Existence, uniqueness, and numerical solutions for two-dimensional nonlinear fractional Volterra and Fredholm integral e
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Existence, uniqueness, and numerical solutions for two-dimensional nonlinear fractional Volterra and Fredholm integral equations in a Banach space Khosrow Maleknejad1 · Jalil Rashidinia1 · Tahereh Eftekhari1 Received: 21 October 2019 / Revised: 2 August 2020 / Accepted: 1 September 2020 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2020
Abstract The purpose of this research is to provide sufficient conditions for the local and global existence of solutions for two-dimensional nonlinear fractional Volterra and Fredholm integral equations, based on the Schauder’s and Tychonoff’s fixed-point theorems. Also, we provide sufficient conditions for the uniqueness of the solutions. Moreover, we use operational matrices of hybrid of two-dimensional block-pulse functions and two-variable shifted Legendre polynomials via collocation method to find approximate solutions of the mentioned equations. In addition, a discussion on error bound and convergence analysis of the proposed method is presented. Finally, the accuracy and efficiency of the presented method are confirmed by solving three illustrative examples and comparing the results of the proposed method with other existing numerical methods in the literature. Keywords Two-dimensional nonlinear fractional Volterra and Fredholm integral equations · Existence and uniqueness · Banach space · Hybrid functions · Operational matrices · Collocation method · Convergence analysis Mathematics Subject Classification 26A33 · 33C45 · 65N35
Communicated by José Tenreiro Machado.
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Khosrow Maleknejad [email protected] Jalil Rashidinia [email protected] Tahereh Eftekhari [email protected]
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School of Mathematics, Iran University of Science and Technology (IUST), Narmak, Tehran 16846 13114, Iran 0123456789().: V,-vol
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1 Introduction The fractional calculus deals with derivatives and integrals to an arbitrary order. In recent years, a large number of scientific and engineering problems involving fractional calculus. It provides more accurate models of systems under consideration. The applications of fractional calculus have been demonstrated by many authors. Many systems in interdisciplinary fields, such as biological systems (Ahmed and Elgazzar 2007; Zalp and Demirci 2011), turbulence (Chen 2006), anomalous diffusion (Chen et al. 2010; Sun et al. 2009), viscoelastic systems (Rossikhin and Shitikova 1997), and partial bed-load transport (Sun et al. 2015), can be described with the help of fractional derivatives. Moreover, various problems in fluid mechanics, biology, physics, physiology, optics, and climatology can be modeled by fractional integral equations (Atanackovic and Stankovic 2004; Evans et al. 2017). In many situations, analytic solutions of fractional integral and differential equations are not available, or may these equations not be directly solvable. Therefore, finding efficient numerical methods to approximate the solutions of these equations has become the main objective of many mathe
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