Mechanics, Materials Science & Engineering, September 2016
ISSN 2412-5954
Numerical Solution of Nonlinear Fredholm Integro-Differential Equations using Leibnitz-Haar Wavelet Collocation Method S. C. Shiralashetti1, a, R. A. Mundewadi1 1 P. G. Department of Studies in Mathematics, Karnatak University, Dharwad-580003, India a
shiralashettisc@gmail.com DOI 10.13140/RG.2.2.31444.19848
Keywords: Leibnitz-Haar wavelet collocation method, operational matrix, nonlinear Fredholm integro-differential equations.
ABSTRACT. In this work, we present a Leibnitz-Haar wavelet collocation method for solving nonlinear Fredholm integro-differential equation of the second kind. Haar wavelet and its Operational matrix are utilized to convert the differential equations into a system of algebraic equations, solving these equations using MATLAB to compute the required Haar coefficients. The numerical results obtained by the present method have been compared with those obtained by [3, 4] of the illustrative examples, which shows the efficiency of the method.
1. Introduction. In recent years, there has been a growing interest in the integro-differential equations (IDEs) which are a combination of differential and Fredholm-Volterra integral equations. IDEs play an important role in many branches of linear and nonlinear functional analysis and their applications in the theory of engineering, mechanics, physics, chemistry, astronomy, biology, economics, potential theory and electrostatics. The mentioned integro-differential equations are usually difficult to solve analytically, so a numerical method is required. There are several numerical methods for approximating the solution of nonlinear Fredholm integro-differential equations are known and many different basic functions have been used [1-4]. Wavelets theory is a relatively new and an emerging tool in applied mathematical research area. It has been applied in a wide range of engineering disciplines; particularly, signal analysis for waveform representation and segmentations, time-frequency analysis and fast algorithms for easy implementation. Wavelets permit the accurate representation of a variety of functions and operators. Moreover, wavelets establish a connection with fast numerical algorithms [5, 6]. Since from 1991 the various types of wavelet method have been applied for the numerical solution of different kinds of integral equations, a detailed survey on these papers can be found in [7]. The solutions are often quite complicated and the advantages of the wavelet method get lost. Therefore, any kind of simplification is welcome. One possibility for it is to make use of the Haar wavelets, which are mathematically the simplest wavelets. Haar wavelet methods are applied for different type of problems in [8-14]. Sirajul-Islam et al. [15], V. Mishra et al. [13] and Lepik [16] and Tamme [17] have applied the Haar wavelet method for solving nonlinear Fredholm integro-differential equations. In the present work, a Leibnitz-Haar wavelet collocation method for solving nonlinear Fredholm integro-differential equations of the second kind is proposed. The article is organized as follows: In Section 2, the basic formulation of Haar wavelets and its operational matrix is given. Section 3 is devoted to the method of solution. In section 4, we report our numerical results and demonstrated the accuracy of the proposed scheme. Conclusion is discussed in section 5. 2. Properties of Haar Wavelets. 2.1. Haar wavelets MMSE Journal. Open Access www.mmse.xyz
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