From () we get = 0 and that for some positive integer C n, This solves the heat equation in the special case that the dependence of has the special form (). u In general, the sum of solutions to () that satisfy the boundary conditions () also satisfies () and (). We can show that the solution to (), () and () is given by where Generalizing the solution technique The solution technique used above can be greatly extended to many other types of equations. The idea is that the operator with the zero boundary conditions can be represented in terms of its . This leads uxx naturally to one of the basic ideas of the of linear . Consider the = The infinite sequence of functions uxx. Δu for are eigenvectors of n ≥ 1 Δ. Indeed Moreover, any eigenvector of with the boundary conditions = = 0 is of the form for some f f(0) f(L) en n Δ 1. The functions for 1 form an sequence with respect to a certain on the space of real-valued en n ≥ ≥ functions on [0, This means L]. Finally, the sequence {en}n spans a dense linear subspace of This shows that in effect we L2((0, L)). N ∈ have the operator Δ. Heat conduction in non-homogeneous anisotropic media In general, the study of heat conduction is based on several principles. Heat flow is a form of flow, and as such it is meaningful to speak of the time rate of flow of heat into a region of space. The time rate of heat flow into a region is given by a time-dependent quantity We assume V qt(V). q has a , so that
From () we get C = 0 and that for some positive integer n, This solves the heat equation in the special case that the dependence of u has the special form (). In general, the sum of solutions to () that satisfy the boundary conditions () also satisfies () and (). We can show that the solution to (), () and () is given by where Generalizing the solution technique The solution technique used above can be greatly extended to many other types of equations. The idea is that the operator uxx with the zero boundary conditions can be represented in terms of its . This leads naturally to one of the basic ideas of the of linear . Consider the Δu = uxx. The infinite sequence of functions for n ≥ 1 are eigenvectors of Δ. Indeed Moreover, any eigenvector f of Δ with the boundary conditions f(0) = f(L) = 0 is of the form en for some n ≥ 1. The functions en for n ≥ 1 form an sequence with respect to a certain on the space of real-valued functions on [0, L]. This means Finally, the sequence {en}n ∈ N spans a dense linear subspace of L2((0, L)). This shows that in effect we have the operator Δ. Heat conduction in non-homogeneous anisotropic media In general, the study of heat conduction is based on several principles. Heat flow is a form of flow, and as such it is meaningful to speak of the time rate of flow of heat into a region of space. The time rate of heat flow into a region V is given by a time-dependent quantity qt(V). We assume q has a , so that
Heat%20equation(Wikipedia).docx