Application of the homotopy perturbation method for weakly singular Volterra integral equations

Year 2024, Volume: 13 Issue: 3, 201 - 213, 31.12.2024

Ahmet Altürk

https://doi.org/10.54187/jnrs.1560535

Abstract

In this paper, we study a weakly singular Volterra integral equation of the second kind with
the kernel $\displaystyle K(x,t) = \left (\frac{t}{x}\right )^\nu\frac{1}{t}$, for some $\nu >0$ and $x\in[0,X]$. The powerful homotopy perturbation method (HPM) is initially applied to find a solution to the integral equation for $\nu > 1$. We then consider the interesting case where $0< \nu < 1$. Applying the homotopy perturbation method constructed by a convex homotopy or other series-related methods produces unwanted results for this case. In this study, we propose conditions to be imposed to overcome this issue. In addition, for completeness, we investigate all cases where $\nu\in \mathbb{R}$. Some numerical examples are provided to confirm the simplicity and applicability of the applied methods.

Keywords

Volterra integral equations, Homotopy perturbation method, Weakly singular kernel, Perturbation

References

• T. Diogo, P. Lima, Superconvergence of collocation methods for a class of weakly singular Volterra integral equations, Journal of Computational and Applied Mathematics 218 (2) (2008) 307–316.
• T. Diogo, Collocation and iterated collocation methods for a class of weakly singular Volterra integral equations, Journal of Computational and Applied Mathematics 229 (2) (2009) 363–372.
• H. J. J. Te Riele, Collocation methods for weakly singular second-kind Volterra integral equations with non-smooth solution, IMA Journal of Numerical Analysis 2 (4) (1982) 437–449.
• M. A. Bartoshevich, A heat conduction problem, Journal of Engineering Physics 28 (2) (1975) 240–244.
• T. Diogo, N. B. Franco, P. Lima, High order product integration methods for a Volterra integral equation with logarithmic singular kernel, Communications on Pure and Applied Analysis 3 (2) (2004) 217–235.
• J. H. He, Homotopy perturbation technique, Computer Methods in Applied Mechanics and Engineering 178 (3-4) (1999) 257–262.
• S. Momani, G. H. Erjaee, M. H. Alnasr, The modified homotopy perturbation method for solving strongly nonlinear oscillators, Computers and Mathematics with Applications 58 (11-12) (2009) 2209–2220.
• A. Altürk, H. Arabacıoğlu, A new modification to homotopy perturbation method for solving Schl¨omilch’s integral equation, International Journal of Advances in Applied Mathematics and Mechanics 5 (1) (2017) 40–48.
• A. I. Alaje, M. O. Olayiwola, K. A. Adedokun, J. A. Adedeji, A. O. Oladapo, Modified homotopy perturbation method and its application to analytical solitons of fractional-order Korteweg-de Vries equation, Beni-Suef University Journal of Basic and Applied Sciences 11 (139) (2022) 1–17.
• S. J. Liao, An approximate solution technique not depending on small parameters: A special example, International Journal of Non-Linear Mechanics 30 (3) (1995) 371–380.
• C. Chun, Integration using He’s homotopy perturbation method, Chaos, Solitons & Fractals 34 (4) (2007) 1130–1134.
• F. Geng, F. Shen, Solving a Volterra integral equation with weakly singular kernel in the reproducing kernel space, Mathematical Sciences 4 (2) (2010) 159–170.
• T. Diogo, P. Lima, Collocation solutions of a weakly singular Volterra integral equation, Trends in Computational and Applied Mathematics 8 (2) (2007) 229–238.
• W. Han, Existence, uniqueness and smoothness results for second-kind Volterra equations with weakly-singular kernels, Journal of Integral Equations and Applications 6 (3) (1994) 365–384.
• L. Zhu, Y. Wang, Numerical solutions of Volterra integral equation with weakly singular kernel using SCW method, Applied Mathematics and Computation 260 (2015) 63–70.
• M. Nosrati, H. Afshari, Triangular functions in solving weakly singular Volterra integral equations, Advances in the Theory of Nonlinear Analysis and its Applications 7 (1) (2023) 195–204.
• A. Wazwaz, R. Ranch, Two reliable methods for solving the Volterra integral equation with a weakly singular kernel, Journal of Computational and Applied Mathematics 302 (2016) 71–80.
• H. Brunner, The numerical solution of weakly singular Volterra integral equations by collocation on graded meshes, Mathematics of Computation 45 (172) (1985) 417–437.
• S. Szufla, On the Volterra integral equation with weakly singular kernel, Mathematica Bohemica 131 (3) (2006) 225–231.
• M. R. Ali, M. M. Mousa, W-X. Ma, Solution of nonlinear Volterra integral equations with weakly singular kernel by using the HOBW method, Advances in Mathematical Physics 2019 (2019) Article ID 705651 10 pages.
• S. Micula, A numerical method for weakly singular nonlinear Volterra integral equations of the second kind, Symmetry 12 (11) (2020) 1–15.
• Z. Chen, W. Jiang, Piecewise homotopy perturbation method for solving linear and nonlinear weakly singular VIE of second kind, Applied Mathematics and Computation 217 (19) (2011) 7790–7798.
• G. Vainikko, Multidimensional weakly singular integral equations, 1st Edition, Springer, Berlin, 1993.
• J. H. He, A coupling method of a homotopy technique and a perturbation technique for non-linear problems, International Journal of Non-Linear Mechanics 35 (1) (2000) 37–43.
• J. H. He, Homotopy perturbation method: a new nonlinear analytical technique, Applied Mathematics and Computation 135 (1) (2003) 73–79.
• S. Gupta, D. Kumar, J. Singh, Application of He’s homotopy perturbation method for solving nonlinear wave-like equations with variable coefficients, International Journal of Advances in Applied Mathematics and Mechanics 1 (2) (2013) 65–79.
• F. Akta¸s, H. Koklu, Application of the homotopy perturbation method to the neutron diffusion equation, Kafkas Üniversitesi Fen Bilimleri Enstitüsü Dergisi 16 (2) (2023) 70–84.
• J. H. He, Homotopy perturbation method for solving boundary value problems, Physics Letters A 350 (1-2) (2006) 87–88.
• D. D. Ganji, M. Nourollahi, E. Mohseni, Application of He’s methods to nonlinear chemistry problems, Computers & Mathematics with Applications 54 (7-8) (2007) 1122–1132.
• A. Wazwaz, Linear and nonlinear integral equations methods and applications, 1st Edition, Springer, Berlin, 2011.