Approximate solution for quadratic Riccati differential equation

Abstract

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.

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

“…The study is implemented on three model examples with exact solutions by taking different values for N, and the computational results are presented in the Tables (1)(2)(3)(4)(5)(6)(7)(8). The results obtained by the present method are compared with the results of Gemechis and Tasfaye [4], Vinod and Dimple [5], Fateme and Esmaile [6] and shows betterment. Furthermore, from the Tables (1-5) it is significant that all of the absolute errors decrease rapidly as h decreases, which in turn shows the convergence of the computed solution.…”

“…Fateme and Esmaile [6] presented approximate solution for quadratic Riccati differential equations. Other authors [7][8][9][10][11][12][13] also discussed the quadratic Riccati differential equations.…”

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

“…x Error of y 0.1 0.01576255020 0.2 0.01957152970 0.3 0.01520160000 0.4 0.007259874300 0.5 7.000000000 × 10 −10 0.6 0.003753795400 0.7 0.003263092300 0.8 3.000000000 × 10 −10 0.9 0.002664043000 1.0 0.0 Example 2. Second, the following RDE is considered (see [15])…”

“…The approximate and exact solution for y(x) are shown in Figure 2. Table 2 demonstrates the absolute error of the this technique and stated technique in [15]. [15])…”

A computational method for solving differential equations with quadratic nonlinearity by using Bernoulli polynomials

“…Non-linear differential equations are of great importance as a model of many physical, chemical, and technical problems such as fluid dynamics, optimal control, diffusion problems, electrical networks, nuclear physics, financial mathematics, and gas dynamics [1][2][3]. Researchers have developed a number of approximate solution methods since it is difficult to find the exact solutions of such problems in recent years [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18].…”