ABSTRACT
The method of abstract differential equations provides proper guidelines for solving various problems with partial differential equations involved. Under the approved interpretation a partial differential equation is treated as an ordinary differential equation in a Banach space. We give below one possible example. Let n be a bounded domain in the space R n , whose boundary is sufficiently smooth. The initial boundary value problem for the heat conduction equation can add interest and aid in understanding. Its statement is as follows:
Ut = du+f(x,t), (6.1.1) u(x,O) = <p(x) ,
(6.1.2)
where G = 11 X [0, 1') is a cylindrical domainj When adopting X = L,(11) as a basic Banach space, we introduce in the space X a linear (unbounded)
operator A = a with the domain 'D(A) = W;(O) n W,l(O) and call it the Laplace operator. The function u(x, t) is viewed as an abstract function u(t) of the variable t with values irt the space X. Along similar lines, the function j(x,t) regards as a function with values in the space X, while the function cp(x) is an element cp E XI, making it possible to treat the direct problem (6.1.1) as the Cauchy pro~lem in the Banach space X for the ordinary differential equation
{ u'(t) = A u(t) + j(t) , u(O) = cp.
