936–A1960, 2018. https://doi.org/10.1137/17M114371X
E. Babolian, Z. Masouri and S. Hatamzadeh-Varmazyar. New direct method to solve nonlinear Volterra-Fredholm integral and integro differential equation using operational matrix with block-pulse functions. Progress in Electromagnetics Research B, 8:59–76, 2008. https://doi.org/10.2528/pierb08050505
E. Babolian, Z. Masouri and S. Hatamzadeh-Varmazyar. Numerical solution of nonlinear Volterra-Fredholm integro-differential equations via direct method using triangular functions. Computers Mathematics With Applications, 58(2):239– 247, 2009. https://doi.org/10.1016/j.camwa.2009.03.087
H.O. Bakodah, M. Almazmumy, S.O. Almuhalbedi and L. Abdullah. Laplace discrete Adomian decomposition method for solving nonlinear integro differential equations. Journal of Applied Mathematics and Physics, 7(6):1388–1407, 2019. https://doi.org/10.4236/jamp.2019.76093
J. Biazar, B. Ghanbari, M.G. Porshokouhi and M.G. Porshokouhi. Hes homotopy perturbation method: A strongly promising method for solving non-linear systems of the mixed VolterraFredholm integral equations. Computers Mathematics with Applications, 61(4):1016–1023, 2011. https://doi.org/10.1016/j.camwa.2010.12.051
N. Bildik. Modified decomposition method for nonlinear VolterraFredholm integral equations. Chaos Solitons Fractals, 33:308–313, 2007. https://doi.org/10.1016/j.chaos.2005.12.058
M.G. Cui and Y.Z. Lin. Nonlinear Numerical Analysis in the Reproducing Kernel Space. Nova Science Publishers, New York, London, 2009.
M. Ghasemi, M.T. Kajani and E. Babolian. Application of Hes homotopy perturbation method to nonlinear integro-differential equations. Applied Mathematics and Computation, 188(1):538–548, 2007. https://doi.org/10.1016/j.amc.2006.10.016
K. Maleknejad, B. Basirat and E. Hashemizadeh. Hybrid legendre polynomials and block-pulse functions approach for nonlinear V0olterra-Fredholm integrodifferential equations. Computers Mathematics with Applications, 61(9):2821– 2828, 2011. https://doi.org/10.1016/j.camwa.2011.03.055
K. Maleknejad and E. Saeedipoor. Convergence analysis of hybrid functions method for two-dimensional nonlinear VolterraFredholm integral equations. Journal of Computational and Applied Mathematics, 368:1–10, 2020. https://doi.org/10.1016/j.cam.2019.112533
J. Niu, Y.Z. Lin and C.P. Zhang. Approximate solution of nonlinear multi-point boundary value problem on the half-line. Mathematical Modelling and Analysis, 17(2):190–202, 2012. https://doi.org/10.3846/13926292.2012.660889
J. Niu, L.X. Sun, M.Q. Xu and J.J. Hou. A reproducing kernel method for solving heat conduction equations with delay. Applied Mathematics Letters, 100:106036, 2019. https://doi.org/10.1016/j.aml.2019.106036
J. Niu, M.Q. Xu and G.M. Yao. An efficient reproducing kernel method for solving the AllenCahn equation. Applied mathematics letters, 89:78–84, 2019. https://doi.org/10.1016/j.aml.2018.09.013
K. Ozen and K. Orucoglu. Approximate solution to a multi-point boundary value problem involving nonlocal integral conditions by reproducing kernel method. Mathematical Modelling and Analysis, 18(4):529–536, 2013. https://doi.org/10.3846/13926292.2013.840867
M.Q. Xu, L.F. Zhang and E. Tohidi. A fourth-order least-squares based reproducing kernel method for one-dimensional elliptic interface problems. Applied Numerical Mathematics, 162:124–136, 2021. https://doi.org/10.1016/j.apnum.2020.12.015
S.A. Yousefi and M. Razzaghi. Legendre wavelets method for the nonlinear Volterra-Fredholm integral equations. Mathematics and Computers in Simulation, 70(1):1–8, 2005. https://doi.org/10.1016/j.matcom.2005.02.035
Z.H. Zhao, Y.Z. Lin and J. Niu. Convergence order of the reproducing kernel method for solving boundary value problems. Mathematical Modelling and Analysis, 21(4):466–477, 2016. https://doi.org/10.3846/13926292.2016.1183240
H. Zhu, J. Niu, R.M. Zhang and Y.Z. Lin. A new approach for solving nonlinear singular boundary value problems. Mathematical Modelling and Analysis, 23(1):33–43, 2018. https://doi.org/10.3846/mma.2018.003
E. Babolian, Z. Masouri and S. Hatamzadeh-Varmazyar. New direct method to solve nonlinear Volterra-Fredholm integral and integro differential equation using operational matrix with block-pulse functions. Progress in Electromagnetics Research B, 8:59–76, 2008. https://doi.org/10.2528/pierb08050505
E. Babolian, Z. Masouri and S. Hatamzadeh-Varmazyar. Numerical solution of nonlinear Volterra-Fredholm integro-differential equations via direct method using triangular functions. Computers Mathematics With Applications, 58(2):239– 247, 2009. https://doi.org/10.1016/j.camwa.2009.03.087
H.O. Bakodah, M. Almazmumy, S.O. Almuhalbedi and L. Abdullah. Laplace discrete Adomian decomposition method for solving nonlinear integro differential equations. Journal of Applied Mathematics and Physics, 7(6):1388–1407, 2019. https://doi.org/10.4236/jamp.2019.76093
J. Biazar, B. Ghanbari, M.G. Porshokouhi and M.G. Porshokouhi. Hes homotopy perturbation method: A strongly promising method for solving non-linear systems of the mixed VolterraFredholm integral equations. Computers Mathematics with Applications, 61(4):1016–1023, 2011. https://doi.org/10.1016/j.camwa.2010.12.051
N. Bildik. Modified decomposition method for nonlinear VolterraFredholm integral equations. Chaos Solitons Fractals, 33:308–313, 2007. https://doi.org/10.1016/j.chaos.2005.12.058
M.G. Cui and Y.Z. Lin. Nonlinear Numerical Analysis in the Reproducing Kernel Space. Nova Science Publishers, New York, London, 2009.
M. Ghasemi, M.T. Kajani and E. Babolian. Application of Hes homotopy perturbation method to nonlinear integro-differential equations. Applied Mathematics and Computation, 188(1):538–548, 2007. https://doi.org/10.1016/j.amc.2006.10.016
K. Maleknejad, B. Basirat and E. Hashemizadeh. Hybrid legendre polynomials and block-pulse functions approach for nonlinear V0olterra-Fredholm integrodifferential equations. Computers Mathematics with Applications, 61(9):2821– 2828, 2011. https://doi.org/10.1016/j.camwa.2011.03.055
K. Maleknejad and E. Saeedipoor. Convergence analysis of hybrid functions method for two-dimensional nonlinear VolterraFredholm integral equations. Journal of Computational and Applied Mathematics, 368:1–10, 2020. https://doi.org/10.1016/j.cam.2019.112533
J. Niu, Y.Z. Lin and C.P. Zhang. Approximate solution of nonlinear multi-point boundary value problem on the half-line. Mathematical Modelling and Analysis, 17(2):190–202, 2012. https://doi.org/10.3846/13926292.2012.660889
J. Niu, L.X. Sun, M.Q. Xu and J.J. Hou. A reproducing kernel method for solving heat conduction equations with delay. Applied Mathematics Letters, 100:106036, 2019. https://doi.org/10.1016/j.aml.2019.106036
J. Niu, M.Q. Xu and G.M. Yao. An efficient reproducing kernel method for solving the AllenCahn equation. Applied mathematics letters, 89:78–84, 2019. https://doi.org/10.1016/j.aml.2018.09.013
K. Ozen and K. Orucoglu. Approximate solution to a multi-point boundary value problem involving nonlocal integral conditions by reproducing kernel method. Mathematical Modelling and Analysis, 18(4):529–536, 2013. https://doi.org/10.3846/13926292.2013.840867
M.Q. Xu, L.F. Zhang and E. Tohidi. A fourth-order least-squares based reproducing kernel method for one-dimensional elliptic interface problems. Applied Numerical Mathematics, 162:124–136, 2021. https://doi.org/10.1016/j.apnum.2020.12.015
S.A. Yousefi and M. Razzaghi. Legendre wavelets method for the nonlinear Volterra-Fredholm integral equations. Mathematics and Computers in Simulation, 70(1):1–8, 2005. https://doi.org/10.1016/j.matcom.2005.02.035
Z.H. Zhao, Y.Z. Lin and J. Niu. Convergence order of the reproducing kernel method for solving boundary value problems. Mathematical Modelling and Analysis, 21(4):466–477, 2016. https://doi.org/10.3846/13926292.2016.1183240
H. Zhu, J. Niu, R.M. Zhang and Y.Z. Lin. A new approach for solving nonlinear singular boundary value problems. Mathematical Modelling and Analysis, 23(1):33–43, 2018. https://doi.org/10.3846/mma.2018.003