Quadratic Riccati Differential Equation Research Articles

Efficient Numerical Method for Solving a Quadratic Riccati Differential Equation

This study presents families of the fourth-order Runge–Kutta methods for solving a quadratic Riccati differential equation. From these families, the England version is more efficient than other fourth-order Runge–Kutta methods and practically well-suited for solving initial value problems in general and quadratic Riccati differential equation in particular. The stability analysis of the present method is well-established. In order to verify the accuracy, we compared the numerical solutions obtained using the England version of fourth-order Runge–Kutta method with the recently published works reported in the literature. Several counter examples are solved using the present methods to demonstrate their reliability and efficiency.

Cubic B-Spline method for the solution of the quadratic Riccati differential equation

<abstract> <p>The quadratic Riccati equations are first-order nonlinear differential equations with numerous applications in various applied science and engineering areas. Therefore, several numerical approaches have been derived to find their numerical solutions. This paper provided the approximate solution of the quadratic Riccati equation via the cubic b-spline method. The convergence analysis of the method is discussed. The efficiency and applicability of the proposed approach are verified through three numerical test problems. The obtained results are in good settlement with the exact solutions. Moreover, the numerical results indicate that the proposed cubic b-spline method attains a superior performance compared with some existing methods.</p> </abstract>

Approximate solutions for the fractional order quadratic Riccati and Bagley-Torvik differential equations

The Galerkin method is presented and applied for getting semi-analytical solutions of quadratic Riccati and Bagley-Torvik differential equations in fractional order. New theorems are proved to minimize the generated residual after invoking the Legendre polynomials as a basis in the Galerkin method. The proposed method is compared with other methods by solving some initial value problems of different fractional orders. The comparisons and results are illustrated via tables and figures. It can be concluded that the Legendre-Galerkin method is convenient for these problems due to its efficiency and reliability.

Eighth order Predictor-Corrector Method to Solve Quadratic Riccati Differential Equations

In this paper, the eighth-order predictor-corrector method is presented for solving quadratic Riccati differential equations. First, the interval is discretized and then the method is formulated by using Newton’s backward difference interpolation formula. The stability and convergence of the method have been investigated. To validate the applicability of the proposed method, two model examples with exact solutions have been considered and numerically solved. Maximum absolute errors are presented in tables and figures for different values of mesh size h and the present method gives better results than some existing numerical methods reported in the literature.

A New Approach to Solving Fuzzy Quadratic Riccati Differential Equations

Moa’ath N. Oqielat, Ahmad El-Ajou, Zeyad Al-Zhour, Tareq Eriqat, and Mohammed Al-Smadi;. International Journal of Fuzzy Logic and Intelligent Systems 2022;22:23-47. https://doi.org/10.5391/IJFIS.2022.22.1.23

Analysis of Nonlinear Active Noise Behavior of Fuzzy Controller Using Non-Perturbation Methods

In this paper, a Fuzzy controller model has been converted into a time-dependent nonlinear model and then quadratic Riccati differential equation was analyzed to satisfy the solution of the nonlinear active noise behavior of Fuzzy controller. Further, the approximate solutions of this equation using non-perturbation methods i.e., adomian decomposition method (ADM), variational iterational method (VIM) and homotopy perturbation method (HPM) were investigated. A comparison of these methods has also been given with tabular and graphical presentations. Our results reveal that VIM provides the closest approximate solution and fast convergence for the proposed model as compare to ADM and HPM.

Intrinsical localization of both topological (anti-kink) envelope and gray (black) gap solitons of the condensed bosons in deep optical lattices.

By developing quasi-discrete multiple-scale method combined with tight-binding approximation, a novel quadratic Riccati differential equation is first derived for the soliton dynamics of the condensed bosons trapped in the optical lattices. For a lack of exact solutions, the trial solutions of the Riccati equation have been analytically explored for the condensed bosons with various scattering length as. When the lattice depth is rather shallow, the results of sub-fundamental gap solitons are in qualitative agreement with the experimental observation. For the deeper lattice potentials, we predict that in the case of as>0, some novel intrinsically localized modes of symmetrical envelope, topological (kink) envelope, and anti-kink envelope solitons can be observed within the bandgap in the system, of which the amplitude increases with the increasing lattice spacing and (or) depth. In the case of as<0, the bandgap brings out intrinsically localized gray or black soliton. This well provides experimental protocols to realize transformation between the gray and black solitons by reducing light intensity of the laser beams forming optical lattice.

Solving a Quadratic Riccati Differential Equation, Multi-Pantograph Delay Differential Equations, and Optimal Control Systems with Pantograph Delays

An effective algorithm for solving quadratic Riccati differential equation (QRDE), multipantograph delay differential equations (MPDDEs), and optimal control systems (OCSs) with pantograph delays is presented in this paper. This technique is based on Genocchi polynomials (GPs). The properties of Genocchi polynomials are stated, and operational matrices of derivative are constructed. A collocation method based on this operational matrix is used. The findings show that the technique is accurate and simple to use.

Solution of Nonlinear Fractional Quadratic Riccati Differential Equations Using Perturbation Method

Homotopy perturbation transform method (HPSTM) in conjunction of Adomian and He’s polynomials has been devoted to obtain approximate solution of fractional order quadratic Riccati equations for arbitrary order $$ \alpha \in (0,1] $$. The suggested method finds the solution without any discontinuation and averts the round-off errors. We have taken three examples to justify the accuracy of the method with the help of Adomian polynomials based on Newton–Raphson’s scheme. The approximate solutions are compared with exact solutions.

APPROXIMATE SOLUTION TO FRACTIONAL RICCATI DIFFERENTIAL EQUATIONS

In this paper, quadratic Riccati differential equation of fractional order has been solved by employing the optimal homotopy asymptotic method (Optimal HAM) with application to random processes, optimal control and diffusion problems. Optimal HAM uses simple computations with quite acceptable approximate solutions which have close agreement with exact solutions as compared to other techniques. To illustrate the efficiency and reliability of the method, some examples are provided and the results are discussed with tables.

An Application of Picard Iteration Method to Fractional Quadratic Ricccati Differential Equations

In this paper, we present an algorithm of the Picard's Iteration method to solve Fractional Quadratic Riccati differential equations with Caputo-derivatives. The non-linear terms are easily and simply expanded using the traditional method of expansion. Four (4) illustrative examples are given to verify the reliability and efficiency of the method. The approximate solutions obtained compare favorably with the exact solutions and the approximate solutions obtained by other numerical methods in the literature.

An Application of Picard Iteration Method to Fractional Quadratic Ricccati Differential Equations

In this paper, we present an algorithm of the Picard's Iteration method to solve Fractional Quadratic Riccati differential equations with Caputo-derivatives. The non-linear terms are easily and simply expanded using the traditional method of expansion. Four (4) illustrative examples are given to verify the reliability and efficiency of the method. The approximate solutions obtained compare favorably with the exact solutions and the approximate solutions obtained by other numerical methods in the literature.

Fifth Order Predictor-Corrector Method for Solving Quadratic Riccati Differential Equations

In this paper, fifth order predictor-corrector method is presented for solving quadratic Riccati differential equations. First, the interval is discretized and then the method is formulated by using the Newton’s backward difference interpolation formula. The stability and convergence of the method have been investigated. To validate the applicability of the proposed method, three model examples with exact solutions have been considered and numerically solved by using MATLAB software. The numerical results are presented in tables and figures for different values of mesh size h. Pointwise absolute errors and maximum absolute errors are also estimated. Concisely, the present method gives better result than some existing numerical methods reported in the literature.

Analytical approximate solutions for quadratic Riccati differential equation of fractional order using the Polynomial Least Squares Method

In this paper the Polynomial Least Squares Method is applied in order to compute analytical approximate polynomial solutions for the fractional Riccati type differential equations. The accuracy of the method is tested by means of the comparison with previous results for several applications.

Numerical solution of quadratic Riccati differential equations

The quadratic Riccati differential equations are part of non-linear differential equations which have many applications. This paper introduces the classical fourth order Runge Kutta method (RK4) for solving the numerical solution of the quadratic Riccati differential equations. To validate the applicability of the method on the proposed equation, some model examples have been solved for different values of mesh sizes. The numerical results in terms of point wise absolute errors presented in tables and graphs show that the present method approximates the exact solution very well.

Newton-Raphson based modified Laplace Adomian decomposition method for solving quadratic Riccati differential equations

Numerical Laplace transform method is applied to approximate the solution of nonlinear (quadratic) Riccati differential equations mingled with Adomian decomposition method. A new technique is proposed in this work by reintroducing the unknown function in Adomian polynomial with that of well known Newton-Raphson formula. The solutions obtained by the iterative algorithm are exhibited in an infinite series. The simplicity and efficacy of method is manifested with some examples in which comparisons are made among the exact solutions, ADM (Adomian decomposition method), HPM (Homotopy perturbation method), Taylor series method and the proposed scheme.

Approximate solution for quadratic Riccati differential equation

The quadratic Riccati differential equations are a class of nonlinear differential equations of much importance, and play a significant role in many fields of applied science. This paper introduces an efficient method for solving the quadratic Riccati differential equation and the Riccati differential-difference equation. In this technique, the Bezier curves method is considered as an algorithm to find the approximate solution of the nonlinear Riccati equation. Some examples in different cases are given to demonstrate simplicity and efficiency of the proposed method.

Modification Of The Hpm By Using Optimal Newton Interpolation Polynomial For Quadratic Riccati Differential Equation

In this work, an efficient modification of the homotopy analysis method by using optimal Newton interpolation polynomials is given for the approximate solutions of the Riccati differential equations. This presented method can be applied to linear and nonlinear models. Examples show that the method is effective.

A new efficient method for solving quadratic Riccati differential equation

A new efficient method called the multistage variational iteration method (MVIM) is applied to the solution of quadratic Riccati differential equations. A comparison between MVIM solution with classical variational iteration method (VIM) and exact solution has been made and show that the MVIM is a powerful method to the solution of nonlinear differential equations.

An efficient computational intelligence approach for solving fractional order Riccati equations using ANN and SQP

A new computational intelligence technique is presented for solution of non-linear quadratic Riccati differential equations of fractional order based on artificial neural networks (ANNs) and sequential quadratic programming (SQP). The power of feed forward ANNs in an unsupervised manner is exploited for mathematical modeling of the equation; training of weights is carried out with an efficient constrained optimization technique based on the SQP algorithm. The proposed scheme is evaluated on two initial value problems of the Riccati fractional order equation with integer and non-integer derivatives. Comparison of results with the exact solution, and with reference numerical methods demonstrates the correctness of the proposed methodology. Performance of the proposed scheme is also validated using results of statistical analysis based on a sufficiently large number of independent runs.