scholarly journals On the solvability of spatial nonlocal boundary value problemsfor one-dimensional pseudoparabolic and pseudohyperbolic equations

2017 ◽  
Vol 21 (3) ◽  
pp. 29-43
Author(s):  
N.S. Popov
Keyword(s):  
Integral Equations ◽  
Boundary Value ◽  
Nonlocal Boundary ◽  
Nonlocal Boundary Conditions ◽  
Integral Conditions ◽  
Regular Solutions ◽  
System Of Integral Equations ◽  
One Dimensional ◽  
Systems Of Integral Equations ◽  
Constant Coefficients

In the present work we study the solvability of spatial nonlocal boundary value problems for linear one-dimensional pseudoparabolic and pseudohyperbolic equations with constant coefficients, but with general nonlocal boundary conditions by A.A. Samarsky and integral conditions with variables coefficients. The proof of the theorems of existence and uniqueness of regular solutions is carried out by the method of Fourier. The study of solvability in the classes of regular solutions leads to the study of a system of integral equations of Volterra of the second kind. In particular cases nongeneracy conditions of the obtained systems of integral equations in explicit form are given.

Existence and monotonicity of nonlocal boundary value problems: the one-dimensional case

2020 ◽  
pp. 1-27
Author(s):  
Christopher Goodrich ◽  
Carlos Lizama
Keyword(s):  
Boundary Value ◽  
Nonlocal Boundary ◽  
Nonlocal Boundary Conditions ◽  
Convolution Equation ◽  
Nonlocal Term ◽  
Qualitative Properties ◽  
One Dimensional ◽  
The One ◽  
Image Position ◽  
Convolution Equations

We consider nonlocal equations of the general form \begin{equation} \left(a*u''\right)(\cdot)+\lambda f\big(\cdot,u(\cdot)\big)=0.\nonumber \end{equation} By developing a Green's function representation for the solution of the boundary value problem we study existence, uniqueness, and qualitative properties (e.g., positivity or monotonicity) of solutions to these problems. We apply our methods to fractional order differential equations. We also demonstrate an application of our methodology both to convolution equations with nonlocal boundary conditions as well as those with a nonlocal term in the convolution equation itself.

Initial-boundary value problems for a time-fractional differential equation with involution perturbation

2019 ◽  
Vol 14 (3) ◽  
pp. 312 ◽  
Author(s):  
Nasser Al-Salti ◽  
Sebti Kerbal ◽  
Mokhtar Kirane
Keyword(s):  
Boundary Value Problems ◽  
Separation Of Variables ◽  
Boundary Value ◽  
Nonlocal Boundary ◽  
Initial Boundary ◽  
Nonlocal Boundary Conditions ◽  
Initial Boundary Value Problems ◽  
Initial Boundary Value ◽  
Formal Solutions ◽  
The Given

Direct and inverse initial-boundary value problems of a time-fractional heat equation with involution perturbation are considered using both local and nonlocal boundary conditions. Results on existence of formal solutions to these problems are presented. Solutions are expressed in a form of series expansions using appropriate orthogonal basis obtained by separation of variables. Convergence of series solutions are obtained by imposing certain conditions on the given data. Uniqueness of the obtained solutions are also discussed. The obtained general solutions are illustrated by an example using an appropriate choice of the given data.

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