scholarly journals Rational Solutions of Riccati Differential Equation with Coefficients Rational

2011 ◽  
Vol 2011 ◽  
pp. 1-44
Author(s):  
Nadhem Echi
Keyword(s):  
Differential Equation ◽  
Galois Group ◽  
Efficient Method ◽  
Rational Solution ◽  
Partial Fraction ◽  
Rational Solutions ◽  
Riccati Differential Equation ◽  
Differential Galois Group ◽  
Partial Fraction Decomposition ◽  
Decomposition Of Solution

This paper presents a simple and efficient method for determining the rational solution of Riccati differential equation with coefficients rational. In case the differential Galois group of the differential equation , is reducible, we look for the rational solutions of Riccati differential equation , by reducing the number of checks to be made and by accelerating the search for the partial fraction decomposition of the solution reserved for the poles of which are false poles of . This partial fraction decomposition of solution can be used to code . The examples demonstrate the effectiveness of the method.

scholarly journals Calculating Galois groups of third-order linear differential equations with parameters

2018 ◽  
Vol 20 (04) ◽  
pp. 1750038
Author(s):  
Andrei Minchenko ◽  
Alexey Ovchinnikov
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Galois Group ◽  
Galois Groups ◽  
Linear Differential Equations ◽  
Unipotent Radical ◽  
Third Order ◽  
Differential Galois Group ◽  
Bessel Differential Equation ◽  
Linear Differential

Motivated by developing algorithms that decide hypertranscendence of solutions of extensions of the Bessel differential equation, algorithms computing the unipotent radical of a parameterized differential Galois group have been recently developed. Extensions of Bessel’s equation, such as the Lommel equation, can be viewed as homogeneous parameterized linear differential equations of the third order. In this paper, we give the first known algorithm that calculates the differential Galois group of a third-order parameterized linear differential equation.

scholarly journals Hybrid functions approach for the fractional Riccati differential equation

Filomat ◽  
2016 ◽  
Vol 30 (9) ◽  
pp. 2453-2463 ◽  
Author(s):  
Khosrow Maleknejad ◽  
Leila Torkzadeh
Keyword(s):  
Differential Equation ◽  
Fractional Differential Equations ◽  
Efficient Method ◽  
Chebyshev Polynomials ◽  
Numerical Experiments ◽  
Operational Matrices ◽  
Riccati Differential Equation ◽  
Hybrid Functions ◽  
Block Pulse Functions ◽  
Fractional Differential

In this paper, we state an efficient method for solving the fractional Riccati differential equation. This equation plays an important role in modeling the various phenomena in physics and engineering. Our approach is based on operational matrices of fractional differential equations with hybrid of block-pulse functions and Chebyshev polynomials. Convergence of hybrid functions and error bound of approximation by this basis are discussed. Implementation of this method is without ambiguity with better accuracy than its counterpart other approaches. The reliability and efficiency of the proposed scheme are demonstrated by some numerical experiments.

scholarly journals Rational solution to a shallow water wave-like equation

2016 ◽  
Vol 20 (3) ◽  
pp. 875-880
Author(s):  
Hong-Cai Ma ◽  
Ke Ni ◽  
Guo-Ding Ruan ◽  
Ai-Ping Deng
Keyword(s):  
Differential Equation ◽  
Linear Differential Equation ◽  
Shallow Water ◽  
Prime Number ◽  
Water Wave ◽  
Rational Solution ◽  
Rational Solutions ◽  
Shallow Water Wave ◽  
Linear Differential ◽  
New Equation

Two classes of rational solutions to a shallow water wave-like non-linear differential equation are constructed. The basic object is a generalized bilinear differential equation based on a prime number, p = 3. Through this new transformation and with the help of symbolic computation with MAPLE, both the new equation and its rational solutions are obtained.

scholarly journals Reproduction of Fuchs Relation for the Group $S_4^{SL_2}$ by Using Groebner Bases

2018 ◽  
Vol 10 (4) ◽  
pp. 49
Author(s):  
Noura Okko
Keyword(s):  
Differential Equation ◽  
Galois Group ◽  
Invariant Theory ◽  
Minimal Polynomial ◽  
Second Order ◽  
Algebraic Solution ◽  
Groebner Basis ◽  
Groebner Bases ◽  
Differential Galois Group

In 1872, Lazarus Fuchs used a new tool which is The Invariant Theory to construct the minimal polynomial of an algebraic solution of a differential equation of second order. He expressed the coefficient of the equation in terms of the (semi-)invariants of its differential Galois group. In this paper we will give another method to obtain Fuchs Relation: for the octahedral groupe $S_4^{SL_2}$ by using Groebner Basis; a tool which is introduced in 1965 nearly two century after Fuchs Relation.

Sign in / Sign up

Export Citation Format

Share Document