Section 7.9The Nonhomogeneous Heat Equation133Solutions to Exercises 7.91.Proceed as in Example 1 withc=1/2. Equation (3) becomes in this caseu(x, t)=2√2te-x2/t*δ1(x)=1√πte-(x-1)2/t,since the eﬀect of convolution byδ1is to shift the function by 1 unit to the rightand multiply by1√2π.5.We use the superposition principle (see the discussion preceeding Example 4). Ifφis the solution ofut=14uxx+δ0,u(x,0) = 0 andψis the solution ofut=14uxx,u(x,0) =U0(x), then you can check thatφ+ψis the solution ofut=14uxx+δ0,u(x,0) =U0(x). By Examples 1,φ(x, t2√t√πe-x2/t-2|x|√πΓ±12,x2t²and by Exercise 20, Section 7.4,ψ(x, t12erf±x√t².9.Apply Theorem 2 with
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