2.2 Existence and uniqueness theorem
We shall prove the existence and uniqueness of the solution of Poisson’s equation with new non-local boundary conditions, i.e. we prove the existence and uniqueness of the solution of second order partial differential equation
Δu = f in Ω
(2.2.1)
with the following non-local boundary condition
u(x) = h(x) u(F(x))+y(x), x 붽(2.2.2)
where , Ω Ìđn bounded domain ; (a) h is a given continuous function on the boundary ∂Ω such that
sup| h |<1 and F:∂Ω →Ω is a continuous mapping .
Definition 2.2.1:
The classical non-local first boundary value problem for the Poisson’s equation is the following:
To find a function uÎC2(Ω)∩C (Ω ) such that
Δu = f in Ω
u(x) = h(x) u(F(x))+y(x), x 붽
(2.2.3)
We assume that f and ∂Ω are sufficiently smooth such that solution of the usual Dirichlet problem
ΔV = f in Ω
V = j on ∂Ω
(2.2.4)
exists for any arbitrary jÎC(∂Ω).
Theorem 2.2.1:
Assume that the condition (a) are fulfilled, then the classical non-local boundary value problems (2.2.3) has a unique solution.
Proof:-
Denote by G(j) the unique solution V of problem (2.2.4).
Further define an operator A by :
A(j) (x):= h(x) (G(j))(F(x))+y(x) ; x 붽(2.2.5)
Then A : C(∂Ω) → C(∂Ω) is a nonlinear mapping,
C(∂Ω) is a complete metric space with the metric
ρ(j1,j2) :≤ sup |j1- j2| .
(2.2.6)
If jÎC(∂Ω) is a fixed point of A i.e. A(j) = j, then u = G(j) is a solution of problem (2.2.3).
Conversely, if u is a solution of problem (2.2.3), then j =u on ∂Ω is a fixed point of A.
Therefore, to prove existence and uniqueness solution for problem (2.2.3), it is sufficient to show that A has exactly one fixed point. This will be a consequence of Banach’s fixed point theorem
ارجوا منكم المساعدة وشرحها لي بالعربي لاني في اشياء لم افهمها ولكم كل الاجر والتواب ان شاء الله
We shall prove the existence and uniqueness of the solution of Poisson’s equation with new non-local boundary conditions, i.e. we prove the existence and uniqueness of the solution of second order partial differential equation
Δu = f in Ω
(2.2.1)
with the following non-local boundary condition
u(x) = h(x) u(F(x))+y(x), x 붽(2.2.2)
where , Ω Ìđn bounded domain ; (a) h is a given continuous function on the boundary ∂Ω such that
sup| h |<1 and F:∂Ω →Ω is a continuous mapping .
Definition 2.2.1:
The classical non-local first boundary value problem for the Poisson’s equation is the following:
To find a function uÎC2(Ω)∩C (Ω ) such that
Δu = f in Ω
u(x) = h(x) u(F(x))+y(x), x 붽
(2.2.3)
We assume that f and ∂Ω are sufficiently smooth such that solution of the usual Dirichlet problem
ΔV = f in Ω
V = j on ∂Ω
(2.2.4)
exists for any arbitrary jÎC(∂Ω).
Theorem 2.2.1:
Assume that the condition (a) are fulfilled, then the classical non-local boundary value problems (2.2.3) has a unique solution.
Proof:-
Denote by G(j) the unique solution V of problem (2.2.4).
Further define an operator A by :
A(j) (x):= h(x) (G(j))(F(x))+y(x) ; x 붽(2.2.5)
Then A : C(∂Ω) → C(∂Ω) is a nonlinear mapping,
C(∂Ω) is a complete metric space with the metric
ρ(j1,j2) :≤ sup |j1- j2| .
(2.2.6)
If jÎC(∂Ω) is a fixed point of A i.e. A(j) = j, then u = G(j) is a solution of problem (2.2.3).
Conversely, if u is a solution of problem (2.2.3), then j =u on ∂Ω is a fixed point of A.
Therefore, to prove existence and uniqueness solution for problem (2.2.3), it is sufficient to show that A has exactly one fixed point. This will be a consequence of Banach’s fixed point theorem
ارجوا منكم المساعدة وشرحها لي بالعربي لاني في اشياء لم افهمها ولكم كل الاجر والتواب ان شاء الله
