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

Our 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.

Polynomial Solutions of Equivariant Polynomial Abel Differential Equations

Let {a(x)} be non-constant and let {b_{j}(x)} , for {j=0,1,2,3} , be real or complex polynomials in the variable x. Then the real or complex equivariant polynomial Abel differential equation {a(x)\dot{y}=b_{1}(x)y+b_{3}(x)y^{3}} , with {b_{3}(x)\neq 0} , and the real or complex polynomial equivariant polynomial Abel differential equation of the second kind {a(x)y\dot{y}=b_{0}(x)+b_{2}(x)y^{2}} , with {b_{2}(x)\neq 0} , have at most 7 polynomial solutions. Moreover, there exist equations of this type having this maximum number of polynomial solutions.

Solution of the problem of the theoretical profile of non-dimensional speed on the thickness of the boundary layer at the turbulent flow in the boundary layer based on the solution of the differential equation of Abel of the second generation with the application of the Lambert function

В статье было найдено точное аналитическое решение дифференциального уравнения для касательных напряжений в турбулентном пограничном слое, являющихся частным случаем т.н. дифференциального уравнения Абеля второго рода, полученное с помощью специальной функции Ламберта, в то время как ранее считалось, что оно не разрешимо в квадратурах. Кроме этого, были получены ещё несколько важных решённых частных случаев этого уравнения. Полученные в статье аналитические решения преимущественно отличаются от имеющихся ранее либо численных, либо приближённых решений задачи. Полученное решение в безразмерном виде представляет собой теоретический профиль безразмерной скорости по толщине пограничного слоя при турбулентном течении в пограничном слое. An exact analytical solution of the differential equation for tangential stresses in a turbulent boundary layer, which is a special case of the so-called " of the Abel differential equation of the second kind, obtained with the help of the special Lambert function, whereas previously it was assumed that it is not solvable in quadratures. In addition, several more important solved special cases of this equation were obtained. The analytic solutions obtained in the paper are predominantly different from the previously available either numerical or approximate solutions of the problem. The solution obtained in dimensionless form is the theoretical profile of the dimensionless velocity along the thickness of the boundary layer for turbulent flow in the boundary layer.

The Integrating Factors for Riccati and 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 $O(h^2)$.