scholarly journals Generalized Green's functions for higher order boundary value matrix differential systems

1992 ◽  
Vol 15 (3) ◽  
pp. 523-535 ◽  
Author(s):  
R. J. Villanueva ◽  
L. Jodar
Keyword(s):  
Boundary Value ◽  
Existence Condition ◽  
Higher Order ◽  
Closed Form Expression ◽  
Problem Dimension ◽  
Green’S Matrix ◽  
Matrix Differential ◽  
Matrix Problems ◽  
Differential Matrix ◽  
Well Posed

In this paper, a Green's matrix function for higher order two point boundary value differential matrix problems is constructed. By using the concept of rectangular co-solution of certain algebraic matrix equation associated to the problem, an existence condition as well as an explicit closed form expression for the solution of possibly not well-posed boundary value problems is given avoiding the increase of the problem dimension.

scholarly journals Solving higher order Fuchs type differential systems avoiding the increase of the problem dimension

1994 ◽  
Vol 17 (1) ◽  
pp. 91-102
Author(s):  
E. Navarro ◽  
L. Jódar ◽  
R. Company
Keyword(s):  
Homogeneous Problem ◽  
Higher Order ◽  
Closed Form Expression ◽  
Power Series Solution ◽  
Matrix Equations ◽  
Form Expression ◽  
Initial Value ◽  
Generalized Power Series ◽  
The Matrix ◽  
Problem Dimension

In this paper, we develop a Frobenius matrix method for solving higher order systems of differential equations of the Fuchs type. Generalized power series solution of the problem are constructed without increasing the problem dimension. Solving appropriate algebraic matrix equations a closed form expression for the matrix coefficient of the series are found. By means of the concept of ak-fundamental set of solutions of the homogeneous problem an explicit solution of initial value problems are given.

Boundary value problems for the inhomogeneous polyanalytic equation I

Analysis ◽  
2005 ◽  
Vol 25 (1) ◽  
Author(s):  
Heinrich Begehr ◽  
Ajay Kumar
Keyword(s):  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Complex Analysis ◽  
Boundary Value ◽  
Higher Order ◽  
Solvability Conditions ◽  
Neumann Type ◽  
Well Posed ◽  
Basic Boundary

AbstractThe three basic boundary value problems in complex analysis are of Schwarz, of Dirichlet and of Neumann type. When higher order equations are investigated all kind of combinations of these boundary conditions are proper to determine solutions. However, not all of these conditions are leading to well-posed problems. Some are overdetermined so that solvability conditions have to be found. Some of these boundary value problems are treated here for the inhomogeneous polyanalytic equation.

scholarly journals Initial boundary value problem for a class of higher-order n-dimensional nonlinear pseudo-parabolic equations

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Liming Xiao ◽  
Mingkun Li
Keyword(s):  
Weak Solution ◽  
Boundary Value Problem ◽  
Parabolic Equations ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Higher Order ◽  
Evolution Problem ◽  
Positive Energy ◽  
Initial Boundary Value

AbstractIn this paper, we study the initial boundary value problem for a class of higher-order n-dimensional nonlinear pseudo-parabolic equations which do not have positive energy and come from the soil mechanics, the heat conduction, and the nonlinear optics. By the mountain pass theorem we first prove the existence of nonzero weak solution to the static problem, which is the important basis of evolution problem, then based on the method of potential well we prove the existence of global weak solution to the evolution problem.

scholarly journals Polyanalytic boundary value problems for planar domains with harmonic Green function

2021 ◽  
Vol 11 (3) ◽  
Author(s):  
Heinrich Begehr ◽  
Bibinur Shupeyeva
Keyword(s):  
Green Function ◽  
Boundary Value Problems ◽  
Arbitrary Order ◽  
Boundary Value ◽  
Planar Domains ◽  
Schwarz Problem ◽  
Bianalytic Functions ◽  
Neumann Type ◽  
The Neumann Problem ◽  
Well Posed

AbstractThere are three basic boundary value problems for the inhomogeneous polyanalytic equation in planar domains, the well-posed iterated Schwarz problem, and further two over-determined iterated problems of Dirichlet and Neumann type. These problems are investigated in planar domains having a harmonic Green function. For the Schwarz problem, treated earlier [Ü. Aksoy, H. Begehr, A.O. Çelebi, AV Bitsadze’s observation on bianalytic functions and the Schwarz problem. Complex Var Elliptic Equ 64(8): 1257–1274 (2019)], just a modification is mentioned. While the Dirichlet problem is completely discussed for arbitrary order, the Neumann problem is just handled for order up to three. But a generalization to arbitrary order is likely.

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