Numerical Solution of Non-linear Riccati Differential Equations with Fractional Order

2010 ◽  
Vol 11 (Supplement) ◽  
Author(s):  
H. Jafari ◽  
H. Tajadodi ◽  
H. Nazari ◽  
C. M. Khalique
Keyword(s):  
Differential Equations ◽  
Numerical Solution ◽  
Fractional Order ◽  
Riccati Differential Equations ◽  
Non Linear

scholarly journals Numerical Solution of Nonlinear Fractional Volterra Integro-Differential Equations via Bernoulli Polynomials

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Emran Tohidi ◽  
M. M. Ezadkhah ◽  
S. Shateyi
Keyword(s):  
Differential Equations ◽  
Numerical Solution ◽  
Linear Combination ◽  
Fractional Order ◽  
Bernoulli Polynomials ◽  
Approximate Solutions ◽  
Computational Approach ◽  
Numerical Results ◽  
Algebraic Equations

This paper presents a computational approach for solving a class of nonlinear Volterra integro-differential equations of fractional order which is based on the Bernoulli polynomials approximation. Our method consists of reducing the main problems to the solution of algebraic equations systems by expanding the required approximate solutions as the linear combination of the Bernoulli polynomials. Several examples are given and the numerical results are shown to demonstrate the efficiency of the proposed method.

Numerical solution of fractional-order ordinary differential equations using the reformulated infinite state representation

2019 ◽  
Vol 22 (5) ◽  
pp. 1321-1350 ◽  
Author(s):  
Matthias Hinze ◽  
André Schmidt ◽  
Remco I. Leine
Keyword(s):  
Differential Equations ◽  
Numerical Solution ◽  
Ordinary Differential Equations ◽  
Fractional Order ◽  
Benchmark Problems ◽  
State Representation ◽  
Weakly Singular Kernel ◽  
Integration Interval ◽  
Novel Approach ◽  
Infinite State

Abstract In this paper, we propose a novel approach for the numerical solution of fractional-order ordinary differential equations. The method is based on the infinite state representation of the Caputo fractional differential operator, in which the entire history of the state of the system is considered for correct initialization. The infinite state representation contains an improper integral with respect to frequency, expressing the history dependence of the fractional derivative. The integral generally has a weakly singular kernel, which may lead to problems in numerical computations. A reformulation of the integral generates a kernel that decays to zero at both ends of the integration interval leading to better convergence properties of the related numerical scheme. We compare our method to other schemes by considering several benchmark problems.

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