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Year 2018, Volume: 47 Issue: 1, 107 - 127, 01.02.2018

Abstract

References

  • Lamb, H., Hydrodynamics, 6th ed., Cambridge Univ. Press, London/New York, 1932.
  • Carslaw, H.S. and Jaeger, J.C. Conduction of Heat in Solids, 2nd ed., Oxford Univ. (Clarendon) Press, London/New York, 1953.
  • Ursell, F. J. Fluid Mech. 19, 305, 1964.
  • Prosperett, A. Phys. Fluids. 19, 195, 1976.
  • Prosperett, A. and Plesse, M.S. J. Fluid Mech. 85, 349, 1978.
  • Tang, B.Q. and Li, X.F. Solution of a class of Volterra integral equations with singular and weakly singular kernels, Appl. Math. Comput. 199, 406–413, 2008.
  • Zozulya, V.V. and Gonzalez–Chi, P.I. Weakly singular, singular and hypersingular integrals in 3–D elasticity and fracture mechanics, J. Chin. Inst. Eng. 22, 763–775, 1999.
  • Kythe, P.K. and Puri, P. Computational Methods for Linear Integral Equations, Birkhäuser, Boston, 2002.
  • Pedas, A. and Tamme, E. Spline collocation method for integro–differential equations with weakly singular kernels, J. Comput. Appl. Math. 197, 253–269, 2006.
  • Cao, Y., Huang, M., Liu, L. and Xu, Y. Hybrid collocation methods for Fredholm integral equations with weakly singular kernels, Appl. Numer. Math. 57, 549–561, 2007.
  • Pedas, A. and Tamme, E. A discrete collocation method for Fredholm integro–differential equations with weakly singular kernels, Appl. Numer. Math. 61, 738–751, 2011.
  • Pedas, A. and Tamme, E. Product integration for weakly singular integro–differential equations, Math. Model. Anal. 16, 153–172, 2011.
  • Kangro, R. and Tamme, E. On fully discrete collocation methods for solving weakly singular integro–differential equations, Math. Model. Anal. 15, 69–8, 2010.
  • Pedas, A. and Tamme, E. Discrete Galerkin method for Fredholm integro–differential equations with weakly singular kernels, J. Comput. Appl. Math. 213, 111–126, 2008.
  • Pedas, A. and Tamme, E. Fully discrete Galerkin method for Fredholm integro–differential equations with weakly singular kernels, Comput. Methods Appl. Math. 8, 294–30, 2008.
  • Lakestani, M., Nemati Saray, B. and Dehghan, M. Numerical solution for the weakly singular Fredholm integro–differential equations using Legendre multiwavelets, J. Comput. Appl. Math. 235, 3291–3303, 2011.
  • Parts, I., Pedas, A. and Tamme, E. Piecewise polynomial collocation for Fredholm integro– differential equations with weakly singular kernels, SIAM J. Numer. Anal. 43, 1897–1911, 2005.
  • Chen, Z. and Cheng, X. An efficient algorithm for solving Fredholm integro–differential equations with weakly singular kernels, J. Comput. Appl. Math. 257, 57–64, 2014.
  • Agarwal, R.P. Boundary value problems for higher order integro–differential equations, Nonlinear Anal. 7, 259–270, 1983.
  • Morchalo, J. On two point boundary value problem for integro–differential equation of second order, Fasc. Math. 9, 51–56, 1975.
  • Wazwaz, A.M. A reliable algorithm for solving boundary value problems for higher–order integro–differential equations, Appl. Math. Comput. 118, 327–342, 2001.
  • Babolian, E. and Shamloo, A.S. Numerical solution of Volterra integral and integro– differential equations of convolution type by using operational matrices of piecewise constant orthogonal functions, J. Comput. Appl. Math. 214, 495–508, 2008.
  • Borhanifar, A. and Sadri, K. A new operational approach for numerical solution of generalized functional integro–differential equations, J. Comput. Appl. Math. 279, 80–9 2015.
  • Rasit Isik, O., Sezer, M. and Guney, Z. Bernstein series solution of a class of linear integro– differential equations with weakly singular kernel, Appl. Math. Comput. 217, 7009–7020, 2011.
  • Huang, Y. and Li, X.F. Approximate solution of a class of linear integro–differential equations by Taylor expansion method, Int. J. Comput. Math. 1–12, 2009.

A new operational approach for solving weakly singular integro-differential equations

Year 2018, Volume: 47 Issue: 1, 107 - 127, 01.02.2018

Abstract

Based on Jacobi polynomials, an operational method is proposed to solve weakly singular integro–differential equations. These equations appear in various fields of science such as physics and engineering, the motion of a plate in a viscous fluid under the action of external forces,
problems of heat transfer, and surface waves. To solve the weakly singular integro–differential equations, a fast algorithm is used for simplifying
the problem under study. The Laplace transform and Jacobi collocation methods are merged, and thus, a novel approach is presented. Some theorems are given and established to theoretically support the computational simplifications which reduce costs. In order to show the efficiency and accuracy of the proposed method some numerical results are provided. It is found that the proposed method has lesser computational size compared to other common methods, such as Adomian decomposition, Taylor expansion, and Bernstein operational methods. It is further found that the absolute errors are almost constant in the studied interval.

References

  • Lamb, H., Hydrodynamics, 6th ed., Cambridge Univ. Press, London/New York, 1932.
  • Carslaw, H.S. and Jaeger, J.C. Conduction of Heat in Solids, 2nd ed., Oxford Univ. (Clarendon) Press, London/New York, 1953.
  • Ursell, F. J. Fluid Mech. 19, 305, 1964.
  • Prosperett, A. Phys. Fluids. 19, 195, 1976.
  • Prosperett, A. and Plesse, M.S. J. Fluid Mech. 85, 349, 1978.
  • Tang, B.Q. and Li, X.F. Solution of a class of Volterra integral equations with singular and weakly singular kernels, Appl. Math. Comput. 199, 406–413, 2008.
  • Zozulya, V.V. and Gonzalez–Chi, P.I. Weakly singular, singular and hypersingular integrals in 3–D elasticity and fracture mechanics, J. Chin. Inst. Eng. 22, 763–775, 1999.
  • Kythe, P.K. and Puri, P. Computational Methods for Linear Integral Equations, Birkhäuser, Boston, 2002.
  • Pedas, A. and Tamme, E. Spline collocation method for integro–differential equations with weakly singular kernels, J. Comput. Appl. Math. 197, 253–269, 2006.
  • Cao, Y., Huang, M., Liu, L. and Xu, Y. Hybrid collocation methods for Fredholm integral equations with weakly singular kernels, Appl. Numer. Math. 57, 549–561, 2007.
  • Pedas, A. and Tamme, E. A discrete collocation method for Fredholm integro–differential equations with weakly singular kernels, Appl. Numer. Math. 61, 738–751, 2011.
  • Pedas, A. and Tamme, E. Product integration for weakly singular integro–differential equations, Math. Model. Anal. 16, 153–172, 2011.
  • Kangro, R. and Tamme, E. On fully discrete collocation methods for solving weakly singular integro–differential equations, Math. Model. Anal. 15, 69–8, 2010.
  • Pedas, A. and Tamme, E. Discrete Galerkin method for Fredholm integro–differential equations with weakly singular kernels, J. Comput. Appl. Math. 213, 111–126, 2008.
  • Pedas, A. and Tamme, E. Fully discrete Galerkin method for Fredholm integro–differential equations with weakly singular kernels, Comput. Methods Appl. Math. 8, 294–30, 2008.
  • Lakestani, M., Nemati Saray, B. and Dehghan, M. Numerical solution for the weakly singular Fredholm integro–differential equations using Legendre multiwavelets, J. Comput. Appl. Math. 235, 3291–3303, 2011.
  • Parts, I., Pedas, A. and Tamme, E. Piecewise polynomial collocation for Fredholm integro– differential equations with weakly singular kernels, SIAM J. Numer. Anal. 43, 1897–1911, 2005.
  • Chen, Z. and Cheng, X. An efficient algorithm for solving Fredholm integro–differential equations with weakly singular kernels, J. Comput. Appl. Math. 257, 57–64, 2014.
  • Agarwal, R.P. Boundary value problems for higher order integro–differential equations, Nonlinear Anal. 7, 259–270, 1983.
  • Morchalo, J. On two point boundary value problem for integro–differential equation of second order, Fasc. Math. 9, 51–56, 1975.
  • Wazwaz, A.M. A reliable algorithm for solving boundary value problems for higher–order integro–differential equations, Appl. Math. Comput. 118, 327–342, 2001.
  • Babolian, E. and Shamloo, A.S. Numerical solution of Volterra integral and integro– differential equations of convolution type by using operational matrices of piecewise constant orthogonal functions, J. Comput. Appl. Math. 214, 495–508, 2008.
  • Borhanifar, A. and Sadri, K. A new operational approach for numerical solution of generalized functional integro–differential equations, J. Comput. Appl. Math. 279, 80–9 2015.
  • Rasit Isik, O., Sezer, M. and Guney, Z. Bernstein series solution of a class of linear integro– differential equations with weakly singular kernel, Appl. Math. Comput. 217, 7009–7020, 2011.
  • Huang, Y. and Li, X.F. Approximate solution of a class of linear integro–differential equations by Taylor expansion method, Int. J. Comput. Math. 1–12, 2009.
There are 25 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Mathematics
Authors

Khadijeh Sadri This is me

Zainab Ayati This is me

Publication Date February 1, 2018
Published in Issue Year 2018 Volume: 47 Issue: 1

Cite

APA Sadri, K., & Ayati, Z. (2018). A new operational approach for solving weakly singular integro-differential equations. Hacettepe Journal of Mathematics and Statistics, 47(1), 107-127.
AMA Sadri K, Ayati Z. A new operational approach for solving weakly singular integro-differential equations. Hacettepe Journal of Mathematics and Statistics. February 2018;47(1):107-127.
Chicago Sadri, Khadijeh, and Zainab Ayati. “A New Operational Approach for Solving Weakly Singular Integro-Differential Equations”. Hacettepe Journal of Mathematics and Statistics 47, no. 1 (February 2018): 107-27.
EndNote Sadri K, Ayati Z (February 1, 2018) A new operational approach for solving weakly singular integro-differential equations. Hacettepe Journal of Mathematics and Statistics 47 1 107–127.
IEEE K. Sadri and Z. Ayati, “A new operational approach for solving weakly singular integro-differential equations”, Hacettepe Journal of Mathematics and Statistics, vol. 47, no. 1, pp. 107–127, 2018.
ISNAD Sadri, Khadijeh - Ayati, Zainab. “A New Operational Approach for Solving Weakly Singular Integro-Differential Equations”. Hacettepe Journal of Mathematics and Statistics 47/1 (February 2018), 107-127.
JAMA Sadri K, Ayati Z. A new operational approach for solving weakly singular integro-differential equations. Hacettepe Journal of Mathematics and Statistics. 2018;47:107–127.
MLA Sadri, Khadijeh and Zainab Ayati. “A New Operational Approach for Solving Weakly Singular Integro-Differential Equations”. Hacettepe Journal of Mathematics and Statistics, vol. 47, no. 1, 2018, pp. 107-2.
Vancouver Sadri K, Ayati Z. A new operational approach for solving weakly singular integro-differential equations. Hacettepe Journal of Mathematics and Statistics. 2018;47(1):107-2.