User talk:Zeraoulia Rafik40

Weighted Bouziani space
In mathematics, a Weighted Bouziani space , also called a the space of square integrable weighted primitive functions on$$[0,b]$$ function''', in It is denoted by  $$B_{2,x}^{1,*}(0,b)$$ it is a completion of $$c_0(a,b) )$$ the vector space of continuous functions with compact support in $$(0,b)$$ for the scalar product defined by the bilinear form $$((.,.))_x$$ given by : $$((u,w))_x=\int_0^b \Im_{x}^{*} (\xi u).\Im_{x}^{*}(\xi w)\mathrm{d}x $$ where  $$\Im_{x}^{*}=\int_x^b g(\xi,t)\mathrm{d}\xi $$

Definition
Let $$H$$ be a Hilbert space with a norm $$||.||_H$$. We denote by $$L^2(0,T;H)$$ (resp $$L_r^2(0,T;H)$$ the set of all measurable abstract functions $$u(.,t)$$ from $$(0,T)$$ into $$H$$ such that $$||u||_{L^2(0,T;H)}= \Bigg \{\int_0^T||u(.,t)||_H^2\mathrm{d}t \Bigg \}^{\frac12} <\infty$$ respectively  $$||u||_{L^2_r(0,T;H)}= \Bigg \{\int_0^T (\exp(\frac{ct}{2})||u(.,t)||_H)^2\mathrm{d}t \Bigg \}^{\frac12} <\infty$$ .Let $$C(0,T;H)$$ be the set of all continuous functions $$u(.,t): (0,T) \to H $$  with $$||u||_{C(0,T;H)}=\sup_{0\leq \tau\leq T}||u(.\tau)_H||$$ We write $$B_{2,\rho}^{1}(0,T;H)$$ for the space of functions from $$(0,T)$$ into $$H$$ which are weighted Bouziani space for the measure $$dt$$. It is a Hilbert space for the norm $$B_{2,\rho}^{1}(0,T;H)= \Bigg \{\int_0^T (\exp(\frac{ct}{2})||u(.,t)||_H)^2\mathrm{d}t \Bigg \}^{\frac12} $$

Application
Weighted Bouziani space used in physics to solve some boundary value problem,  such as to prove  uniqueness, and continuous dependence upon the data of a generalized solution for certain singular parabolic equations with initial and nonlocal boundary conditions,