Journal of Applied and Computational Mechanics

A Simple Approach to Volterra-Fredholm Integral Equations

Document Type : Technical Brief

Author

1 School of Science, Xi'an University of Architecture and Technology, Xi’an, China

2 School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo, China

3 National Engineering Laboratory for Modern Silk, College of Textile and Engineering, Soochow University, Suzhou, China

Abstract

This paper suggests a simple analytical method for Volterra-Fredholm integral equations, the solution process is similar to that by variational-based analytical method, e.g., Ritz method, however, the method requires no establishment of the variational principle for the discussed problem, making the method much attractive for practical applications. The examples show the method is straightforward and effective, and the method can also be extended to other nonlinear problems.

Keywords

Main Subjects

[1] Ji FY, He CH, Zhang JJ, et al. A fractal Boussinesq equation for nonlinear transverse vibration of a nanofiber-reinforced concrete pillar, Applied Mathematical Modelling, 82, 2020, 437-448.
[2] Negarchi N, Nouri K. A New Direct Method for Solving Optimal Control Problem of Nonlinear Volterra-Fredholm Integral Equation via the Muntz-Legendre Polynomials, Bulletin of the Iranian Mathematical Society, 45(3), 2019, 917-934.
[3] Li XX, Li YY, Li Y, et al. Gecko-like adhesion in the electrospinning process, Results in Physics, 16, 2020, 102899.
[4] He JH, Ain QT. New promises and future challenges of fractal calculus: from two-scale Thermodynamics to fractal variational principle, Thermal Science, 24(2A), 2020, 659-681.
[5] He JH. Taylor series solution for a third order boundary value problem arising in architectural engineering, Ain Shams Engineering Journal, 2020, DOI: 10.1016/j.asej.2020.01.016.
[6] Ghorbani A, Saberi-Nadjafi J. Exact solutions for nonlinear integral equations by a modified homotopy perturbation method, Computers & Mathematics with Applications, 56(4), 2008, 1032-1039.
[7] Novin R, Araghi MAF. Hypersingular integral equations of the first kind: A modified homotopy perturbation method and its application to vibration and active control, Journal of Low Frequency Noise Vibration and Active Control, 38(2),2019, 706-727.
[8] Deniz S. Optimal perturbation iteration technique for solving nonlinear Volterra-Fredholm integral equations, Mathematical Methods in the Applied Sciences, 2020; 1-7. DOI: 10.1002/mma.6312
[9] He JH. Iteration perturbation method for strongly nonlinear oscillations, Journal of Vibration and Control, 7(5), 2001, 631-642.  
[10] Tian Y. Markov chain Monte Carlo method to solve Fredholm integral equations, Thermal Science, 22(4), 2018, 1673-1678.
[11] Tian Y. Monte Carlo method with control variate for integral equations, Thermal Science, 22(4), 2018, 1765-1771.
[12] He JH. A short review on analytical methods for to a fully fourth-order nonlinear integral boundary value problem with fractal derivatives, International Journal of Numerical Methods for Heat and Fluid Flow, 2020, DOI: 10.1108/HFF-01-2020-0060
[13] He JH. Variational principle and periodic solution of the Kundu–Mukherjee–Naskar equation, Results in Physics, 17, 2020, 103031.
[14] He JH. The simpler, the better: Analytical methods for nonlinear oscillators and fractional oscillators, Journal of Low Frequency Noise, Vibration and Active Control, 38, 2019, 1252-1260.
Journal of Applied and Computational Mechanics
Volume 6, Special Issue
December 2020
Pages 1184-1186