Reproducing kernel method for solving Fredholm integro-differential equations with weakly singularity. (English) Zbl 1291.65379
Summary: Singular integral equations (SIEs) are often encountered in certain contact and fracture problems in solid mechanics. Numerical methods for solving SIEs have been the focus of much research, including reproducing kernel methods. However, there are no reports on reproducing kernel methods for solving differential-integral equations with weakly singular kernels. We developed a reproducing kernel method for solving Fredholm integro-differential equations with weakly singular kernels in reproducing kernel Hilbert space. This involves changing a weakly singular kernel to a logarithm kernel to a Kalman kernel. Weak singularity is removed by applying a smooth transformation to the Kalman kernel. Solution representations are obtained in reproducing kernel Hilbert space. Numerical experiments show that our reproducing kernel method is efficient.
MSC:
| 65R20 | Numerical methods for integral equations |
| 65L05 | Numerical methods for initial value problems involving ordinary differential equations |
| 45J05 | Integro-ordinary differential equations |
References:
| [1] | Hunter, D. B., Some Gauss-type formulae for the evaluation of Cauchy principle values of integrals, Numer. Math., 19, 419-424 (1972) · Zbl 0231.65028 |
| [2] | Paget, D. F.; Elliott, D., An algorithm for the numerical evaluation of certain Cauchy principle values of integrals, Numer. Math., 19, 373-385 (1972) · Zbl 0229.65091 |
| [3] | Lu, C. K., The approximate of Cauchy type integral by some kinds of interpolatory splines, Numer. Math., 36, 197-212 (1982) · Zbl 0497.41006 |
| [4] | Krenk, S., Numerical quadrature of periodic singular integral equations, J. Inst. Math. Appl., 21, 181-187 (1978) · Zbl 0376.65045 |
| [5] | Pedas, A.; Tamme, E., Discrete Galerkin method for Fredholm integro-differential equations with weakly singular kernels, J. Comput. Appl. Math., 213, 111-126 (2008) · Zbl 1137.65073 |
| [6] | Badr, A. A., Numerical solution of Fredholm-Volterra integral equation in one dimension with time dependent, Appl. Math. Comput., 167, 1156-1161 (2005) · Zbl 1082.65593 |
| [7] | Aleksandrov, V. M.; Kovalenko, E. V., Mathematical method in the displacement problem, Inz. Zh. Mekh. Tverd. Tela, 2, 77-89 (1984) · Zbl 0588.73199 |
| [8] | Darwish, M. A., Fredholm-Volterra integral equation with singular kernel, Kor. J. Comput. Appl. Math., 6, 163-174 (1999) · Zbl 0921.65094 |
| [9] | Abdou, M. A.; Mohamed, K. I.; Ismail, A. S., On the numerical solutions of Fredholm-Volterra integral equation, Appl. Math. Comput., 146, 713-728 (2003) · Zbl 1033.65117 |
| [10] | Lu, J.; Du, J., Numerical methods for singular integral equations, Adv. Math., 20, 2, 278-291 (1991) · Zbl 0756.65151 |
| [11] | Du, J., Some systems of orthogonal trigonometric polynomials associated with singular integrals with Hilbert kernel, J. Wuhan Univ. (Nat. Sc. Ed.), 2, 2, 17-35 (1988) · Zbl 0695.45006 |
| [12] | Jin, X.; Keer, L. M.; Wang, Q., A practical method for singular integral equations of the second kind, Eng. Fracture Mech., 206, 189-195 (2007) |
| [13] | Du, J., On the numerical solution for singular integral equations with Hilbert kernel, Chin. J. Numer. Math. Appl., 11, 2, 9-27 (1989) |
| [14] | Chen, Z.; Zhou, Y. F., A new method for solving Hilbert type singular integral equations, Appl. Math. Comput., 218, 406-412 (2011) · Zbl 1228.65246 |
| [15] | Chen, Z.; Zhou, Y. F., A new method for solving hypersingular integral equations of the first kind, Appl. Math. Lett., 24, 636-641 (2011) · Zbl 1213.45003 |
| [16] | Du, H.; Shen, J. H., Reproducing kernel method of solving singular integral equation with cosecant kernel, J. Math. Anal. Appl., 348, 308-314 (2008) · Zbl 1152.45007 |
| [17] | Cui, M. G., Numerical Analysis in Reproducing Kernel Space, 15-27 (2004), Science Publishers: Science Publishers Beijing |
| [18] | Du, H.; Cui, M. G., Representation of the exact solution and a stability analysis on the Fredholm integral equation of the first kind in reproducing kernel space, Appl. Math. Comput., 182, 2, 1608-1614 (2006) · Zbl 1108.65126 |
| [19] | Chen, S.; Chen, S., Higher derivative formula of a compound function and its application, Chin. J. Nanyang Normal Univ., 9, 12, 22-24 (2010) · Zbl 1221.26003 |
| [20] | Chen, Z.; Wang, C. H.; Zhou, Y. F., A new method for solving Cauchy type singular integral equations of the second kind, Int. J. Comput. Math., 87, 9, 2076-2087 (2010) · Zbl 1204.65150 |
| [21] | Cui, M. G.; Lin, Y., Nonlinear Numerical Analysis in the Reproducing Kernel Space, 12 (2008), Nova Science Publishers |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
