scholarly journals Solution of the parametric center problem for the Abel differential equation

2017 ◽  
Vol 19 (8) ◽  
pp. 2343-2369 ◽  
Author(s):  
Fedor Pakovich
Keyword(s):  
Differential Equation ◽  
Center Problem ◽  
Abel Differential Equation

scholarly journals An attractive numerical algorithm for solving nonlinear Caputo–Fabrizio fractional Abel differential equation in a Hilbert space

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Mohammed Al-Smadi ◽  
Nadir Djeddi ◽  
Shaher Momani ◽  
Shrideh Al-Omari ◽  
Serkan Araci
Keyword(s):  
Differential Equation ◽  
Hilbert Space ◽  
Reproducing Kernel ◽  
Numerical Approach ◽  
Accurate Solution ◽  
Stability And Convergence ◽  
Type Model ◽  
Schmidt Orthogonalization ◽  
And Performance ◽  
Abel Differential Equation

AbstractOur aim in this paper is presenting an attractive numerical approach giving an accurate solution to the nonlinear fractional Abel differential equation based on a reproducing kernel algorithm with model endowed with a Caputo–Fabrizio fractional derivative. By means of such an approach, we utilize the Gram–Schmidt orthogonalization process to create an orthonormal set of bases that leads to an appropriate solution in the Hilbert space $\mathcal{H}^{2}[a,b]$ H 2 [ a , b ] . We investigate and discuss stability and convergence of the proposed method. The n-term series solution converges uniformly to the analytic solution. We present several numerical examples of potential interests to illustrate the reliability, efficacy, and performance of the method under the influence of the Caputo–Fabrizio derivative. The gained results have shown superiority of the reproducing kernel algorithm and its infinite accuracy with a least time and efforts in solving the fractional Abel-type model. Therefore, in this direction, the proposed algorithm is an alternative and systematic tool for analyzing the behavior of many nonlinear temporal fractional differential equations emerging in the fields of engineering, physics, and sciences.

THE CENTER PROBLEM FOR DISCONTINUOUS LIÉNARD DIFFERENTIAL EQUATION

1999 ◽  
Vol 09 (09) ◽  
pp. 1751-1761 ◽  
Author(s):  
B. COLL ◽  
R. PROHENS ◽  
A. GASULL
Keyword(s):  
Differential Equation ◽  
Vector Field ◽  
Differential Equations ◽  
General Expression ◽  
The Other ◽  
Center Problem ◽  
Homogeneous Polynomials ◽  
Lyapunov Constants

We prove that the Lyapunov constants for differential equations given by a vector field with a line of discontinuities are quasi-homogeneous polynomials. This property is strongly used to derive the general expression of the Lyapunov constants for two families of discontinuous Liénard differential equations, modulus some unknown coefficients. In one of the families these coefficients are determined and the center problem is solved. In the other family the center problem is just solved by assuming that the coefficients which appear in these expressions are nonzero. This assumption on the coefficients is supported by their computation (analytical and numerical) for several cases.

scholarly journals The Integrating Factors for Riccati and Abel Differential Equations

2015 ◽  
Vol 7 (2) ◽  
pp. 125
Author(s):  
Chein-Shan Liu
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Nonlinear Differential Equations ◽  
Type Systems ◽  
Second Order ◽  
Numerical Schemes ◽  
Riccati Differential Equation ◽  
Group Preserving Scheme ◽  
Abel Differential Equation ◽  
Abel Differential Equations

We can recast the Riccati and Abel differential equationsinto new forms in terms of introduced integrating factors.Therefore, the Lie-type systems endowing with transformation Lie-groups$SL(2,{\mathbb R})$ can be obtained.The solution of second-order linearhomogeneous differential equation is an integrating factorof the corresponding Riccati differential equation.The numerical schemes which are developed to fulfil the Lie-group property have better accuracy and stability than other schemes.We demonstrate that upon applying the group-preserving scheme (GPS) to the logistic differential equation, it is not only qualitatively correct for all values of time stepsize $h$, and is also the most accurate one among all numerical schemes compared in this paper.The group-preserving schemes derived for the Riccati differential equation, second-order linear homogeneous and non-homogeneous differential equations, the Abel differential equation and higher-order nonlinear differential equations all have accuracy better than $O(h^2)$.

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