Analysis and Application of an Iterative Numerical Method for Volterra Integral Equations of the Second Kind
DOI:
https://doi.org/10.32792/jeps.v16i1.749Keywords:
Volterra, Iteration, Numerical Method, Fixed Point, Singular Kernel, Approximation.Abstract
Volterra integral equations of the second kind are important for studying systems that remember past events and change over time. They are often used in physics, biology, and engineering. These equations are usually hard to solve with exact methods because they are complicated, especially when the kernel function or source term includes nonlinearity or cannot be expressed in a simple formula. So, numerical methods are important tools for getting close answers that are very accurate. This study looks at a straightforward and useful method for solving a specific type of math problem called second kind Volterra integral equations. The method divides the time into equal parts and starts with an initial guess for the unknown function. The solution is gradually improved through several rounds until it is accurate enough. The method skips complicated math with matrices and instead uses the way the integral equation is set up to calculate values one by one in a forward way. To check if the method works, we used several test problems that have known answers. The results show that the numbers we calculated mostly match the exact answers, with very small mistakes and quick improvements. The method showed that it works well and is quick, needing only a few tries to get good results. These features make it really good for real-life situations where fast and dependable number solutions are required. In short, the suggested step-by-step numerical method is a trustworthy, precise, and simple way to solve second-kind Volterra integral equations. It is especially helpful when regular ways of analyzing data can't be used. This makes it a valuable tool for researchers and engineers in different scientific and technical areas.
References
R. P. Agarwal and D. O'Regan, "Singular Volterra integral equations," Applied Mathematics Letters, vol. 13, no. 1, pp. 115-120, 2000, doi: https://doi.org/10.1016/S0893-9659(99)00154-8.
R. Gorenflo and S. Vessella, Abel integral equations. Springer, 1991.
J. Wang, C. Zhu, and M. Fečkan, "Analysis of Abel-type nonlinear integral equations with weakly singular kernels," Boundary Value Problems, vol. 2014, pp. 1-16, 2014, doi: https://doi.org/10.1186/1687-2770-2014-20.
L. Becker, "Properties of the resolvent of a linear Abel integral equation: implications for a complementary fractional equation," Electronic Journal of Qualitative Theory of Differential Equations, vol. 2016, no. 64, pp. 1-38, 2016, doi: https://doi.org/10.14232/ejqtde.2016.1.64.
A. Aghili and H. Zeinali, "Solution to Volterra singular integral equations and nonhomogeneous time fractional PDEs," General Mathematics Notes, vol. 14, pp. 6-20, 2013. [Online]. Available: https://www.emis.de/journals/GMN/yahoo_site_admin/assets/docs/2_GMN-2622-V14N1.77231028.pdf.
G.-C. Wu and D. Baleanu, "Variational iteration method for fractional calculus-a universal approach by Laplace transform," Advances in Difference Equations, vol. 2013, pp. 1-9, 2013, doi: https://doi.org/10.1186/1687-1847-2013-18.
S. András, "Weakly singular Volterra and Fredholm-Volterra integral equations," Stud. Univ. Babes-Bolyai Math, vol. 48, no. 3, pp. 147-155, 2003.
B. Bertram and O. Ruehr, "Product integration for finite-part singular integral equations: numerical asymptotics and convergence acceleration," Journal of computational and applied mathematics, vol. 41, no. 1-2, pp. 163-173, 1992, doi: https://doi.org/10.1016/0377-0427(92)90246-T.
H. Brunner, "The numerical solution of weakly singular Volterra integral equations by collocation on graded meshes," Mathematics of computation, vol. 45, no. 172, pp. 417-437, 1985. [Online]. Available: https://www.ams.org/mcom/1985-45-172/S0025-5718-1985-0804933-3/.
T. Diogo, "Collocation and iterated collocation methods for a class of weakly singular Volterra integral equations," Journal of computational and applied mathematics, vol. 229, no. 2, pp. 363-372, 2009, doi: https://doi.org/10.1016/j.cam.2008.04.002.
P. Assari, "Solving weakly singular integral equations utilizing the meshless local discrete collocation technique," Alexandria engineering journal, vol. 57, no. 4, pp. 2497-2507, 2018, doi: https://doi.org/10.1016/j.aej.2017.09.015.
S. Rehman, A. Pedas, and G. Vainikko, "Fast solvers of weakly singular integral equations of the second kind," Mathematical Modelling and Analysis, vol. 23, no. 4, pp. 639-664, 2018, doi: https://doi.org/10.3846/mma.2018.039.
S. Kumar, A. Kumar, D. Kumar, J. Singh, and A. Singh, "Analytical solution of Abel integral equation arising in astrophysics via Laplace transform," Journal of the Egyptian Mathematical Society, vol. 23, no. 1, pp. 102-107, 2015, doi: http://dx.doi.org/10.1016/j.joems.2014.02.004.
P. Mokhtary and F. Ghoreishi, "Convergence analysis of the operational Tau method for Abel-type Volterra integral equations," Electron. Trans. Numer. Anal, vol. 41, pp. 289-305, 2014. [Online]. Available: http://etna.math.kent.edu/.
T. Diogo, N. J. Ford, P. Lima, and S. Valtchev, "Numerical methods for a Volterra integral equation with non-smooth solutions," Journal of computational and applied mathematics, vol. 189, no. 1-2, pp. 412-423, 2006, doi: https://doi.org/10.1016/j.cam.2005.10.019.
M. R. Ali, M. M. Mousa, and W.-X. Ma, "Solution of nonlinear Volterra integral equations with weakly singular kernel by using the HOBW method," Advances in Mathematical Physics, vol. 2019, no. 1, p. 1705651, 2019, doi: https://doi.org/10.1155/2019/1705651.
M. Nadir and B. Gagui, "Quadratic numerical treatment for singular integral equations with logarithmic kernel," International Journal of Computing Science and Mathematics, vol. 10, no. 3, pp. 288-296, 2019, doi: https://doi.org/10.1504/IJCSM.2019.101102.
A. Alvandi and M. Paripour, "Reproducing kernel method for a class of weakly singular Fredholm integral equations," Journal of Taibah University for Science, vol. 12, no. 4, pp. 409-414, 2018, doi: https://doi.org/10.1080/16583655.2018.1474841.
H. Brunner, A. Pedas, and G. Vainikko, "The piecewise polynomial collocation method for nonlinear weakly singular Volterra equations," Mathematics of Computation, vol. 68, no. 227, pp. 1079-1095, 1999, doi: https://www.ams.org/journals/mcom/1999-68-227/S0025-5718-99-01073-X/.
M. A. Darwish, "On monotonic solutions of an integral equation of Abel type," 2007. [Online]. Available: https://www.osti.gov/etdeweb/biblio/20982989.
N. Sidorov, D. Sidorov, and A. Krasnik, "Solution of Volterra operator-integral equations in the nonregular case by the successive approximation method," Differential Equations, vol. 46, pp. 882-891, 2010, doi: https://doi.org/10.1134/S001226611006011X.
D. N. Sidorov and N. A. Sidorov, "Convex majorants method in the theory of nonlinear Volterra equations," Banach Journal of Mathematical Analysis, vol. 6, no. 1, pp. 1-10, 2012, doi: https://doi.org/10.15352/bjma/1337014661.
S. Noeiaghdam et al., "Error estimation of the homotopy perturbation method to solve second kind Volterra integral equations with piecewise smooth kernels: Application of the CADNA library," Symmetry, vol. 12, no. 10, p. 1730, 2020, doi: https://doi.org/10.3390/sym12101730.
K. E. Atkinson, An introduction to numerical analysis. John wiley & sons, 2008.
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