scholarly journals Ideals Generated by Differential Equations

2020 ◽  
pp. 170-186
Author(s):  
OlegV. Kaptsov
Keyword(s):  
Algebraic Geometry ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Differential System ◽  
Commutative Algebra ◽  
Algebraic Approach ◽  
Jet Space ◽  
Passive Systems ◽  
Partial Differential ◽  
Condi Tions

We propose a new algebraic approach to study compatibility of partial differential equations. The approach uses concepts from commutative algebra, algebraic geometry and Gr¨obner bases to clarify crucial notions concerning compatibility such as passivity and reducibility. One obtains sufficient condi- tions for a differential system to be passive and proves that such systems generate manifolds in the jet space. Some examples of constructions of passive systems associated with the sinh-Cordon equation are given

scholarly journals A foundational approach to the Lie theory for fractional order partial differential equations

2017 ◽  
Vol 20 (1) ◽  
Author(s):  
Rosario Antonio Leo ◽  
Gabriele Sicuro ◽  
Piergiulio Tempesta
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Finite Number ◽  
Fractional Order ◽  
Jet Space ◽  
Lie Theory ◽  
Partial Differential ◽  
Infinite Dimensional ◽  
Independent Variables ◽  
Fractional Differential

AbstractWe provide a general theoretical framework allowing us to extend the classical Lie theory for partial differential equations to the case of equations of fractional order. We propose a general prolongation formula for the study of Lie symmetries in the case of an arbitrary finite number of independent variables. We also prove the Lie theorem in the case of fractional differential equations, showing that the proper space for the analysis of these symmetries is the infinite dimensional jet space.

The Mapping of the Main Functions and Different Variations of YH-DIE

2020 ◽  
Author(s):  
Gilbert Mason ◽  
Yuanjue Chou ◽  
Springer Pind
Keyword(s):  
Algebraic Geometry ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Minimal Surface ◽  
Access Point ◽  
Partial Differential ◽  
Surface Equation ◽  
Minimal Surface Equation ◽  
Fusion Point ◽  
Special Case

YH-DIE must have continuity . Given the basic algebraic clusters of homogeneous configurations,we can get the basic three equations: \begin{array}{l} {\mathop{\int}\nolimits_{0}\nolimits^{{x}_{i}}{\frac{{G}\left({{x}_{i}\mathrm{,}s}\right)}{{\left({{x}_{i}\mathrm{{-}}{s}}\right)}^{\mathit{\alpha}}}\mathrm{\varphi}\left({s}\right){ds}}\mathrm{{=}}{f}\left({{x}_{i}}\right)}\ ;\ {\frac{\mathrm{\partial}}{\mathrm{\partial}{x}_{i}}\left({\frac{{\mathrm{\partial}}_{{x}_{i}}G}{\sqrt{{1}\mathrm{{+}}{\left|{\mathrm{\nabla}{G}}\right|}^{2}}}}\right)\mathrm{{=}}{0}}\ ;\ {{i}\mathrm{{=}}\mathop{\sum}\limits_{{x}_{i}\mathrm{{=}}{1}}\limits^{\mathrm{\infty}}{\arccos\hspace{0.33em}\mathrm{\varphi}\left({{x}_{i}}\right)}\mathrm{{=}}{f}\left({\fbox{${Yuh}$}}\right)} \end{array} YH-DIE has become a fusion point and access point in the fields of algebraic geometry and partial differential equations, and its mapping on multidimensional algebraic clusters or manifolds is very special. The minimal surface equation is a special case.

Considerations on Computational Accuracy of the Solution for a Partial Differential Equation

2020 ◽  
Vol 896 ◽  
pp. 59-66
Author(s):  
Ionica Valeriu ◽  
Cosmin Mihai Miriţoiu ◽  
Alexandru Bolcu ◽  
Dan Gheorghe Bagnaru ◽  
Dumitru Bolcu ◽  
...  
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Fourier Transforms ◽  
Algebraic Approach ◽  
Integral Transforms ◽  
Computational Accuracy ◽  
Laplace And Fourier Transforms ◽  
Partial Differential ◽  
Mechanical Models ◽  
Implicit Finite Difference

In this paper, we will compare the methods of solving with explicit or implicit finite difference of the partial differential equations that define the mechanical models of hydrodynamics movements, thermodynamics or those that define the vibration movements with the ones that use integral transforms. By applying the Laplace and Fourier transforms, finite in sine or cosine, depending on the boundary conditions of the real physical problem, it leads to the algebraic approach of the problem, which reduces the difficulty of solving partial differential equations. The errors obtained for the solution of partial differential equations using different methods are within the standard norms. However, in terms of calculus precision, the use of integral transforms is more advantageous.

scholarly journals Linear PDE with constant coefficients

2021 ◽  
pp. 1-26
Author(s):  
Rida Ait El Manssour ◽  
Marc Härkönen ◽  
Bernd Sturmfels
Keyword(s):  
Partial Differential Equations ◽  
Linear System ◽  
Differential Equations ◽  
Commutative Algebra ◽  
Fundamental Principle ◽  
Partial Differential ◽  
Linear Pde ◽  
Constant Coefficients ◽  
The 1960S ◽  
Homogeneous Linear

Abstract We discuss practical methods for computing the space of solutions to an arbitrary homogeneous linear system of partial differential equations with constant coefficients. These rest on the Fundamental Principle of Ehrenpreis–Palamodov from the 1960s. We develop this further using recent advances in computational commutative algebra.

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