Numerical Approach of Fractional Abel Differential Equation by Genocchi Polynomials

2019 ◽  
Vol 5 (5) ◽  
Author(s):  
Fariba Rigi ◽  
Haleh Tajadodi
Keyword(s):  
Differential Equation ◽  
Numerical Approach ◽  
Genocchi Polynomials ◽  
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.

An Algorithm for the Approximate Solution of the Fractional Riccati Differential Equation

2019 ◽  
Vol 20 (6) ◽  
pp. 661-674
Author(s):  
S. S. Ezz-Eldien ◽  
J. A. T. Machado ◽  
Y. Wang ◽  
A. A. Aldraiweesh
Keyword(s):  
Differential Equation ◽  
Approximate Solution ◽  
Chebyshev Polynomials ◽  
Unknown Function ◽  
Numerical Approach ◽  
Continuous Functions ◽  
Basis Functions ◽  
Real Constant ◽  
Riccati Differential Equation ◽  
Suggested Approach

AbstractThis manuscript develops a numerical approach for approximating the solution of the fractional Riccati differential equation (FRDE): $$\begin{align*}D^{\mu}&u(x)+a(x) u^2(x)+b(x) u(x)= g(x),\quad 0\leq \mu \leq 1,\quad 0\leq x \leq t,\\&u(0)=d,\end{align*}$$where u(x) is the unknown function, a(x), b(x) and g(x) are known continuous functions defined in [0,t] and d is a real constant. The proposed method is applied for solving the FRDE with shifted Chebyshev polynomials as basis functions. In addition, the convergence analysis of the suggested approach is investigated. The efficiency of the algorithm is demonstrated by means of several examples and the results compared with those given using other numerical schemes.

Numerical approach to the Caputo derivative of the unknown function

Open Physics ◽  
2013 ◽  
Vol 11 (10) ◽  
Author(s):  
Fanhai Zeng ◽  
Changpin Li
Keyword(s):  
Differential Equation ◽  
Numerical Method ◽  
Differential System ◽  
Numerical Algorithm ◽  
Unknown Function ◽  
Caputo Derivative ◽  
Order Differential Equation ◽  
Numerical Approach ◽  
Numerical Examples ◽  
Integer Order

AbstractIf a function can be explicitly expressed, then one can easily compute its Caputo derivative by the known methods. If a function cannot be explicitly expressed but it satisfies a differential equation, how to seek Caputo derivative of such a function has not yet been investigated. In this paper, we propose a numerical algorithm for computing the Caputo derivative of a function defined by a classical (integer-order) differential equation. By the properties of Caputo derivative derived in this paper, we can change the original typical differential system into an equivalent Caputo-type differential system. Numerical examples are given to support the derived numerical method.

Investigating Thermal Stability in a Two-Step Convective Radiating Cylindrical Pipe

2021 ◽  
Vol 408 ◽  
pp. 99-107
Author(s):  
Ramoshweu Solomon Lebelo ◽  
Radley Kebarapetse Mahlobo ◽  
Samuel Olumide Adesanya
Keyword(s):  
Differential Equation ◽  
Thermal Stability ◽  
Kinetic Parameters ◽  
Numerical Solutions ◽  
Combustion Process ◽  
Numerical Approach ◽  
Cylindrical Pipe ◽  
Surrounding Environment ◽  
Exothermic Chemical Reaction ◽  
Governing Differential Equation

Thermal stability in a stockpile of reactive materials is analyzed in this article. The combustion process is modelled in a long cylindrical pipe that is assumed to lose heat to the surrounding environment by convection and radiation. The study of effects of different kinetic parameters embedded on the governing differential equation, makes it easier to investigate the complicated combustion process. The combustion process results with nonlinear molecular interactions and as a result it is not easy to solve the differential equation exactly, and therefore the numerical approach by using the Finite Difference Method (FDM) is applied. The numerical solutions are depicted graphically for each parameter’s effect on the temperature of the system. In general, the results indicate that kinetic parameters like the reaction rate promote the exothermic chemical reaction process by increasing the temperature profiles, whilst kinetic parameters such as the order of the reaction show the tendency to retard the combustion process by lowering the temperature of the system.

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