On the numerical solutions for nonlinear Volterra-Fredholm integral equations
DOI:
https://doi.org/10.5269/bspm.42815Abstract
In this note, we study a class of multistep collocation methods for the numerical integration of nonlinear Volterra-Fredholm Integral Equations (V-FIEs). The derived method is characterized by a lower triangular or diagonal coefficient matrix of the nonlinear system for the computation of the stages which, as it is known, can be exploited to get an efficient implementation. Convergence analysis and linear stability estimates are investigated. Finally numerical experiments are given, which confirm our theoretical results.
References
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11. K. Maleknejad, H. Almasieh, M. Roodaki, Triangular functions (TF) method for the solution of nonlinear VolterraFredholm integral equations, Commun Nonlinear Sci Numer Simulat, 15 (2010) 3293-3298. https://doi.org/10.1016/j.cnsns.2009.12.015
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13. T. Tang, X. Xu and J. Cheng, On the spectral methods for Volterra type integral equations and the convergence analysis, J. Comput. Math., 26 (2008) 825-837.
14. H. R. Thieme, A model for the spatio spread of an epidemic, J. Math. Biol. 4 (1977) 337-351. https://doi.org/10.1007/BF00275082
15. S. Youse and M. Razzaghi, Legendre, Wavelets method for the nonlinear Volterra-Fredholm integral equations, Math Comput Simulat, 70 (2005) 1-8. https://doi.org/10.1016/j.matcom.2005.02.035
16. A. M. Wazwaz, A reliable treatment for mixed Volterra-Fredholm integral equations, Appl. Math. Comput., 127 (2002) 405-414. https://doi.org/10.1016/S0096-3003(01)00020-0
2. Jafar Ahmadi Shali, P. Darania and A.A Jodayeri Akbarfam, Collocation method for nonlinear Volterra-Fredholm integral equations, Open Journal of Applied Sciences, 2 (2012) 115-121. https://doi.org/10.4236/ojapps.2012.22016
3. Sz. Andras, Fredholm-Volterra equations, Pure Math. Appl. (PU.M.A.), 13 (2002) 21-30.
4. K. E. Atkinson and W. Han, Theoretical Numerical Analysis, Springer, 2009.
5. D. Conte and B. Paternoster, Multistep collocation methods for Volterra integral equations, Appl. Numer. Math., 59 (2009) 1721-1736. https://doi.org/10.1016/j.apnum.2009.01.001
6. P. Darania and K. Ivaz, Numerical solution of nonlinear Volterra-Fredholm integro-differential equations, Computers and Mathematics with Applications, 56 (2008) 2197-2209. https://doi.org/10.1016/j.camwa.2008.03.045
7. O. Diekman, Thresholds and traveling waves for the geographical spread of infection, J. Math. Biol. 6 (1978) 109-130. https://doi.org/10.1007/BF02450783
8. L. Hocia, On Approximate Solution for Integral Equation of the Mixed Type, ZAMM, vol. 76 1996.
9. J. P. Kauthen, Countiunous time collocation for Volterra-Fredholm integral equations, Numerishe Math, 56 (1989) 409-424. https://doi.org/10.1007/BF01396646
10. W. Liniger and R.A. Willoughby, Efficient numerical integration of stiff systems of ordinary differential equations, Technical Report RC-1970, Thomas J. Watson Research Center, Yorktown Heihts, New York, 1976. https://doi.org/10.1137/0707002
11. K. Maleknejad, H. Almasieh, M. Roodaki, Triangular functions (TF) method for the solution of nonlinear VolterraFredholm integral equations, Commun Nonlinear Sci Numer Simulat, 15 (2010) 3293-3298. https://doi.org/10.1016/j.cnsns.2009.12.015
12. K. Maleknejad and M. Hadizadeh, A New computational method for Volterra-Fredholm integral equations, Comput. Math. Appl., 37 (1999) 1-8. https://doi.org/10.1016/S0898-1221(99)00107-8
13. T. Tang, X. Xu and J. Cheng, On the spectral methods for Volterra type integral equations and the convergence analysis, J. Comput. Math., 26 (2008) 825-837.
14. H. R. Thieme, A model for the spatio spread of an epidemic, J. Math. Biol. 4 (1977) 337-351. https://doi.org/10.1007/BF00275082
15. S. Youse and M. Razzaghi, Legendre, Wavelets method for the nonlinear Volterra-Fredholm integral equations, Math Comput Simulat, 70 (2005) 1-8. https://doi.org/10.1016/j.matcom.2005.02.035
16. A. M. Wazwaz, A reliable treatment for mixed Volterra-Fredholm integral equations, Appl. Math. Comput., 127 (2002) 405-414. https://doi.org/10.1016/S0096-3003(01)00020-0
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2021-12-16
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