Regularization technique and numerical analysis of the mixed system of first and second-kind Volterra–Fredholm integral equations

Iranian Journal of Numerical Analysis and Optimization

مقاله 7، دوره 9، شماره 1 - شماره پیاپی 15، خرداد 2019، صفحه 127-150   

نوع مقاله: Research Article

شناسه دیجیتال (DOI): 10.22067/ijnao.v9i1.63182

نویسندگان

S. Pishbin^* ؛ J. Shokri

Urmia University, Urmia, Iran.

چکیده

‎‎ It is important to note that mixed systems of first and second-kind Volterra–Fredholm integral equations are ill-posed problems, so that solving discretized system of such problems has a lot of difficulties. We will apply the regularization method to convert this mixed system (ill-posed problem) to system of the second kind Volterra–Fredholm integral equations (well-posed problem). A numerical method based on Chebyshev wavelets is suggested for solving the obtained well-posed problem, and convergence analysis of the method is discussed. For showing efficiency of the method, some test problems, for which the exact solution is known, are considered.

کلیدواژه‌ها

Mixed systems of first and second-kind Volterra-Fredholm integral ‎equations‎؛ ‎Regularization‎ ‎method‎؛ Chebyshev wavelets؛ ‎Convergence ‎analysis‎‎

مراجع

1. M.A. Abdou, F.A. Salama, Volterra–Fredholm integral equation of the first kind and spectral relationships, Appl. Math. Comput. 153 (2004), 141–153.

2. J. Ahmadi Shali, P. Darania, A.A. Jodayree, Collocation method for nonlinear Volterra–Fredholm integral equations, Open J. Appl. Sci. 2(2012),115–121.

3. H. Brunner, Collocation methods for Volterra integral and related functional equations, Cambridge Monographs on Applied and Computational Mathematics, vol. 15, Cambridge University Press, Cambridge, 2004.

4. O. Diekman, Thresholds and traveling waves for the geographical spread of infection, J. Math. Biol. 6 (1978), 109–130.

5. M. Hamina, J. Saranen , On the spline collocation method for the single layer heat operator equation, Math. Comp. 62 (1994), 41–64.

6. L. Hocia, On Approximate Solution for Integral Equation of the Mixed Type, ZAMM J. Appl. Math. Mech. 76 (1996), 415–416.

7. Y, Iso, K. Onishi, On the stability of the boundary element collocation method applied to the linear heat equation, J. Comput. Appl. Math. 38(1991), 201–209

8. J.P. Kauthen, Continuous time collocation for Volterra–Fredholm integral equations, Numer. Math. 56 (1989), 409–424.

9. A. Kirsch, An introduction to the mathematical theory of inverse problem, Second edition. Applied Mathematical Sciences, 120. Springer, New York, 2011.

10. K. Maleknejad, H. Almasieh, M. Roodaki, Triangular functions (TF) method for the solution of nonlinear Volterra–Fredholm integral equations, Commun. Nonlinear Sci. Numer. Simul. 15 (2010), 3293–3298

11. K. Maleknejad, M. Hadizadeh, A New computational method for Volterra–Fredholm integral equations, Comput. Math. Appl., 37 (1999), 1–8.

12. D.L. Phillips, A technique for the numerical solution of certain integral equations of the first kind, J. Assoc. Comput. Mach. 9 (1962), 84–97.

13. H. R. Thieme, A model for the spatio spread of an epidemic, J. Math. Biol. 4, (1977), 337–351.

14. A. N. Tikhonov, On the solution of incorrectly posed problem and the method of regularization, Soviet Math. 4 (1963), 1035–1038.

15. S. Yousefi, M. Razzaghi, Legendre Wavelets method for the nonlinear Volterra–Fredholm integral equations, Math. Comput. Simul. 70 (2005), 1–8.

16. A.M. Wazwaz, A reliable treatment for mixed Volterra–Fredholm integral equations, Appl. Math. Comput. 127 (2002), 405–414.