scholarly journals Elementary Quadrature for the Riccati equation

2020 ◽  
Author(s):  
Ji-Xiang Zhao
Keyword(s):  
Differential Equations ◽  
Riccati Equation ◽  
Analytical Method ◽  
Nonlinear Differential Equations ◽  
Integrability Condition ◽  
Type Equation ◽  
Abel Equation ◽  
General Solutions ◽  
Suitable Transformation ◽  
Elementary Quadrature

Abstract Using suitable transformation in combination with a specific Riccati-type equation solvable, the problem of solving Riccati equation can be transformed into that of a quasi-Abel equation of the second kind. By the extended Julia’s integrability condition, the general solutions of Riccati equation in the form of elementary quadrature are obtained, which contains numerous. This method opens up a new prospect for the study of nonlinear differential equations by analytical method.

scholarly journals The Riccati Equation and the Hidden Parameter &Variable

2021 ◽  
Author(s):  
Ji-Xiang Zhao
Keyword(s):  
General Solution ◽  
Differential Equations ◽  
Riccati Equation ◽  
Stokes Equation ◽  
Type Equation ◽  
Navier Stokes ◽  
Navier Stokes Equation ◽  
Elementary Quadrature ◽  
Linear Odes ◽  
Hidden Parameter

Abstract Using suitable function transformation in combination with a specific Riccati-type equation solvable, general solution of the Riccati equation in the form of elementary quadrature is given. In the process of solving the Riccati equation, the hidden parameter and variable are discovered. This indicates that hidden parameter & variable exist in all differential equations associated with the Riccati equation, such as the second-order linear ODEs, the Schrödinger equation and the Navier–Stokes equation.

Sine–cosine wavelets operational matrix method for fractional nonlinear differential equation

2019 ◽  
Vol 17 (04) ◽  
pp. 1950026 ◽  
Author(s):  
Umer Saeed
Keyword(s):  
Differential Equations ◽  
Nonlinear Differential Equations ◽  
Operational Matrix ◽  
Type Equation ◽  
Operational Matrices ◽  
Algebraic Equations ◽  
Operational Matrix Method ◽  
Fractional Differential ◽  
Quasilinearization Technique ◽  
Distribution Equation

In this paper, we present a solution method for fractional nonlinear ordinary differential equations. We propose a method by utilizing the sine–cosine wavelets (SCWs) in conjunction with quasilinearization technique. The fractional nonlinear differential equations are transformed into a system of discrete fractional differential equations by quasilinearization technique. The operational matrices of fractional order integration for SCW are derived and utilized to transform the obtained discrete system into systems of algebraic equations and the solutions of algebraic systems lead to the solution of fractional nonlinear differential equations. Convergence analysis and procedure of implementation for the proposed method are also considered. To illustrate the reliability and accuracy of the method, we tested the method on fractional nonlinear Lane–Emden type equation and temperature distribution equation.

Analytical solution of nonlinear differential equations two oscillators mechanism using Akbari–Ganji method

2021 ◽  
pp. 2150462 ◽  
Author(s):  
Saman Hosseinzadeh ◽  
Kh. Hosseinzadeh ◽  
M. Rahai ◽  
D. D. Ganji
Keyword(s):  
Differential Equations ◽  
Analytical Method ◽  
Homotopy Perturbation Method ◽  
Nonlinear Differential Equations ◽  
Variational Iteration Method ◽  
Approximate Analytic Solution ◽  
Energy Balance Method ◽  
Wide Range ◽  
Vibration Problems ◽  
Mechanical Oscillators

In the last decade, many potent analytical methods have been utilized to find the approximate solution of nonlinear differential equations. Some of these methods are energy balance method (EBM), homotopy perturbation method (HPM), variational iteration method (VIM), amplitude frequency formulation (AFF), and max–min approach (MMA). Besides the methods mentioned above, the Akbari–Ganji method (AGM) is a highly efficient analytical method to solve a wide range of nonlinear equations, including heat transfer, mass transfer, and vibration problems. In this study, it was constructed the approximate analytic solution for movement of two mechanical oscillators by employing the AGM. In the derived analytical method, both oscillator motion equations and the sensitivity analysis of the frequency were included. The AGM was validated through comparison against Runge–Kutta fourth-order numerical method and an excellent agreement was achieved. Based on the results, the highest sensitivity of the oscillation frequency is related to the mass. As [Formula: see text] and [Formula: see text] increase, the slope of the system velocity and acceleration will increase.

scholarly journals On the Generalized Simplest Equations: Toward the Solution of Nonlinear Differential Equations with Variable Coefficients

2021 ◽  
Author(s):  
Gunawan Nugroho ◽  
Purwadi Agus Darwito ◽  
Ruri Agung Wahyuono ◽  
Murry Raditya
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Riccati Equation ◽  
Nonlinear Differential Equation ◽  
Nonlinear Differential Equations ◽  
Variable Coefficients ◽  
Higher Order ◽  
First Order ◽  
Generalized Riccati Equation ◽  
Polynomial Differential Equation

The simplest equations with variable coefficients are considered in this research. The purpose of this study is to extend the procedure for solving the nonlinear differential equation with variable coefficients. In this case, the generalized Riccati equation is solved and becomes a basis to tackle the nonlinear differential equations with variable coefficients. The method shows that Jacobi and Weierstrass equations can be rearranged to become Riccati equation. It is also important to highlight that the solving procedure also involves the reduction of higher order polynomials with examples of Korteweg de Vries and elliptic-like equations. The generalization of the method is also explained for the case of first order polynomial differential equation.

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