Projected Iterations of Fixed-Point Type to Solve Nonlinear Partial Volterra Integro-Differential Equations
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Projected Iterations of Fixed-Point Type to Solve Nonlinear Partial Volterra Integro-Differential Equations M. I. Berenguer1 · D. Gámez1 Received: 5 March 2019 / Revised: 19 December 2019 © Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2020
Abstract In this paper, we propose a method to approximate the fixed point of an operator in a Banach space. Using biorthogonal systems, this method is applied to build an approximation of the solution of a class of nonlinear partial integro-differential equations. The theoretical findings are illustrated with several numerical examples, confirming the reliability, validity and precision of the proposed method. Keywords Operators in Banach spaces · fixed-point theorem · Biorthogonal systems · Numerical methods · Nonlinear partial integro-differential equations Mathematics Subject Classification 45A05 · 45L05 · 45N05 · 65R20
1 Introduction The fixed-point theory of operators is a major research area in nonlinear analysis and one of the most powerful and fruitful tools of modern mathematics. This flourishing area of research in pure and applied mathematics can be used to establish the existence and uniqueness of solutions to problems that arise naturally in applications. However, the explicit calculation of the fixed point of an operator is only possible in some simple cases, and in most cases, it is essential to approximate this fixed point by a computational method.
Communicated by Ali Hassan Mohamed Murid.
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M. I. Berenguer [email protected] D. Gámez [email protected]
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E.T.S. Ingeniería Edificación, Departamento de Matemática Aplicada, Universidad de Granada, c/ Severo Ochoa s/n, 18071 Granada, Spain
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M.I. Berenguer, D. Gámez
The study of nonlinear partial integro-differential equations (NPIDEs) has become a subject of considerable interest. These equations and similar ones arise in a variety of science and technology fields such as reaction-diffusion problems [13], theory of elasticity [23], heat conduction [1], mechanic of solids [2], population dynamics [20], transient, conductive and radiative transport [14] and other applications. Various numerical methods have been developed for approximating the solution of NPIDEs. For example in [3], the method is based on radial basis functions and in [11] the authors generalize the Lomov’s regularization method. In [16], a Laguerre collocation method is presented to solve certain nonlinear partial integro-differential equations, in [19] on the fractional differential transform method, in [24] the numerical solution is computed using the 2D shifted orthogonal Legendre polynomial system, and in [17] the Newton–Kantorovich method is used for similar equations. Our purpose in this paper is: (i) to present a method to approximate the fixed point of a operator defined in a Banach space, by means of a composition of operators defined in said space and (ii) develop and apply the method presented for get an approximation of the solution of an nonlinear partial Volterra integro-differential equation (NP
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