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 Journal of Computational Methods in Physics , 2014, DOI: 10.1155/2014/207916 Abstract: A formulation for the fractional Legendre functions is constructed to find the solution of the fractional Riccati equation. The fractional derivative is described in the Caputo sense. The method is based on the Tau Legendre and path following methods. Theoretical and numerical results are presented. Analysis for the presented method is given. 1. Introduction Recently, many papers on fractional boundary value problems have been studied extensively. Several forms of them have been proposed in standard models, and there has been significant interest in developing numerical schemes for their solutions. Several numerical techniques are used to solve such problems such as Laplace and Fourier transforms [1, 2], Adomian decomposition and variational iteration methods [3, 4], eigenvector expansion , differential transform and finite differences methods [6, 7], power series method , collocation method , and wavelet method [10, 11]. Many applications of fractional calculus on various branches of science such as engineering, physics, and economics can be found in [12, 13]. Considerable attention has been given to the theory of fractional ordinary differential equations and integral equations [14, 15]. Additionally, the existence of solutions of ordinary and fractional boundary value problems using monotone iterative sequences has been investigated by several authors [16–20]. We consider the Riccati equation with fractional orders of the form where , and are continuous functions on and？？ is a constant. Riccati equation with fractional order has been discussed by many researchers using different techniques such as collocation method based on Muntz polynomials , homotopy perturbation method , and series solution method . In this paper we study the Tau-path following method for solving the Riccati equation with fractional order. We organize this paper as follows. In Section 2, we present basic definitions and results of fractional derivatives. We extend basic results to path following method for the fractional case. In Section 3, we introduce the fractional-order Legendre Tau method with path following method for solving the Riccati equation with fractional order. In Section 4, we present some numerical results to illustrate the efficiency of the presented method. Finally, we conclude with some comments in the last section. 2. Preliminaries In this section, we review the definition and some preliminary results of the Caputo fractional derivatives, as well as, the definition of the fractional-order Legendre functions and their properties. Definition
 Journal of Computational Methods in Physics , 2014, DOI: 10.1155/2014/381074 Abstract: A formulation of the fractional Legendre functions is constructed to solve the generalized time-fractional diffusion equation. The fractional derivative is described in the Caputo sense. The method is based on the collection Legendre and path following methods. Analysis for the presented method is given and numerical results are presented. 1. Introduction We consider the generalized time-fractional diffusion equation of the form with initial and boundary conditions where , , , , , , and . For , the fractional diffusion equation is reduced to a conventional diffusion-reaction equation which is well studied, so we focus on . Some existence and uniqueness results of Problem (1)-(2) were established in . In recent years, great interests were devoted to the analytical and numerical treatments of fractional differential equations (FDEs). Usually, FDEs appear as generalizations to existing models with integer derivative and they also present new models for some physical problems [2, 3]. In general, FDEs do not possess exact solutions in closed forms, and, therefore, numerical methods such as the variational iteration (VIM) [4, 5], the homotopy analysis method (HAM) [6, 7], and the Adomian decomposition method (ADM) [8, 9] have been implemented for several types of FDEs. Also, the maximum principle and the method of lower and upper solutions have been extended to deal with FDEs and obtain analytical and numerical results [10, 11]. The Tau method, the pseudospectral method, and the wavelet method based on the Legendre polynomials have been implemented for several types of FDEs [12–14]. Kazem et al.  have constructed the Legendre functions of fractional order and discussed some of their properties. The resulting Legendre function operational and product matrices, together with the Tau method, have been implemented to solve linear and nonlinear fractional differential equations. The effectiveness of the approach has been examined through several examples. In , a fractional diffusion equation is considered, where the fractional derivative of order refers to the spatial variable . The Legendre pseudospectral method is implemented to solve the problem, where the solution is expanded with regular Legendre polynomials. As a result, a system of linear equation has been obtained and integrated using the finite difference method. However, in solving fractional differential equations of order using series expansions, it is common and more efficient to expand the solution with fractional functions of the form . Rawashdeh  has implemented the Legendre wavelets
 Open Access Library Journal (OALib Journal) , 2018, DOI: 10.4236/oalib.1104565 Abstract: In this paper, block procedure for some k-step linear multi-step methods, using the Legendre polynomials as the basis functions, is proposed. Discrete methods were given which were used in block and implemented for solving the initial value problems, being continuous interpolant derived and collocated at grid points. Some numerical examples of ordinary differential equations were solved using the derived methods to show their validity and the accuracy. The numerical results obtained show that the proposed method can also be efficient in solving such problems.
 American Journal of Computational Mathematics (AJCM) , 2017, DOI: 10.4236/ajcm.2017.74028 Abstract: In this paper, we build the integral collocation method by using the second shifted Chebyshev polynomials. The numerical method solving the model non-linear such as Riccati differential equation, Logistic differential equation and Multi-order ODEs. The properties of shifted Chebyshev polynomials of the second kind are presented. The finite difference method is used to solve this system of equations. Several numerical examples are provided to confirm the reliability and effectiveness of the proposed method.
 International Journal of Mathematics and Mathematical Sciences , 2011, DOI: 10.1155/2011/673085 Abstract: We extend a collocation method for solving a nonlinear ordinary differential equation (ODE) via Jacobi polynomials. To date, researchers usually use Chebyshev or Legendre collocation method for solving problems in chemistry, physics, and so forth, see the works of (Doha and Bhrawy 2006, Guo 2000, and Guo et al. 2002). Choosing the optimal polynomial for solving every ODEs problem depends on many factors, for example, smoothing continuously and other properties of the solutions. In this paper, we show intuitionally that in some problems choosing other members of Jacobi polynomials gives better result compared to Chebyshev or Legendre polynomials. 1. Introduction The Jacobi polynomials with respect to parameters , (see, e.g., [1, 2]) are sequences of polynomials satisfying the following relation where These polynomials are eigenfunctions of the following singular Sturm-Liouville equation: A consequence of this is that spectral accuracy can be achieved for expansions in Jacobi polynomials so that The Jacobi polynomials can be obtained from Rodrigue's formula as Furthermore, we have that The Jacobi polynomials are normalized such that An important consequence of the symmetry of weight function and the orthogonality of Jacobi polynomial is the symmetric relation that is, the Jacobi polynomials are even or odd depending on the order of the polynomial. In this form the polynomials may be generated using the starting form such that which is obtained from Rodrigue's formula as follows: Following the two seminal papers of Doha [3, 4] let be an infinitely differentiable function defined on [？1, 1]; then we can write and, for the th derivative of , Then, where For the proof of the above, see . The formula for the expansion coefficients of a general-order derivative of an infinitely differentiable function in terms of those of the function is available for expansions in ultraspherical and Jacobi polynomials in Doha . Another interesting formula is with where For the proof see . Chebyshev, Legendre, and ultraspherical polynomials are particular cases of the Jacobi polynomials. These polynomials have been used both in the solution of boundary value problems  and in computational fluid dynamics [7, 8]. For the ultraspherical coefficients of the moments of a general-order derivative of an infinitely differentiable function, see . Collocation method is a kind of spectral method that uses the delta function as a test function. Test functions have an important role because these functions are applied to obtain the minimum value for residual by using inner
 Yin Yang Applied Mathematics (AM) , 2013, DOI: 10.4236/am.2013.41020 Abstract: In this paper, we apply the Legendre spectral-collocation method to obtain approximate solutions of nonlinear multi-order fractional differential equations (M-FDEs). The fractional derivative is described in the Caputo sense. The study is conducted through illustrative example to demonstrate the validity and applicability of the presented method. The results reveal that the proposed method is very effective and simple. Moreover, only a small number of shifted Legendre polynomials are needed to obtain a satisfactory result.
 Applied Mathematics (AM) , 2013, DOI: 10.4236/am.2013.412224 Abstract: In this paper we present a proposal using Legendre polynomials approximation for the solution of the second order linear partial differential equations. Our approach consists of reducing the problem to a set of linear equations by expanding the approximate solution in terms of shifted Legendre polynomials with unknown coefficients. The performance of presented method has been compared with other methods, namely Sinc-Galerkin, quadratic spline collocation and LiuLin method. Numerical examples show better accuracy of the proposed method. Moreover, the computation cost decreases at least by a factor of 6 in this method.
 S. Balaji International Journal of Mathematics and Mathematical Sciences , 2014, DOI: 10.1155/2014/304745 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. 1. Introduction In recent years, use of fractional-order derivative goes very strongly in engineering and life sciences and also in other areas of science. One of the best advantages of use of fractional differential equation is modeling and control of many dynamic systems. Fractional-order derivatives are used in fruitful way to model many remarkable developments in those areas of science such as quantum chemistry, quantum mechanics, damping laws, rheology, and diffusion processes [1–5] described through the models of fractional differential equations (FDEs). Modeling of a physical phenomenon depends on two parameters such as the time instant and the prior time history; because of this reason, reasonable modeling through fractional calculus was successfully achieved. The abovementioned advantages and applications of FDEs attracted researchers to develop efficient methods to solve FDEs in order to get accurate solutions to such problems and more active research is still going on in those areas. Most of the FDEs are complicated in their structure; hence finding exact solutions for them cannot be simple. Therefore, one can approach the best accurate solution of FDEs through analytical and numerical methods. Designing accurate or best solution to FDEs, many methods are developed in recent years; each method has its own advantages and limitations. This paper aims to solve a FDE called fractional-order Riccati differential equation, one of the important equations in the family of FDEs. The Riccati equations play an important role in engineering and applied science , especially in quantum mechanics  and quantum chemistry [8, 9]. Therefore, solutions to the Riccati differential equations are important to scientists and engineers. Solving fractional-order Riccati differential equation,
 Journal of Applied Mathematics , 2012, DOI: 10.1155/2012/542401 Abstract: The operational matrices of fractional-order integration for the Legendre and Chebyshev wavelets are derived. Block pulse functions and collocation method are employed to derive a general procedure for forming these matrices for both the Legendre and the Chebyshev wavelets. Then numerical methods based on wavelet expansion and these operational matrices are proposed. In this proposed method, by a change of variables, the multiorder fractional differential equations (MOFDEs) with nonhomogeneous initial conditions are transformed to the MOFDEs with homogeneous initial conditions to obtain suitable numerical solution of these problems. Numerical examples are provided to demonstrate the applicability and simplicity of the numerical scheme based on the Legendre and Chebyshev wavelets.
 Journal of Applied Sciences and Environmental Management , 2005, Abstract: In this paper, we proved the superiority of Legendre polynomial to Chebyshev polynomial in solving first order ordinary differential equation with rational coefficient. We generated shifted polynomial of Chebyshev, Legendre and Canonical polynomials which deal with solving differential equation by first choosing Chebyshev polynomial T*n (X), defined with the help of hypergeometric series T*n (x) =F ( -n, n, ;X) and later choosing Legendre polynomial P*n (x) define by the series P*n (x) = F ( -n, n+1, 1;X); with the help of an auxiliary set of Canonical polynomials Qk in order to find the superiority between the two polynomials. Numerical examples are given which show the superiority of Legendre polynomials to Chebyshev polynomials. Journal of Applied Sciences and Environmental Management Vol. 9(3) 2005: 121-124
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