A Cauchy problem for the heat equation

SummaryLet u(x, t) satisfy the heat equation in 0<x<1, 0<t≤T. Let u(x, 0)=0 for 0<x<1 and let |u(0, t)|<ε, | ux(0, t) |<ε, and | u(1, t) |<M for 0≤t≤T. Then, $$\left| {u\left( {x, t} \right)} \right|< M_{1^{1 - \beta \left( x \right)\varepsilon \beta \left( x \right)} } $$ , where M1 and β(x) are given explicitly by simple formulas. The application of the a priori bound to obtain error estimates for a numerical solution of the Cauchy problem for the heat equation with u(x, 0)=h(x), u(0, t)=f(t), and ux(0, t)=g(t) is discussed.