A hierarchy of generalized Jaulent–Miodek equations and their explicit solutions

2017 ◽  
Vol 15 (01) ◽  
pp. 1850002
Author(s):  
Xianguo Geng ◽  
Liang Guan ◽  
Bo Xue
Keyword(s):  
Meromorphic Function ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Spectral Problem ◽  
Theta Function ◽  
Algebraic Curves ◽  
Explicit Solutions ◽  
Two Systems ◽  
Energy Dependent

A hierarchy of generalized Jaulent–Miodek (JM) equations related to a new spectral problem with energy-dependent potentials is proposed. Depending on the Lax matrix and elliptic variables, the generalized JM hierarchy is decomposed into two systems of solvable ordinary differential equations. Explicit theta function representations of the meromorphic function and the Baker–Akhiezer function are constructed, the solutions of the hierarchy are obtained based on the theory of algebraic curves.

scholarly journals Algebro-Geometric Solutions of the Coupled Chaffee-Infante Reaction Diffusion Hierarchy

2021 ◽  
Vol 2021 ◽  
pp. 1-21
Author(s):  
Chao Yue ◽  
Tiecheng Xia
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Spectral Problem ◽  
Theta Function ◽  
Meromorphic Functions ◽  
Reaction Diffusion ◽  
Arithmetic Genus ◽  
Trigonal Curve ◽  
Matrix Spectral Problem ◽  
Geometric Solutions

The coupled Chaffee-Infante reaction diffusion (CCIRD) hierarchy associated with a 3 × 3 matrix spectral problem is derived by using two sets of the Lenard recursion gradients. Based on the characteristic polynomial of the Lax matrix for the CCIRD hierarchy, we introduce a trigonal curve K m − 2 of arithmetic genus m − 2 , from which the corresponding Baker-Akhiezer function and meromorphic functions on K m − 2 are constructed. Then, the CCIRD equations are decomposed into Dubrovin-type ordinary differential equations. Furthermore, the theory of the trigonal curve and the properties of the three kinds of Abel differentials are applied to obtain the explicit theta function representations of the Baker-Akhiezer function and the meromorphic functions. In particular, algebro-geometric solutions for the entire CCIRD hierarchy are obtained.

scholarly journals A Fractional Equation with Left-Sided Fractional Bessel Derivatives of Gerasimov–Caputo Type

Mathematics ◽  
2019 ◽  
Vol 7 (12) ◽  
pp. 1216 ◽  
Author(s):  
Elina Shishkina ◽  
Sergey Sitnik
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Integral Transform ◽  
Bessel Operator ◽  
Explicit Solutions ◽  
Solutions To Equations ◽  
Fractional Powers ◽  
Derivatives Of ◽  
Fractional Equation

In this article we propose and study a method to solve ordinary differential equations with left-sided fractional Bessel derivatives on semi-axes of Gerasimov–Caputo type. We derive explicit solutions to equations with fractional powers of the Bessel operator using the Meijer integral transform.

scholarly journals New Neumann System Associated with a 3 × 3 Matrix Spectral Problem

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Fang Li ◽  
Liping Lu
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Spectral Problem ◽  
Hamiltonian Systems ◽  
Lax Pair ◽  
Completely Integrable ◽  
Neumann System ◽  
Matrix Spectral Problem

The nonlinearization approach of Lax pair is applied to the case of the Neumann constraint associated with a 3 × 3 matrix spectral problem, from which a new Neumann system is deduced and proved to be completely integrable in the Liouville sense. As an application, solutions of the first nontrivial equation related to the 3 × 3 matrix spectral problem are decomposed into solving two compatible Hamiltonian systems of ordinary differential equations.

scholarly journals Analytical Solutions of Fuzzy Linear Differential Equations in the Conformable Setting

2021 ◽  
Vol 2 (2) ◽  
pp. 13-30
Author(s):  
Awais Younus ◽  
Muhammad Asif ◽  
Usama Atta ◽  
Tehmina Bashir ◽  
Thabet Abdeljawad
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Analytical Solutions ◽  
Higher Order ◽  
Linear Differential Equations ◽  
Conformable Derivative ◽  
Systems Of Differential Equations ◽  
Two Systems ◽  
Linear Differential

In this paper, we provide the generalization of two predefined concepts under the name fuzzy conformable differential equations. We solve the fuzzy conformable ordinary differential equations under the strongly generalized conformable derivative. For the order $\Psi$, we use two methods. The first technique is to resolve a fuzzy conformable differential equation into two systems of differential equations according to the two types of derivatives. The second method solves fuzzy conformable differential equations of order $\Psi$ by a variation of the constant formula. Moreover, we generalize our results to solve fuzzy conformable ordinary differential equations of a higher order. Further, we provide some examples in each section for the sake of demonstration of our results.

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