Factoring linear partial differential operators and the Darboux method for integrating nonlinear partial differential equations

2000 ◽  
Vol 122 (1) ◽  
pp. 121-133 ◽  
Author(s):  
S. P. Tsarev
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Differential Operators ◽  
Nonlinear Partial Differential Equations ◽  
Partial Differential ◽  
Partial Differential Operators ◽  
Darboux Method ◽  
Linear Partial Differential Operators

scholarly journals Infinite-dimensional linear algebra and solvability of partial differential equations

2021 ◽  
Vol 13 ◽  
Author(s):  
Todor D. Todorov
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Linear Algebra ◽  
Differential Operators ◽  
Test Functions ◽  
Partial Differential ◽  
Infinite Dimensional ◽  
Partial Differential Operators ◽  
Linear Partial Differential Operators ◽  
Regular Operators

  We discuss linear algebra of infinite-dimensional vector spaces in terms of algebraic (Hamel) bases. As an application we prove the surjectivity of a large class of linear partial differential operators with smooth ($\mathcal C^\infty$-coefficients) coefficients, called in the article \emph{regular}, acting on the algebraic dual $\mathcal D^*(\Omega)$ of the space of test-functions $\mathcal D(\Omega)$. The surjectivity of the partial differential operators guarantees solvability of the corresponding partial differential equations within $\mathcal D^*(\Omega)$. We discuss our result in contrast to and comparison with similar results about the restrictions of the regular operators on the space of Schwartz distribution $\mathcal D^\prime(\Omega)$, where these operators are often non-surjective. 

scholarly journals Improvement of the Modified Decomposition Method for Handling Third-Order Singular Nonlinear Partial Differential Equations with Applications in Physics

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Nemat Dalir
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Decomposition Method ◽  
Differential Operators ◽  
Nonlinear Partial Differential Equations ◽  
Initial Value ◽  
Third Order ◽  
Partial Differential ◽  
The Third

The modified decomposition method (MDM) is improved by introducing new inverse differential operators to adapt the MDM for handling third-order singular nonlinear partial differential equations (PDEs) arising in physics and mechanics. A few case-study singular nonlinear initial-value problems (IVPs) of third-order PDEs are presented and solved by the improved modified decomposition method (IMDM). The solutions are compared with the existing exact analytical solutions. The comparisons show that the IMDM is effectively capable of obtaining the exact solutions of the third-order singular nonlinear IVPs.

scholarly journals Newton polygons and gevrey indices for linear partial differential operators

1992 ◽  
Vol 128 ◽  
pp. 15-47 ◽  
Author(s):  
Masatake Miyake ◽  
Yoshiaki Hashimoto
Keyword(s):  
Differential Operator ◽  
Power Series ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Differential Operators ◽  
Newton Polygon ◽  
Convergent Power Series ◽  
Partial Differential ◽  
Partial Differential Operators ◽  
Linear Partial Differential Operators

This paper is a continuation of Miyake [7] by the first named author. We shall study the unique solvability of an integro-differential equation in the category of formal or convergent power series with Gevrey estimate for the coefficients, and our results give some analogue in partial differential equations to Ramis [10, 11] in ordinary differential equations.In the study of analytic ordinary differential equations, the notion of irregularity was first introduced by Malgrange [3] as a difference of indices of a differential operator in the categories of formal power series and convergent power series. After that, Ramis extended his theory to the category of formal or convergent power series with Gevrey estimate for the coefficients. In these studies, Ramis revealed a significant meaning of a Newton polygon associated with a differential operator.

scholarly journals Some sufficient conditions for the comparability of two differential operators

Filomat ◽  
2002 ◽  
pp. 57-61 ◽  
Author(s):  
Raid Al-Momani ◽  
Qassem Al-Hassan ◽  
Ali Al-Jarrah ◽  
Ghanim Momani
Keyword(s):  
Partial Differential Equations ◽  
Differential Operators ◽  
Complete Solution ◽  
Sufficient Conditions ◽  
Order Relation ◽  
Partial Differential ◽  
Partial Differential Operators ◽  
The Sixties ◽  
Constant Coefficients ◽  
Linear Partial Differential Operators

The comparison of differential operators is a problem of the theory of partial differential operators with constant coefficients. This problem up to now doesn't have a complete solution. It was formulated in the sixties by Lars Hormander in his monograph "The Analysis of Linear Partial Differential Operators". Many facts of the theory of partial differential equations can be formulated by using the concept of pre-order relation over the set of differential operators, however it is too complicated to check the comparability condition of two differential operators. In this paper we get some sufficient conditions for the comparability of two differential operators.

scholarly journals A Parabolic Transform and Averaging Methods for General Partial Differential Equations

2019 ◽  
Vol 17 ◽  
pp. 352-361
Author(s):  
Mahmoud Mohammed Mostafa El-Borai ◽  
Hamed Kamal Awad Awad ◽  
Randa Hamdy. M. Ali Ali
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Differential Operators ◽  
Averaging Method ◽  
Partial Differential ◽  
Averaging Methods ◽  
Partial Differential Operators ◽  
Existence And Stability ◽  
Special Case ◽  
Stability Results

Averaging method of the fractional general partial differential equations and a special case of these equations are studied, without any restrictions on the characteristic forms of the partial differential operators. We use the parabolic transform, existence and stability results can be obtained.

Lyapunov-type inequalities for partial differential equations with 𝑝-Laplacian

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Robert Stegliński
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Nonlinear Partial Differential Equations ◽  
Dirichlet Boundary Conditions ◽  
Partial Differential ◽  
Linear Partial Differential Equations ◽  
Lyapunov Inequalities ◽  
Funct Anal

Abstract The aim of this paper is to extend results from [A. Cañada, J. A. Montero and S. Villegas, Lyapunov inequalities for partial differential equations, J. Funct. Anal. 237 (2006), 1, 176–193] about Lyapunov-type inequalities for linear partial differential equations to nonlinear partial differential equations with 𝑝-Laplacian with zero Neumann or Dirichlet boundary conditions.

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