Separation of variables for partial differential equations by Gunter H. Meyer George Cain
By Gunter H. Meyer George Cain
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1). 1) with variable coefficients and additional terms is usually called the diffusion equation. 8) because the qualitative behavior of its solution is generally a good guide to the behavior of the solution of a general diffusion equation. 2). 3 for Poisson’s equation. 8). In addition, the application will usually provide an initial condition at some time t 0 (henceforth set to t 0=0). It is possible in applications that the domain D for the spacial variable changes with time. However, separation of variables will require a time-independent domain.
The resulting problem, even if formally solvable, tends to have an unstable solution. 8). 8) is the mathematical model for the (scaled) temperature u in a homogeneous body D which changes through conduction in space and time. F represents a heat source when F ≤0 and a sink when F >0. , [7]). 1). 1) with variable coefficients and additional terms is usually called the diffusion equation. 8) because the qualitative behavior of its solution is generally a good guide to the behavior of the solution of a general diffusion equation.
If then the proposition is obviously true, so assume number. Then Let α be a complex Now Next, let where t is any real number. Then This expression is quadratic in t and so the fact that it is never negative means that In other words, which completes the proof. The inequality is known as Schwarz’s inequality. 5 Suppose X is an inner product space. Then the function F defined by norm on X. Proof. The proofs that F(f)≥0 and F(αf)=|α|F(f) are simple and omitted. We prove the triangle inequality. html[22/02/2009 23:51:22] is a next page > page_29 < previous page Page 29 Hence F(f+g)≤F(f)+F(g), and we see that In an inner product space, the norm page_29 next page > is indeed a norm on X .