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Asian-European Journal of Mathematics Vol. 12, No. 1 (2019) 1950055–1950070 c World Scientific Publishing Company
DOI: 10.1142/10.1142/S179355711950055
Solving of nonlinear Fredholm integro-differential equation in a complex plane with rationalized Haar wavelet bases∗
Majid Erfanian† Department of Science, School of Mathematical Sciences, University of Zabol, Iran
[email protected] Hamed. Zeidabadi Faculty of Engineering, Sabzevar University of New Technology, Sabzevar, Iran.
[email protected]
Received (March 3, 2018) Revised (March 12, 201) Accepted (March 16, 2018) Everyone knows about the complicatedness solution of the non-linear Fredholm integrodifferential equation in general. Hence often, Authors attempt to obtain the approximation solution. In this paper, a numerical method for the solutions of the non-linear Fredholm integro-differential equation (NFIDE) of the second kind in the complex plane is presented. In fact, by using the properties of Rationalized Haar (RH) wavelet we try to give the solution of the problem. So far, as we know, no study has yet been attempted for solving the NFIDE in the complex plane. For this purpose, we introduce the continuous integral operator and real valued function. The Banach fixed point theorem guarantees that, under certain assumptions, the integral operator has a unique solution. Furthermore, we give an upper bound for the error analysis. An algorithm is presented to compute and illustrated the solutions for some numerical examples. Keywords: Nonlinear integro Fredholm integral equation, Haar Wavelet, Error estimation, complex plane. AMS Subject Classification: 47A56; 45B05; 47H10; 42C40
1. Introduction Wavelet analysis is a new branch of mathematics and widely applied in signal analysis, image processing, numerical analysis, and Etc . The wavelet methods have proven to be the very effective and efficient tool for solving problems of mathematical calculus. Among the wavelet families, which are defined by an analytical ∗ nonlinear
Fredholm integro-differential equation.
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Fig. 4. Absolute errors of Example 6.2.
bounded for equations discussed and with using of Banach fixed point theorem we proved the convergence theorem of our method. By two numerical examples, it has been observed that the approximation solutions are obtained by using the i for i = 1, ..., 9 are very suitable. Haar wavelet basis in the nodes specified xi = 10 Moreover, the CPU runs times in seconds are presented. According to the reported results in tables and the shown figures we can say that the approach is effective and efficient in solving some kinds of the NIDEs. In addition, we introduce an algorithm, based on the method presented in Section 2, and has been used to solve for all examples. References 1. Atkinson KE. The Numerical Solution of Integral Equations of the Second Kind. Cambridge University Press, 1997. 2. Chen, CF, Hsiao CH, Haar wavelet method for solving lumped and distributed parameter systems. IEEE Proceedings D 1997;144 (1):87-94. 3. El-Kady M, Biomy M. Efficient Legendre pseudospectral method for solving integral and integro-differential equations.Communications in Nonlinear Science and Numerical Simulation 2010;15(7):1724-39. 4. El-Kalla IL. A new approach for solving a class of nonlinear integro-differential equations. Communications in Nonlinear Science and Numerical Simulation 2012;17(12):4634-41. 5. Erfanian M, Gachpazan M, Beiglo M. A new sequential approach for solving the integro-differential equation via Haar wavelet bases. Computational Mathematics and Mathematical Physics 2017;57(2):297-305. 6. Erfanian M, Gachpazan M, Beiglo M. Rationalized Haar wavelet bases to approximate solution of nonlinear Fredholm integral equations with error analysis. Applied Mathematics and Computation 2015;265:304-12. 7. Euler WD. The Master of Us All. The Dolciani Mathematical Expositions, Number 22, Mathematical Association of America, 1999. 8. Forbes LK, Crozier S, Doddrell DM. Calculating current densities and fields produced by shielded magnetic resonance imaging probes. SIAM Journal on Applied Mathematics 1997;57(2):401-25.
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