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.

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

Fractional Calculus and Applied Analysis ◽

10.1515/fca-2015-0025 ◽

2015 ◽

Vol 18 (2) ◽

Cited By ~ 12

Author(s):

Hossein Jafari ◽

Haleh Tajadodi ◽

Dumitru Baleanu

Keyword(s):

Differential Equation ◽

Numerical Solutions ◽

Operational Matrix ◽

Numerical Approach ◽

Algebraic Equations ◽

Riccati Differential Equation ◽

Suggested Approach ◽

B Spline ◽

Riccati Differential Equations ◽

The Given

AbstractIn 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.

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A Pseudospectral Algorithm for Solving Multipantograph Delay Systems on a Semi-Infinite Interval Using Legendre Rational Functions

Abstract and Applied Analysis ◽

10.1155/2014/816473 ◽

2014 ◽

Vol 2014 ◽

pp. 1-9 ◽

Cited By ~ 3

Author(s):

E. H. Doha ◽

D. Baleanu ◽

A. H. Bhrawy ◽

R. M. Hafez

Keyword(s):

Numerical Solutions ◽

Rational Functions ◽

Absolute Error ◽

Delay Systems ◽

Infinite Interval ◽

Algebraic Equations ◽

Collocation Points ◽

Nonlinear Algebraic Equations ◽

Linear And Nonlinear ◽

Pseudospectral Scheme

A new Legendre rational pseudospectral scheme is proposed and developed for solving numerically systems of linear and nonlinear multipantograph equations on a semi-infinite interval. A Legendre rational collocation method based on Legendre rational-Gauss quadrature points is utilized to reduce the solution of such systems to systems of linear and nonlinear algebraic equations. In addition, accurate approximations are achieved by selecting few Legendre rational-Gauss collocation points. The numerical results obtained by this method have been compared with various exact solutions in order to demonstrate the accuracy and efficiency of the proposed method. Indeed, for relatively limited nodes used, the absolute error in our numerical solutions is sufficiently small.

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Orthonormal Ultraspherical Operational Matrix Algorithm for Fractal–Fractional Riccati Equation with Generalized Caputo Derivative

Fractal and Fractional ◽

10.3390/fractalfract5030100 ◽

2021 ◽

Vol 5 (3) ◽

pp. 100

Author(s):

Youssri Hassan Youssri

Keyword(s):

Differential Equation ◽

Operational Matrix ◽

Initial Guess ◽

Test Problems ◽

Tau Method ◽

Algebraic Equations ◽

Differential Problem ◽

Riccati Differential Equation ◽

Ultraspherical Polynomials ◽

Expansion Coefficients

Herein, we developed and analyzed a new fractal–fractional (FF) operational matrix for orthonormal normalized ultraspherical polynomials. We used this matrix to handle the FF Riccati differential equation with the new generalized Caputo FF derivative. Based on the developed operational matrix and the spectral Tau method, the nonlinear differential problem was reduced to a system of algebraic equations in the unknown expansion coefficients. Accordingly, the resulting system was solved by Newton’s solver with a small initial guess. The efficiency, accuracy, and applicability of the developed numerical method were checked by exhibiting various test problems. The obtained results were also compared with other recent methods, based on the available literature.

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The Bernstein Operational Matrices for Solving the Fractional Quadratic Riccati Differential Equations with the Riemann-Liouville Derivative

Abstract and Applied Analysis ◽

10.1155/2013/461970 ◽

2013 ◽

Vol 2013 ◽

pp. 1-7 ◽

Cited By ~ 8

Author(s):

Dumitru Baleanu ◽

Mohsen Alipour ◽

Hossein Jafari

Keyword(s):

Differential Equation ◽

Approximate Analytical Solution ◽

Fractional Integration ◽

Bernstein Polynomials ◽

Operational Matrix ◽

Operational Matrices ◽

Algebraic Equations ◽

Riccati Differential Equation ◽

Riccati Differential Equations ◽

Liouville Derivative

We obtain the approximate analytical solution for the fractional quadratic Riccati differential equation with the Riemann-Liouville derivative by using the Bernstein polynomials (BPs) operational matrices. In this method, we use the operational matrix for fractional integration in the Riemann-Liouville sense. Then by using this matrix and operational matrix of product, we reduce the problem to a system of algebraic equations that can be solved easily. The efficiency and accuracy of the proposed method are illustrated by several examples.

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OPTIMAL CHOICE OF THE METHOD FOR INVESTIGATING MULTIPARAMETER SISTEMS OF LINEAR AND NONLINEAR ALGEBRAIC EQUATIONS

Scientific and Technical Volga region Bulletin ◽

10.24153/2079-5920-2018-8-5-13-16 ◽

2018 ◽

Vol 8 (5) ◽

pp. 13-16

Author(s):

I.Х. Enikeev ◽

Keyword(s):

Optimal Choice ◽

Algebraic Equations ◽

Nonlinear Algebraic Equations ◽

Linear And Nonlinear

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On the properties of numerical solutions of dynamical systems obtained using the midpoint method

Discrete and Continuous Models and Applied Computational Science ◽

10.22363/2658-4670-2019-27-3-242-262 ◽

2019 ◽

Vol 27 (3) ◽

pp. 242-262 ◽

Cited By ~ 1

Author(s):

Vladimir P. Gerdt ◽

Mikhail D. Malykh ◽

Leonid A. Sevastianov ◽

Yu Ying

Keyword(s):

Difference Scheme ◽

Coupled Oscillators ◽

Numerical Solutions ◽

Nonlinear Oscillators ◽

Approximate Solutions ◽

Algebraic Equations ◽

Integrals Of Motion ◽

Midpoint Method ◽

Nonlinear Algebraic Equations ◽

The Difference

The article considers the midpoint scheme as a finite-difference scheme for a dynamical system of the form ̇ = (). This scheme is remarkable because according to Cooper’s theorem, it preserves all quadratic integrals of motion, moreover, it is the simplest scheme among symplectic Runge-Kutta schemes possessing this property. The properties of approximate solutions were studied in the framework of numerical experiments with linear and nonlinear oscillators, as well as with a system of several coupled oscillators. It is shown that in addition to the conservation of all integrals of motion, approximate solutions inherit the periodicity of motion. At the same time, attention is paid to the discussion of introducing the concept of periodicity of an approximate solution found by the difference scheme. In the case of a nonlinear oscillator, each step requires solving a system of nonlinear algebraic equations. The issues of organizing computations using such schemes are discussed. Comparison with other schemes, including those symmetric with respect to permutation of and .̂

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On the continuous analogue of the Seidel method

Zhurnal Srednevolzhskogo Matematicheskogo Obshchestva ◽

10.15507/2079-6900.20.201804.364-377 ◽

2018 ◽

Vol 20 (4) ◽

pp. 364-377

Author(s):

I. V. Boikov ◽

A. I. Boikova

Keyword(s):

Differential Equations ◽

Systems Of Nonlinear Equations ◽

Algebraic Equations ◽

Seidel Method ◽

Linear Algebraic Equations ◽

Nonlinear Algebraic Equations ◽

Continuous Analogue ◽

Solution Methods ◽

Linear And Nonlinear ◽

Equations With Delay

Continuous Seidel method for solving systems of linear and nonlinear algebraic equations is constructed in the article, and the convergence of this method is investigated. According to the method discussed, solving a system of algebraic equations is reduced to solving systems of ordinary differential equations with delay. This allows to use rich arsenal of numerical ODE solution methods while solving systems of algebraic equations. The main advantage of the continuous analogue of the Seidel method compared to the classical one is that it does not require all the elements of the diagonal matrix to be non-zero while solving linear algebraic equations’ systems. The continuous analogue has the similar advantage when solving systems of nonlinear equations.

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A modified variational iteration method for solving fractional Riccati differential equation by Adomian polynomials

Fractional Calculus and Applied Analysis ◽

10.2478/s13540-013-0008-9 ◽

2013 ◽

Vol 16 (1) ◽

Cited By ~ 33

Author(s):

Hossein Jafari ◽

Hale Tajadodi ◽

Dumitru Baleanu

Keyword(s):

Differential Equation ◽

Iteration Method ◽

Variational Iteration Method ◽

Approximate Solutions ◽

Convergence Region ◽

Riccati Differential Equation ◽

Adomian Polynomials ◽

Riccati Differential Equations ◽

Variational Iteration ◽

Modified Variational Iteration Method

AbstractIn this paper, we introduce a modified variational iteration method (MVIM) for solving Riccati differential equations. Also the fractional Riccati differential equation is solved by variational iteration method with considering Adomians polynomials for nonlinear terms. The main advantage of the MVIM is that it can enlarge the convergence region of iterative approximate solutions. Hence, the solutions obtained using the MVIM give good approximations for a larger interval. The numerical results show that the method is simple and effective.

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Numerical Solutions for Multi-Term Fractional Order Differential Equations with Fractional Taylor Operational Matrix of Fractional Integration

Mathematics ◽

10.3390/math8010096 ◽

2020 ◽

Vol 8 (1) ◽

pp. 96 ◽

Cited By ~ 1

Author(s):

İbrahim Avcı ◽

Nazim I. Mahmudov

Keyword(s):

Differential Equations ◽

Numerical Solutions ◽

Fractional Integration ◽

Operational Matrix ◽

Main Idea ◽

System Of Equations ◽

Algebraic Equations ◽

Fractional Order Differential Equations ◽

Fractional Differential ◽

The Given

In this article, we propose a numerical method based on the fractional Taylor vector for solving multi-term fractional differential equations. The main idea of this method is to reduce the given problems to a set of algebraic equations by utilizing the fractional Taylor operational matrix of fractional integration. This system of equations can be solved efficiently. Some numerical examples are given to demonstrate the accuracy and applicability. The results show that the presented method is efficient and applicable.

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Existence of Solutions of Riccati Differential Equations

Journal of Dynamic Systems Measurement and Control ◽

10.1115/1.4005496 ◽

2012 ◽

Vol 134 (3) ◽

Cited By ~ 8

Author(s):

Sinan Kilicaslan ◽

Stephen P. Banks

Keyword(s):

Differential Equation ◽

Nonlinear Systems ◽

Existence Of Solutions ◽

Linear Time ◽

Sufficient Conditions ◽

Necessary Condition ◽

Time Varying ◽

Riccati Differential Equation ◽

Riccati Differential Equations ◽

Time Varying Systems

A necessary condition for the existence of the solution of the Riccati differential equation for both linear, time varying systems and nonlinear systems is introduced. First, a necessary condition for the existence of the solution of the Riccati differential equation for linear, time varying systems is proposed. Then, the sufficient conditions to satisfy the necessary condition are given. After that, the existence of the solution of the Riccati differential equation is generalized for nonlinear systems.

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