(1) where = is a function of two variables and Here u u(x, t) x t. is the space variable, so [0, where is the length of the rod. x x L], L ∈ is the time variable, so t t ≥ 0. We assume the initial condition (2) where the function is given, and the boundary conditions f . (3) Let us attempt to find a solution of () that is not identically zero satisfying the boundary conditions () but with the following property: is a product in which the dependence of on is separated, that is: u u x, t (4) This solution technique is called . Substituting back into equation (), u Since the right hand side depends only on and the left hand side only on both sides are equal to some x t, constant value −λ. Thus: (5) and (6) We will now show that nontrivial solutions for () for values of 0 cannot occur: λ ≤ Suppose that < 0. Then there exist real numbers such that B, C λ From () we get = 0 = and therefore = 0 = which implies is identically 0. X(0) X(L) B C u Suppose that = 0. Then there exist real numbers such that = + From equation () B, C X(x) Bx C. λ we conclude in the same manner as in 1 that is identically 0. u Therefore, it must be the case that > 0. Then there exist real numbers such that A, B, C λ and
(1) where u = u(x, t) is a function of two variables x and t. Here x is the space variable, so x ∈ [0, L], where L is the length of the rod. t is the time variable, so t ≥ 0. We assume the initial condition (2) where the function f is given, and the boundary conditions . (3) Let us attempt to find a solution of () that is not identically zero satisfying the boundary conditions () but with the following property: u is a product in which the dependence of u on x, t is separated, that is: (4) This solution technique is called . Substituting u back into equation (), Since the right hand side depends only on x and the left hand side only on t, both sides are equal to some constant value −λ. Thus: (5) and (6) We will now show that nontrivial solutions for () for values of λ ≤ 0 cannot occur: Suppose that λ < 0. Then there exist real numbers B, C such that From () we get X(0) = 0 = X(L) and therefore B = 0 = C which implies u is identically 0. Suppose that λ = 0. Then there exist real numbers B, C such that X(x) = Bx + C. From equation () we conclude in the same manner as in 1 that u is identically 0. Therefore, it must be the case that λ > 0. Then there exist real numbers A, B, C such that and
Heat%20equation(Wikipedia).docx