Numerical treatment for solving fractional SIRC model and influenza A
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Numerical treatment for solving fractional SIRC model and influenza A M. M. Khader · Mohammed M. Babatin
Received: 24 May 2013 / Revised: 30 August 2013 / Accepted: 18 September 2013 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2013
Abstract This paper presents an accurate numerical method for solving fractional SIRC model. In this work, we propose a method so called fractional Chebyshev finite difference method. In this technique, we approximate the proposed model with a finite dimensional problem. The method is based on the combination of the useful properties of Chebyshev polynomials approximation and finite difference method. The Caputo fractional derivative is replaced by a difference quotient and the integral by a finite sum. By this method, the given problem is reduced to a problem for solving a system of algebraic equations, and by solving this system, we obtain the solution of SIRC model. Special attention is given to study the convergence analysis and estimate an error upper bound of the obtained approximate formula. We compare our numerical solutions with those numerical solutions using fourthorder Runge–Kutta method. The obtained numerical results show the simplicity and the efficiency of the proposed method. Keywords Fractional SIRC model · Caputo fractional derivative · Chebyshev approximation · Finite difference method · Fourth-order Runge–Kutta method · Convergence analysis Mathematics Subject Classification
34L16 · 41A10 · 41A45 · 65L05 · 65L07
Communicated by Pablo Jacovkis. M. M. Khader (B) · M. M. Babatin Department of Mathematics and Statistics, College of Science, Al-Imam Mohammed Ibn Saud Islamic University (IMSIU), P.O.Box 65892, Riyadh 11566, Saudi Arabia e-mail: [email protected] M. M. Babatin e-mail: [email protected] M. M. Khader Department of Mathematics, Faculty of Science, Benha University, Benha, Egypt
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M. M. Khader, M. M. Babatin
1 Introduction Fractional differential equations (FDEs) have been the focus of many studies due to their frequent appearance in various applications in fluid mechanics, viscoelasticity, biology, physics and engineering (Podlubny 1999). Fractional calculus is a generalization of ordinary differentiation and integration to an arbitrary non-integer order. Many physical processes appear to exhibit fractional order behavior that may vary with time or space. Most FDEs do not have exact solutions, so approximate and numerical techniques must be used such as variational iteration method (Khader 2012), collocation method (Khader 2011; Khader and Hendy 2012; Khader et al. 2013; Sweilam and Khader 2010; Sweilam et al. 2012, 2013) and finite difference method (Burden and Faires 1993; Khader 2013; Smith 1965; Sweilam et al. 2011, 2012). Mathematical models have become important tools in analyzing the spread and control of infectious diseases. Understanding the transmission characteristics of infectious diseases in communities, regions, and countries can lead to better approaches to decrease the transmission of these diseases (
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