A fractional‐order hybrid system of differential equations: Existence theory and numerical solutions

2021 ◽  
Author(s):  
Anwar Shah ◽  
Rahmat Ali Khan ◽  
Hasib Khan
Keyword(s):  
Differential Equations ◽  
Fractional Order ◽  
Hybrid System ◽  
Numerical Solutions ◽  
Existence Theory ◽  
System Of Differential Equations

scholarly journals Numerical Solutions of Coupled Systems of Fractional Order Partial Differential Equations

2017 ◽  
Vol 2017 ◽  
pp. 1-14 ◽  
Author(s):  
Yongjin Li ◽  
Kamal Shah
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Fractional Order ◽  
Numerical Solutions ◽  
Approximate Solutions ◽  
Coupled Systems ◽  
Test Problems ◽  
Operational Matrices ◽  
Partial Differential ◽  
Established Technique

We develop a numerical method by using operational matrices of fractional order integrations and differentiations to obtain approximate solutions to a class of coupled systems of fractional order partial differential equations (FPDEs). We use shifted Legendre polynomials in two variables. With the help of the aforesaid matrices, we convert the system under consideration to a system of easily solvable algebraic equation of Sylvester type. During this process, we need no discretization of the data. We also provide error analysis and some test problems to demonstrate the established technique.

scholarly journals Numerical behavior of a fractional order dynamical model of RNA silencing

2016 ◽  
Vol 4 (2) ◽  
pp. 52
Author(s):  
A.M.A. El-Sayed ◽  
M. Khalil ◽  
A.A.M. Arafa ◽  
Amaal Sayed
Keyword(s):  
Differential Equations ◽  
Numerical Simulations ◽  
Detailed Analysis ◽  
Fractional Order ◽  
Rna Silencing ◽  
Fractional Derivatives ◽  
Numerical Solutions ◽  
The Stability ◽  
Predictor Corrector ◽  
With Memory

A class of fractional-order differential models of RNA silencing with memory is presented in this paper. We also carry out a detailed analysis on the stability of equilibrium and we show that the model established in this paper possesses non-negative solutions. Numerical solutions are obtained using a predictor-corrector method to handle the fractional derivatives. The fractional derivatives are described in the Caputo sense. Numerical simulations are presented to illustrate the results. Also, the numerical simulations show that, modeling the phenomena of RNA silencing by fractional ordinary differential equations (FODE) has more advantages than classical integer-order modeling.

Numerical Scheme for the Solution of Fractional Differential Equations of Order Greater Than One

2005 ◽  
Vol 1 (2) ◽  
pp. 178-185 ◽  
Author(s):  
Pankaj Kumar ◽  
Om P. Agrawal
Keyword(s):  
Integral Equation ◽  
Differential Equations ◽  
Fractional Order ◽  
Fractional Differential Equations ◽  
Numerical Scheme ◽  
Numerical Solutions ◽  
Volterra Type ◽  
Entire Domain ◽  
Type Integral ◽  
Fractional Differential

This paper presents a numerical scheme for the solutions of Fractional Differential Equations (FDEs) of order α, 1<α<2 which have been expressed in terms of Caputo Fractional Derivative (FD). In this scheme, the properties of the Caputo derivative are used to reduce an FDE into a Volterra-type integral equation. The entire domain is divided into several small domains, and the distribution of the unknown function over the domain is expressed in terms of the function values and its slopes at the node points. These approximations are then substituted into the Volterra-type integral equation to reduce it to algebraic equations. Since the method enforces the continuity of variables at the node points, it provides a solution that is continuous and with a slope that is also continuous over the entire domain. The method is used to solve two problems, linear and nonlinear, using two different types of polynomials, cubic order and fractional order. Results obtained using both types of polynomials agree well with the analytical results for problem 1 and the numerical results obtained using another scheme for problem 2. However, the fractional order polynomials give more accurate results than the cubic order polynomials do. This suggests that for the numerical solutions of FDEs fractional order polynomials may be more suitable than the integer order polynomials. A series of numerical studies suggests that the algorithm is stable.

Third-Kind Chebyshev Wavelet Method for the Solution of Fractional Order Riccati Differential Equations

2019 ◽  
Vol 28 (14) ◽  
pp. 1950247 ◽  
Author(s):  
Sadiye Nergis Tural-Polat
Keyword(s):  
Differential Equations ◽  
Fractional Order ◽  
Numerical Solutions ◽  
Operational Matrix ◽  
Algebraic Equations ◽  
Wavelet Method ◽  
Riccati Differential Equations ◽  
The Third ◽  
Chebyshev Wavelet ◽  
Fractional Orders

In this paper, we derive the numerical solutions of the various fractional-order Riccati type differential equations using the third-kind Chebyshev wavelet operational matrix of fractional order integration (C3WOMFI) method. The operational matrix of fractional order integration method converts the fractional differential equations to a system of algebraic equations. The third-kind Chebyshev wavelet method provides sparse coefficient matrices, therefore the computational load involved for this method is not as severe and also the resulting method is faster. The numerical solutions agree with the exact solutions for non-fractional orders, and also the solutions for the fractional orders approach those of the integer orders as the fractional order coefficient [Formula: see text] approaches to 1.

scholarly journals New Perspective on the Conventional Solutions of the Nonlinear Time-Fractional Partial Differential Equations

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-10 ◽  
Author(s):  
Hijaz Ahmad ◽  
Ali Akgül ◽  
Tufail A. Khan ◽  
Predrag S. Stanimirović ◽  
Yu-Ming Chu
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Fractional Order ◽  
Numerical Solutions ◽  
Variational Iteration Method ◽  
Numerical Results ◽  
Iteration Algorithm ◽  
Partial Differential ◽  
Novel Approach ◽  
New Perspective

The role of integer and noninteger order partial differential equations (PDE) is essential in applied sciences and engineering. Exact solutions of these equations are sometimes difficult to find. Therefore, it takes time to develop some numerical techniques to find accurate numerical solutions of these types of differential equations. This work aims to present a novel approach termed as fractional iteration algorithm-I for finding the numerical solution of nonlinear noninteger order partial differential equations. The proposed approach is developed and tested on nonlinear fractional-order Fornberg–Whitham equation and employed without using any transformation, Adomian polynomials, small perturbation, discretization, or linearization. The fractional derivatives are taken in the Caputo sense. To assess the efficiency and precision of the suggested method, the tabulated numerical results are compared with the standard variational iteration method and the exact solution as well. In addition, numerical results for different cases of the fractional-order α are presented graphically, which show the effectiveness of the proposed procedure and revealed that the proposed scheme is very effective, suitable for fractional PDEs, and may be viewed as a generalization of the existing methods for solving integer and noninteger order differential equations.

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