Numerical solution of a Volterra integral equation

Ahmed Abdulkareem Hadi

doi:10.55927/fjas.v2i5.4038

Authors

Ahmed Abdulkareem Hadi Mathematics Science, Ministry of Education, Directorate of Education Baghdad Al-Rusafa

DOI:

https://doi.org/10.55927/fjas.v2i5.4038

Keywords:

Numerical Solution, Volterra, Integral Equation, Second Kind, Hermite Polynomials

Abstract

This work deals with a generalized nonlinear Volterra integral equation of the second kind, in whose kernel the unknown function occurs with two different arguments. The equation is solved by a collocation approach with piecewise Hermite polynomials. When using polynomials of degree 2 m- 1, m e N, and suitable quadrature formulas, the method has the order 2 m. The collocation points must be chosen in accordance with a stability condition.

Downloads

Download data is not yet available.

References

Atkinson, K. (1972). The numerical solution of Fredholm integral equations of the second kind with singular kernels. Numerische Mathematik, 19(3), 248-259. doi:10.1007/BF01404695

Burton, T. A., & Zhang, B. (2004). Fixed points and stability of an integral equation: nonuniqueness. Applied Mathematics Letters, 17(7), 839-846.

Ciarlet, P. G., Schultz, M. H., & Varga, R. S. (1967). Numerical methods of high-order accuracy for nonlinear boundary value Problems. Numerische Mathematik, 9(5), 394-430. doi:10.1007/BF02162155

Diogo, T., McKee, S., & Tang, T. (1994). Collocation methods for second-kind Volterra integral equations with weakly singular kernels. Proceedings of the Royal Society of Edinburgh Section A: Mathematics, 124(2), 199-210.

Doornenbal, P. (2012). In-beam gamma-ray spectroscopy at the RIBF. Progress of Theoretical and Experimental Physics, 2012(1), 03C004. doi:10.1093/ptep/pts076

Dzyadyk, V. K. (1995). Approximation methods for solutions of differential and integral equations: De Gruyter.

Fahim, A., Fariborzi Araghi, M. A., Rashidinia, J., & Jalalvand, M. (2017). Numerical solution of Volterra partial integro-differential equations based on sinc-collocation method. Advances in Difference Equations, 2017(1), 362. doi:10.1186/s13662-017-1416-7

Golberg, M. A. (2013). Numerical solution of integral equations (Vol. 42): Springer Science & Business Media.

Groetsch, C. W. (2007). Integral equations of the first kind, inverse problems and regularization: a crash course.

Imai, Y., & Aoki, T. (2004). Fourth-order accurate IDO scheme using gradient-staggered interpolation. JSME International Journal Series B Fluids and Thermal Engineering, 47(4), 681-689.

Jajarmi, A., & Baleanu, D. (2020). A New Iterative Method for the Numerical Solution of High-Order Non-linear Fractional Boundary Value Problems. Frontiers in Physics, 8.

Kansy, K. (1973). Elementare Fehlerdarstellung für Ableitungen bei der Hermite-interpolation. Numerische Mathematik, 21(4), 350-354. doi:10.1007/BF01436389

Levesley, J., & Luo, Z. (2003). Error estimates for Hermite interpolation on spheres. Journal of Mathematical Analysis and Applications, 281(1), 46-61. doi:https://doi.org/10.1016/S0022-247X(02)00451-1

Lima, P., & Diogo, T. (1997). An extrapolation method for a Volterra integral equation with weakly singular kernel. Applied Numerical Mathematics, 24(2), 131-148. doi:https://doi.org/10.1016/S0168-9274(97)00016-0

Maleknejad, K., & Aghazadeh, N. (2005). Numerical solution of Volterra integral equations of the second kind with convolution kernel by using Taylor-series expansion method. Applied Mathematics and Computation, 161(3), 915-922.

Meis, T. (1978, 1978//). Eine spezielle Integralgleichung erster Art. Paper presented at the Numerical Treatment of Differential Equations, Berlin, Heidelberg.

Netravali, A. N. (1973). Spline approximation to the solution of the Volterra integral equation of the second kind. mathematics of computation, 27(121), 99-106.

Niu, J., Lin, Y. Z., & Zhang, C. P. (2012). Approximate solution of nonlinear multi-point boundary value problem on the half-line. Mathematical Modelling and Analysis, 17(2), 190-202.

Salim, S., Saeed, R., & Jwamer, K. (2022). SOLVING VOLTERRA-FREDHOLM INTEGRAL EQUATIONS BY LINEAR SPLINE FUNCTION. 9, 99-107.

Steinberg, J. (1972). Numerical solution of Volterra integral equation. Numerische Mathematik, 19(3), 212-217. doi:10.1007/BF01404691

Swartz, B. K., & Varga, R. S. (1972). Error bounds for spline and L-spline interpolation. Journal of approximation theory, 6(1), 6-49.

Tahmasbi, A., & Fard, O. S. (2008). Numerical solution of linear Volterra integral equations system of the second kind. Applied Mathematics and Computation, 201(1), 547-552. doi:https://doi.org/10.1016/j.amc.2007.12.041

Weisner, L. (1959). Generating functions for Hermite functions. Canadian Journal of Mathematics, 11, 141-147.

Zhu, L., Ma, Y.-G., Chen, Q., & Han, D.-D. (2016). Multilayer Network Analysis of Nuclear Reactions. Scientific Reports, 6(1), 31882. doi:10.1038/srep31882