Numerical approach for solving the Riccati and logistic equations via QLM-rational Legendre collocation method
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Numerical approach for solving the Riccati and logistic equations via QLM-rational Legendre collocation method M. M. Khader1,2 · M. Adel3,4 Received: 27 November 2019 / Revised: 20 February 2020 / Accepted: 22 May 2020 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2020
Abstract In the presented study, we are presenting the approximate solutions of two important equations, the Riccati and Logistic equations; the presented technique is based on the rational Legendre function. Since all the studied models are nonlinear, we convert these nonlinear equations to a sequence of linear ordinary differential equations (ODEs), then by applying the quasi-linearization method (QLM) on these resulting ODEs at each iteration, these ODEs will be converted to a simple linear system of algebraic equations which can be solved. Two numerical examples are presented and we compared between the approximate and the exact solutions. Keywords Riccati and Logistic equations · Rational Legendre functions · QLM · Spectral methods Mathematics Subject Classification 65N20 · 41A30
1 Introduction Using the nonlinear ODEs, there are a lot of applications that are modeled mathematically like fluid mechanics, biology (Adel 2018), and engineering (Khader and Adel 2018). As a result special attention is given to solve ODEs of physical interest. Most nonlinear ODEs have a disadvantage which is they do not have an exact solution, so the approximate techniques
Communicated by Masaaki Harada.
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M. M. Khader [email protected]; [email protected] M. Adel [email protected]
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Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 11566, Saudi Arabia
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Department of Mathematics, Faculty of Science, Benha University, Benha, Egypt
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Department of Mathematics, Faculty of Science, Cairo University, Giza, Egypt
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Department of Mathematics, Faculty of Science, Islamic University of Madinah, Medina, Saudi Arabia 0123456789().: V,-vol
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M. M. Khader, M. Adel
must be used (Khader 2011; Khader and Adel 2018). In Reid (1972), there are many details and theories of Riccati equation with applications. In addition to the important applications, such as stochastic realization theory, optimal control, and robust stabilization, the newer applications include such areas as financial mathematics (Lasiecka and Triggiani 1991). The Adomian decomposition method (ADM) (Bahnasawi and El-Tawil 2004) is one of of the approximate techniques which can be used to solve these equations. The continuous Logistic model can be described using 1st order ODE, while the discrete logistic model is a simple iterative equation that reveals the chaotic property in certain regions (Alligood et al. 1996). The Verhulst model is the classic example to illustrate the periodic doubling and chaotic behavior in a dynamical system. A lot of details about the solution for the logistic equation can be found in (Ausloos 2006; Cushing 1998). The main target of the work is to p
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