# Solving Two-Dimensional Nonlinear Fredholm Integral Equations Using Rationalized Haar Functions in the Complex Plane

• Majid Erfanian
• Abbas Akrami
• Mahmmod Parsamanesh
Original Paper

## Abstract

As far as we are aware, no research has been published about two-dimensional integral equations in the complex plane by using Haar bases or any other kinds of wavelets. We introduce a numerical method to solve two-dimensional Fredholm integral equations, using Haar wavelet bases. To attain this purpose, first, an operator and then an orthogonal projection should be defined. Regarding the characteristics of Haar wavelet, we solve an integral equation without using common mathematical methods. We prove the convergence and an upper bound that mentioned in the method by employing the Banach fixed point theorem. Moreover, the rate of convergence our method is $$O(n(2q)^n)$$. We present several examples of different kinds of functions and solve them by this method in this study.

## Keywords

Nonlinear 2-dimensional Fredholm integral equation Haar wavelet Error estimation

45D05 65L60

## References

1. 1.
Krantz, S.G.: A Guide to Complex Variables. ISBN: 978-0-88385-338-2 (2007)Google Scholar
2. 2.
Chew, W.C., Tong, M.S., Hu, B.: Integral equation methods for electromagnetic and elastic waves. In: Synthesis Lectures on Computational Electromagnetics, vol. 3, pp. 1–241. (2008).
3. 3.
Burton, D.M.: The History of Mathematics. McGraw-Hill, New York (1995). ISBN 978-0-07-009465-9Google Scholar
4. 4.
Nahin, P.: The Story of $$\sqrt{-1}$$. Princeton University Press, Princeton (1998)
5. 5.
Kwok, Y.K.: Applied Complex Variables for Scientists and Engineers, 2nd edn. Cambridge University Press, Cambridge (2010)
6. 6.
Jafari, H., Jassim, H.K.: A new approach for solving system of local fractional partial differential equations. Appl. Appl. Math. 11, 162–173 (2016)
7. 7.
Hosseini, V.R., Shivanian, E., Chen, W.: Local radial point interpolation (MLRPI) method for solving time fractional diffusion–wave equation with damping. J. Comput. Phys. 312, 307–332 (2016)
8. 8.
Hosseini, V.R., Chen, W., Avazzadeh, Z.: Numerical solution of fractional telegraph equation by using radial basis functions. Eng. Anal. Bound. Elem. 38, 31–39 (2014)
9. 9.
Zeng, G., Chen, C., Lei, L., Xu, X.: A modified collocation method for weakly singular fredholm integral equations of second kind. J. Comput. Anal. Appl. 27(7), 1091–1102 (2019)Google Scholar
10. 10.
Erfanian, M., Gachpazan, M., Beiglo, M.: A new sequential approach for solving the integro–differential equation via Haar wavelet bases. Comput. Math. Math. Phys. 57(2), 297–305 (2017)
11. 11.
Erfanian, M., Gachpazan, M., Beiglo, M.: Rationalized Haar wavelet bases to approximate solution of nonlinear Fredholm integral equations with error analysis. Appl. Math. Comput. 256, 304–312 (2015)
12. 12.
Panigrahi, B.L., Mandal, M., Nelakanti, G.: Legendre multi-Galerkin methods for fredholm integral equations with weakly singular kernel and the corresponding eigenvalue problem. J. Comput. Appl. Math. 346, 224–236 (2019)
13. 13.
Jafari, H., Jassim, H.K., Moshokoa, S.P., Ariyan, V.M., Tchier, F.: Reduced differential transform method for partial differential equations within local fractional derivative operators. Adv. Mech. Eng. 8(4), 1–6 (2016).
14. 14.
Sharma, V., Setia, A., Agarwal, R.P.: Numerical solution for system of Cauchy type singular integral equations with its error analysis in complex plane. Appl. Math. Comput. 328, 338–352 (2018)
15. 15.
Erfanian, M., Zeidabadi, H.: Solving of nonlinear Fredholm integro-differential equation in a complex plane with rationalized Haar wavelet bases, Asian-Eur. J. Math. (2019).
16. 16.
Erfanian, M.: The approximate solution of nonlinear integral equations with the RH wavelet bases in a complex plane. Int. J. Appl. Comput. Math 4(1), 31 (2018).
17. 17.
Erfanian, M., Zeidabadi, H.: Approximate solution of linear Volterra integro-differential equation by using Cubic B-spline finite element method in the complex plane. Adv. Differ. Equ. 2019(1), 62 (2019).
18. 18.
Erfanian, M.: The approximate solution of nonlinear mixed Volterra–Fredholm–Hammerstein integral equations with RH wavelet bases in a complex plane. Math. Methods Appl. Sci. 41(18), 8942–8952 (2018).
19. 19.
Toutounian, F., Tohidi, E., Shateyi, S.: A collocation method based on the Bernoulli operational matrix for solving high-order linear complex differential equations in a rectangular domain. In: Abstract and Applied Analysis, Art. ID 823098 (2013)Google Scholar
20. 20.
Reis, J.J., Lynch, R.T., Butman, J.: Adaptive Haar transform video bandwidth reduction system for RPVs. In: Proceedings of Annual Meeting of Society of Photo-Optic Institute of Engineering (SPIE), San Dieago, CA, pp. 24–35 (1976)Google Scholar
21. 21.
Wojtaszczyk, P.: A Mathematical Introduction to Wavelets. Cambridge University Press, Cambridge (1997)
22. 22.
Daubechies, I.: Orthonormal bases of compactly supported wavelets. Commun. Pure Appl. Math. 41, 909–996 (1988)
23. 23.
Larson, D.R.: Unitary systems and wavelet sets. In: Wavelet Analysis and Applications, pp. 143–171. Applied and Numerical Harmonic Analysis, Birkhäuser (2007)Google Scholar
24. 24.
Atkinson, K.E.: The Numerical Solution of Integral Equations of the Second Kind. Cambridge University Press, Cambridge (1997)

© Springer Nature India Private Limited 2019

## Authors and Affiliations

• Majid Erfanian
• 1
• Abbas Akrami
• 1
• Mahmmod Parsamanesh
• 1
1. 1.Department of Science, School of Mathematical SciencesUniversity of ZabolZabolIran