1. (10 Points) (Reduction to Zero BC): Solve Ut = Duxx u|x=0= To, u|x=L = T...

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1. (10 Points) (Reduction to Zero BC): Solve Ut = Duxx u|x=0= To, u|x=L = T₁ (T₁, T₂ constants) ult=0 = f(x), in three steps: Step 1. Construct a linear function T(x) of x such that T(0) = T₁ and T(L) = T₂. Step 2. Solve a new auxillary v(t, x) with a zero BC problem and a modified initial conditions: Vt = Dvxx v|x=0 = v|x= L = 0 v|t=0= f(x) - T(x). Step 3. Show u(t, x) = v(t, x) +T(x). 1. (10 Points) (Reduction to Zero BC): Solve Ut = Duxx u|x=0= To, u|x=L = T₁ (T₁, T₂ constants) ult=0 = f(x), in three steps: Step 1. Construct a linear function T(x) of x such that T(0) = T₁ and T(L) = T₂. Step 2. Solve a new auxillary v(t, x) with a zero BC problem and a modified initial conditions: Vt = Dvxx v|x=0 = v|x= L = 0 v|t=0= f(x) - T(x). Step 3. Show u(t, x) = v(t, x) +T(x).