RETRACTED CHAPTER: A Collocation Method for Integral Equations in Terms of Generalized Bernstein Polynomials
In this study, a collocation method based on generalized Bernstein polynomials is presented for approximate solution of Fredholm–Volterra integral equations. While this method is applicable directly to linear integral equations of the first, second and th
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Abstract In this study, a collocation method based on generalized Bernstein polynomials is presented for approximate solution of Fredholm–Volterra integral equations. While this method is applicable directly to linear integral equations of the first, second and third kinds, it is applicable iteratively to nonlinear integral equations using method of quasilinearzation. Error bounds are demonstrated for the Bernstein collocation method, and the convergence of this method is shown, Moreover, some numerical examples are given to illustrate the accuracy, efficiency and applicability of the method.
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Keywords Bernstein polynomials linear and nonlinear integral equations Quasilinearzation technique · Collocation method 2000 Mathematics Subject Classification 42A60 · 42A10
1 Introduction Integral equations are closely related to a number of different areas of mathematics. For instance, many problems included to ordinary and partial differential equations can be converted in the integral equations. In addition, these equations are often used in the engineering, mathematical physics, potential theory, electrostatic and radioactive heat transfer. Therefore, many researchers are interested in numerical methods to get the solution of integral equations. V.K. Singh (B) Department of Applied Mathematics, Raj Kumar Goel Institute of Technology, NH-58, DelhiMeerut Road, Ghaziabad 201003, India e-mail: [email protected] A.K. Singh Department of Science and Technology, Government of India, Technology Bhavan, New Mehrauli Road, New Delhi 110016, India e-mail: [email protected] © Springer Science+Business Media Singapore 2016 V.K. Singh et al. (eds.), Modern Mathematical Methods and High Performance Computing in Science and Technology, Springer Proceedings in Mathematics & Statistics 171, DOI 10.1007/978-981-10-1454-3_23
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V.K. Singh and A.K. Singh
Quasilinearzation [1, 2] is an effective method that solves the nonlinear equations recursively by a sequence of linear equations. The main advantage of this method is that it converges quadratically to the solution of the original equation. Besides, since many problems in system identification and optimization can be reduced to this format, quasilinearzation is a useful computational technique in modern control applications. This method has also been applied the variety of nonlinear equations such as ordinary differential equations [3–7], functional differential equations [8–10], integral equations [11, 12], integro-differential equations [13]. Bernstein polynomials have many useful properties, such as the positivity, continuity, recursion’s relation, symmetry, unity partition of the basis set over the interval [0, 1], and the polynomials are differentiable and integrable. For this reason, these polynomials have been used to numerical solution of Volterra [14–17, 25] and Fredholm [18, 19] integral equations. The definitions of the Bernstein polynomials and their basis from that can be easily generalized on the interval [a, b], are given as follows: Definition 1.1 Ge
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