A Numerical Method for Fredholm Integro-Differential Equation using Positive Definite Radial Kernels and A Study of the Effect of The Shape Parameter



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Submitted Feb 8, 2023
Published Jun 24, 2023
10.24018/ejmath.2023.4.3.213

Abstract viewed = 98 times

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In this paper, we use two radial kernels, the Generalized Multiquadrics and the linear Laguerre-Gaussians for the formulation of radial kernel collocation method for solving problems involving Fredholm integro-differential equations. The effect of the shape parameter contained in each of the kernels, on the accuracy of the method is investigated. The method is demonstrated using two examples the numerical results displayed in form of tables and graphs. MATLAB 2018a was used for the implementation.

Keywords: Integro-differential equation radial kernel Gauss-Legendre quadrature formula collocation method

References

  1. Aboiyar, T ., Kuhe, D. A. and Soomiyol, M. C. (2010). On Laguerre-Gaussians and Fredholm integral equation. African Journal of Mathematics and Computer Science Research, 13(10), 220-224.
     Google Scholar
  2. Arqub, O. A., Al-Smadi, M. and Shawagfeh, N. (2013). Solving Fredholm Integro-Differential Equations using Reproducing Kernel Hilbert Space Method. Applied Mathematics and Computation, 219(2013), 8938-8948.
     Google Scholar
  3. Brunner, H. (2004). Collocation Methods for Volterra Integral and Related Functional Equations. Cambridge, UK: Cambridge University Press.
     Google Scholar
  4. Fang, C., He, G. and Xiang, S. (2019). Hermite-Type Collocation Methods to Solve Volterra Integral Equations with Highly Oscillatory Bessel Kernels. Symmetry 2019, 11, 168; doi:10.3390/sym11020168, 11(168), 1-17.
     Google Scholar
  5. Fasshauer, G. E. (2007). Meshfree Approximation Methods with MATLAB, Interdisciplinary Mathematical Sciences (Vol. 6). Singapore: World Scientific Publishing Co. Pte. Ltd, 520pp.
     Google Scholar
  6. Krowiak, A. (2016). Hermite Type Radial Basis Function-Based Differential Quadrature Method for Higher Order Equations. Applied Mathematical Modelling, 40(4), 2421–2430.
     Google Scholar
  7. Liu, J. and Li, X. (2018). A novel Hermite RBF-based differential quadrature method for solving two-dimensional variable-order time fractional advection-diffusion equation with Neumann boundary condition. Computatational Physics (Physics.Comp-ph) arXiv, 1809(230), 1-12.
     Google Scholar
  8. Ma, J. and Liu, H. (2018). On the Convolution Quadrature Rule for Integral Transforms with Oscillatory Bessel Kernels. Symmetry 2018, 10, 239.
     Google Scholar
  9. Thounthong, P., Khan, M. N., Hussain, I., Ahmad, I. and Kumam, P. (2018). Symmetric Radial Basis Function Method for Simulation of Elliptic Partial Differential Equations. Multidisciplinary Digital Publishing Institute, 1-12. Retrieved from www.mdpi.com/journal/mathematics
     Google Scholar
  10. Wang, H. and Xiang, S. (2011). Asymptotic expansion and Filon-type methods for a Volterra integral equation with a highly oscillatory kernel. IMA J. Numer. Anal, 31, 469–490.
     Google Scholar
  11. Wazwaz, A.-M. (2011). Linear and Nonlinear Integral Equations, Methods and Applications. London, New York: Higher Education Press, Beijing and Springer-Verlag Berlin Heidelberg, 658pp.
     Google Scholar