Fractional Order Riccati Differential Equations Research Articles

Asymptotic Decomposition Method for Fractional Order Riccati Differential Equation

In this study, the asymptotic Adomian decomposition method (AADM) is implemented to solve fractional order Riccati differential equations. The product integration method is used to solve the singular integrals, resulted from fractional derivative. Some fractional order Riccati differential equations are presented as examples to illustrate the ability and efficiency of the proposed approach. The approximate solutions of AADM are compared with the results of the Laplace Adomian Pade method (LAPM). Generalizing AADM for solving fractional Riccati differential equations by the far-field approximation indicates the novelty of the paper.

Jacobi collocation method for the approximate solution of some fractional-order Riccati differential equations with variable coefficients

This paper presents a computational method for the approximate solution of arbitrary-order non-linear fractional Riccati differential equations with variable coefficients. Proposed computational method is a combination of the operational matrix of integration method and the collocation method associated with the Jacobi polynomials. Convergence analysis of the proposed method is provided. Numerical results for different fractional orders of the Riccati differential equations are discussed. Figures and tables are used to show the numerical results derived from the proposed computational method for particular cases of Jacobi polynomials such as the Legendre polynomials, the Chebyshev polynomials of the second kind, the Chebyshev polynomials of the third kind, the Chebyshev polynomial of the fourth kind, and the Gegenbauer (or ultraspherical) polynomials. Numerical results from the proposed methods are compared from those derived by using the existing analytical and numerical methods. It is observed that the results from the proposed method are more accurate. Maximum absolute error and the root-mean square error tables are given for all five kinds of polynomials for comparison purposes.

Third-Kind Chebyshev Wavelet Method for the Solution of Fractional Order Riccati Differential Equations

In this paper, we derive the numerical solutions of the various fractional-order Riccati type differential equations using the third-kind Chebyshev wavelet operational matrix of fractional order integration (C3WOMFI) method. The operational matrix of fractional order integration method converts the fractional differential equations to a system of algebraic equations. The third-kind Chebyshev wavelet method provides sparse coefficient matrices, therefore the computational load involved for this method is not as severe and also the resulting method is faster. The numerical solutions agree with the exact solutions for non-fractional orders, and also the solutions for the fractional orders approach those of the integer orders as the fractional order coefficient [Formula: see text] approaches to 1.

Solution of Fractional Order Riccati Differential Equations Using Euler Wavelet Method

The fractional-order differential equations (FDEs) have the ability to model the real-life phenomena better in a variety of applied mathematics, engineering disciplines including diffusive transport, electrical networks, electromagnetic theory, probability and so forth. In most cases, there are no analytical solutions therefore a variety of numerical methods have been developed for the solution of the FDEs. In this paper, we derive the numerical solutions of the various fractional-order Riccati type differential equations using the Euler Wavelet Method (EWM). The Euler wavelet operational matrix method converts the fractional differential equations to a system of algebraic equations. Illustrative examples are included to demonstrate the validity and efficiency of the technique.

A class of Runge\u2013Kutta methods for nonlinear Volterra integral equations of the second kind with singular kernels

This paper aims to obtain an approximate solution for fractional order Riccati differential equations (FRDEs). FRDEs are equivalent to nonlinear Volterra integral equations of the second kind. In order to solve nonlinear Volterra integral equations of the second kind, a class of Runge–Kutta methods has been applied. Runge–Kutta methods have been implemented to solve nonsingular integral equations. In this work Volterra integral equations are singular. The singularity by a suitable subtraction technique will be weakened; then, this method will be applied to gain an approximate solution. Fractional derivatives are defined in the Caputo form of order 0<alphaleq1.

Numerical solution of fractional-order Riccati differential equation by differential quadrature method based on Chebyshev polynomials

We apply the Chebyshev polynomial-based differential quadrature method to the solution of a fractional-order Riccati differential equation. The fractional derivative is described in the Caputo sense. We derive and utilize explicit expressions of weighting coefficients for approximation of fractional derivatives to reduce a Riccati differential equation to a system of algebraic equations. We present numerical examples to verify the efficiency and accuracy of the proposed method. The results reveal that the method is accurate and easy to implement.

Sine-cosine wavelet operational matrix of fractional order integration and its applications in solving the fractional order Riccati differential equations

In this paper, the sine-cosine wavelet method is presented for solving Riccati differential equations. The sine-cosine wavelet operational matrix of fractional integration is derived and utilized to transform the equations to system of algebraic equations. Also, the error analysis of the sine-cosine wavelet bases is given. The proposed method can be used to solve not only the classical Riccati differential equations but also the fractional ones. Some examples are included to demonstrate the validity and applicability of the technique.

A Numerical Approach for Fractional Order Riccati Differential Equation Using B-Spline Operational Matrix

Abstract In this article, we develop an effective numerical method to achieve the numerical solutions of nonlinear fractional Riccati differential equations. We found the operational matrix within the linear B-spline functions. By this technique, the given problem converts to a system of algebraic equations. This technique is used to solve fractional Riccati differential equation. The obtained results are illustrated both applicability and validity of the suggested approach.

APPLICATION OF GENERALIZED DIFFERENTIAL TRANSFORM METHOD TO FRACTIONAL ORDER RICCATI DIFFERENTIAL EQUATION AND NUMERICAL RESULT

In the present paper, we obtain the approximate solutions of Fractional Order Riccati Differential Equation by Generalized Differential Transform Method (GDTM). The fractional derivatives are described in the Caputo sense. Next we give eight numerical examples for sake of illustration of our method. Finally, we demonstrate our findings by means of two graphs by making use of MATHEMATICA software. AMS Subject Classification: 26A33, 35A22, 65L05, 34A08

Legendre wavelet operational matrix method for solution of fractional order Riccati differential equation

Abstract A Legendre wavelet operational matrix method (LWM) presented for the solution of nonlinear fractional order Riccati differential equations, having variety of applications in engineering and applied science. 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.

Legendre Wavelet Operational Matrix Method for Solution of Riccati Differential Equation

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.

New Ultraspherical Wavelets Spectral Solutions for Fractional Riccati Differential Equations

We introduce two new spectral wavelets algorithms for solving linear and nonlinear fractional-order Riccati differential equation. The suggested algorithms are basically based on employing the ultraspherical wavelets together with the tau and collocation spectral methods. The main idea for obtaining spectral numerical solutions depends on converting the differential equation with its initial condition into a system of linear or nonlinear algebraic equations in the unknown expansion coefficients. For the sake of illustrating the efficiency and the applicability of our algorithms, some numerical examples including comparisons with some algorithms in the literature are presented.

Fractional-order Riccati differential equation: Analytical approximation and numerical results

The aim of this article is to introduce the Laplace-Adomian-Pade method (LAPM) to the Riccati differential equation of fractional order. This method presents accurate and reliable results and has a great perfection in the Adomian decomposition method (ADM) truncated series solution which diverges promptly as the applicable domain increases. The approximate solutions are obtained in a broad range of the problem domain and are compared with the generalized Euler method (GEM). The comparison shows a precise agreement between the results, the applicable one of which needs fewer computations.

An efficient approach for solving the Riccati equation with fractional orders

The present study introduces a novel and simple analytical method for the solution of fractional order Riccati differential equation. In this approach, the solution considered as a Taylor series expansion converges rapidly to the nonlinear problem. New homotopy perturbation method (NHPM) depends only on two components of the homotopy series. The method is illustrated by applications and the results obtained are compared with those of the exact solution. Moreover, comparing the methodology with some known techniques shows that the present approach is relatively easy and efficient.

A new stochastic approach for solution of Riccati differential equation of fractional order

In this article, a stochastic technique has been developed for the solution of nonlinear Riccati differential equation of fractional order. Feed-forward artificial neural network is employed for accurate mathematical modeling and learning of its weights is made with heuristic computational algorithm based on swarm intelligence. In this scheme, particle swarm optimization is used as a tool for the rapid global search method, and simulating annealing for efficient local search. The scheme is equally capable of solving the integer order or fractional order Riccati differential equations. Comparison of results was made with standard approximate analytic, as well as, stochastic numerical solvers and exact solutions.