Legendre Wavelet Operational Matrix Method for Solution of Riccati Differential Equation

Abstract

A Legendre wavelet operational matrix method (LWM) is presented for the solution of nonlinear fractional-order Riccati differential equations, having variety of applications in quantum chemistry and quantum mechanics. The fractional-order Riccati differential equations converted into a system of algebraic equations using Legendre wavelet operational matrix. Solutions given by the proposed scheme are more accurate and reliable and they are compared with recently developed numerical, analytical, and stochastic approaches. Comparison shows that the proposed LWM approach has a greater performance and less computational effort for getting accurate solutions. Further existence and uniqueness of the proposed problem are given and moreover the condition of convergence is verified.

Identification of tissue and therapy parameters from surface temperature.

“…Since the basal state provides the initial conditions of the problem, the first step that will be taken is to determine it. If single-tissue model is considered, i.e., equation (21), with its boundary conditions ( 24) and ( 25) with t = 0, It has analytical solution, since A o does not depend on z, so this boundary problem can be studied as an ordinary differential equation with constant coefficients. Its solution has this form:…”

“…If the properties of the wavelets are applied, it is possible to transform a differential equation or a system in algebraic equations whose unknowns are the coefficients by which the basis functions will be multiplied to get the solution of the differential equation. This system will be linear if the original system is linear ( [17], [18], [19], [20] and [21]).…”

“…• Since the stimulus used is discontinuous in time, a resolution method known as Legendre Wavelets is used. Numerous applied examples of this technique have been found in the literature ( [17], [18], [19], [20] and [21]), which have been useful for developing tools.…”

A piecewise spectral-collocation method for solving fractional Riccati differential equation in large domains

“…Because of its emergence in various applications, a significant number of works has been devoted to the analytical and approximate solution of the RDE in the last decades. For example, methods such as the Adomian's decomposition (Momani and Shawagfeh 2006), homotopy perturbation (Hosseinnia et al 2008;Odibat and Momani 2008;Khan et al 2011), variational iteration (Abbasbandy 2007;Geng et al 2009;Jafari and Tajadodi 2010), Bernstein polynomials (Yüzbasi 2013), wavelet (Abd-Elhameed and Youssri 2014; Wang et al 2017;Mohammadi and Hosseini 2011;Momani and Shawagfeh 2006;Balaji 2014), reproducing kernel Hilbert space (Sakar 2017;Sakar et al 2017;, Taylor series expansion (Aminkhah and Hemmatnezhad 2010), fractional Chebyshev finite difference (Khader 2013), Laplace-Adomian-Pade (Tsai and Chen 2010), Taylor matrix (Gülsu and Sezer 2006), artificial neural networks (Raja et al 2015) and Laplace Adomian decomposition (Vahidi et al 2014) are used to find the approximate solution of the FRDE.…”