### Abstract:

Hilbert spaces are the most important tools in the theories of partial diﬀerential equations. The purpose of this paper is how to use the Hilbert space method to solve partial diﬀerential equation. A diﬀerential equation is the most important part of mathematics for understanding many of the basic laws of physical sciences as well as other sciences. Some of the laws are formulated in terms of mathematical equations involving certain known and unknown quantities and their derivative.
The diﬀerential equation can be ODE - a diﬀerential equation that involves only one independent variable and PDE - a diﬀerential equation involves two or more independent variables. This is the mathematical arise of PDE. The other view regarding its arise is the physical problems. From an applied point of view, perhaps the most important manner in which PDE may arise is from a mathematical formulation of physical problems or boundary value problems which includes the problem of the vibrating string and the problem of heat conduction in a metal bar. These PDE has diﬀerent classiﬁcation, for example: Homogenous ﬁrst order linear PDE: ∂u ∂t + t∂u ∂x = 0
Homogenous second order linear constant coeﬃcient PDE of elliptic type - the Laplace equation: ∂2u ∂x2 + ∂2u ∂y2 = 0
Homogeneous third order non-linear PDE: ∂u ∂t = 6u∂u ∂x − ∂3u ∂x3 Due to the mathematical formulation of a physical problem is very diﬃcult, many PDE problems now remain unsolved. In many cases, crude approximations to reality are made in order to produce a formulation, the hope being that any errors thus arising will be small. Therefore, to solve the PDE, we do not have easy way for their solutions. Under Hilbert space methods of solving a second order PDE, we have to cover solutions by BVP, separation of variables, and numerical methods. Inaddition to the solution of PDE, we had the real life physical world application of PDE - which is to solve problems of heat conduction in a metal bar.