The difference between the nonlocal and local conditions problems
$begingroup$
In some of Boundary value problems involving ordinary differential equations,, subsidiary conditions are imposed locally. In some other cases, nonlocal conditions are imposed.
In this paper: Existence and uniqueness of a classical solution to a functional-differential abstract nonlocal Cauchy problem Byszewski studied this form of functional-differential nonlocal problem:
$(1)left{begin{matrix}
u'(t)=f(t,u(t),u(a(t))),::tin I \
u(t_0)+sum_{k=1}^{p}c_ku(t_k)=x_0
end{matrix}right.$
With $I:=[t_0,t_0+T], t_0<t_1<...<t_pleq t_0+T, T>0$ and $f:Itimes E^2rightarrow E :$ and $:a:Irightarrow I :$are given functions satisfying some assumptions; $E$ is a Banach space with norm $:left | . right |; x_0in E, c_kneq 0 ::(k=1,...,p): p in mathbb N$.
And here, in the classical Robin problem: $$u''(t) + f(t,u(t),u'(t)) = 0$$
With local conditions: $u(0)= 0$ and $u'(1) = 0.$
Or
With nonlocal conditions: $u(0)= 0$ and $u(1) = u(eta);:etain(0,1)$
My question is:
-When we say that the boundary conditions are local or nonlocal?
-In which situation we impose local or nonlocal conditions?
Thank you!
real-analysis functional-analysis ordinary-differential-equations pde
asked Jan 9 at 10:30
MotakaMotaka
246111
$endgroup$
add a comment |
$begingroup$
In some of Boundary value problems involving ordinary differential equations,, subsidiary conditions are imposed locally. In some other cases, nonlocal conditions are imposed.
In this paper: Existence and uniqueness of a classical solution to a functional-differential abstract nonlocal Cauchy problem Byszewski studied this form of functional-differential nonlocal problem:
$(1)left{begin{matrix}
u'(t)=f(t,u(t),u(a(t))),::tin I \
u(t_0)+sum_{k=1}^{p}c_ku(t_k)=x_0
end{matrix}right.$
With $I:=[t_0,t_0+T], t_0<t_1<...<t_pleq t_0+T, T>0$ and $f:Itimes E^2rightarrow E :$ and $:a:Irightarrow I :$are given functions satisfying some assumptions; $E$ is a Banach space with norm $:left | . right |; x_0in E, c_kneq 0 ::(k=1,...,p): p in mathbb N$.
And here, in the classical Robin problem: $$u''(t) + f(t,u(t),u'(t)) = 0$$
With local conditions: $u(0)= 0$ and $u'(1) = 0.$
Or
With nonlocal conditions: $u(0)= 0$ and $u(1) = u(eta);:etain(0,1)$
My question is:
-When we say that the boundary conditions are local or nonlocal?
-In which situation we impose local or nonlocal conditions?
Thank you!
real-analysis functional-analysis ordinary-differential-equations pde
asked Jan 9 at 10:30
MotakaMotaka
246111
$endgroup$
add a comment |
$begingroup$
In some of Boundary value problems involving ordinary differential equations,, subsidiary conditions are imposed locally. In some other cases, nonlocal conditions are imposed.
In this paper: Existence and uniqueness of a classical solution to a functional-differential abstract nonlocal Cauchy problem Byszewski studied this form of functional-differential nonlocal problem:
$(1)left{begin{matrix}
u'(t)=f(t,u(t),u(a(t))),::tin I \
u(t_0)+sum_{k=1}^{p}c_ku(t_k)=x_0
end{matrix}right.$
With $I:=[t_0,t_0+T], t_0<t_1<...<t_pleq t_0+T, T>0$ and $f:Itimes E^2rightarrow E :$ and $:a:Irightarrow I :$are given functions satisfying some assumptions; $E$ is a Banach space with norm $:left | . right |; x_0in E, c_kneq 0 ::(k=1,...,p): p in mathbb N$.
And here, in the classical Robin problem: $$u''(t) + f(t,u(t),u'(t)) = 0$$
With local conditions: $u(0)= 0$ and $u'(1) = 0.$
Or
With nonlocal conditions: $u(0)= 0$ and $u(1) = u(eta);:etain(0,1)$
My question is:
-When we say that the boundary conditions are local or nonlocal?
-In which situation we impose local or nonlocal conditions?
Thank you!
real-analysis functional-analysis ordinary-differential-equations pde
asked Jan 9 at 10:30
MotakaMotaka
246111
$endgroup$
In some of Boundary value problems involving ordinary differential equations,, subsidiary conditions are imposed locally. In some other cases, nonlocal conditions are imposed.
In this paper: Existence and uniqueness of a classical solution to a functional-differential abstract nonlocal Cauchy problem Byszewski studied this form of functional-differential nonlocal problem:
$(1)left{begin{matrix}
u'(t)=f(t,u(t),u(a(t))),::tin I \
u(t_0)+sum_{k=1}^{p}c_ku(t_k)=x_0
end{matrix}right.$
With $I:=[t_0,t_0+T], t_0<t_1<...<t_pleq t_0+T, T>0$ and $f:Itimes E^2rightarrow E :$ and $:a:Irightarrow I :$are given functions satisfying some assumptions; $E$ is a Banach space with norm $:left | . right |; x_0in E, c_kneq 0 ::(k=1,...,p): p in mathbb N$.
And here, in the classical Robin problem: $$u''(t) + f(t,u(t),u'(t)) = 0$$
With local conditions: $u(0)= 0$ and $u'(1) = 0.$
Or
With nonlocal conditions: $u(0)= 0$ and $u(1) = u(eta);:etain(0,1)$
My question is:
-When we say that the boundary conditions are local or nonlocal?
-In which situation we impose local or nonlocal conditions?
Thank you!
real-analysis functional-analysis ordinary-differential-equations pde
real-analysis functional-analysis ordinary-differential-equations pde
asked Jan 9 at 10:30
MotakaMotaka
246111
asked Jan 9 at 10:30
MotakaMotaka
246111
edited Jan 9 at 10:49
Motaka
asked Jan 9 at 10:30
MotakaMotaka
246111
asked Jan 9 at 10:30
MotakaMotaka
246111
asked Jan 9 at 10:30
MotakaMotaka
246111
246111
add a comment |
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