Section 7.4The Heat Equation and Gauss’s Kernel121You can easily check that this solution verifes the heat equation andu(x,0) =x3.IFn=4,u(x, t)=2±j=0²42j³tj(2j)!j!·x4-2j=x4+12tx2t2.Here too, you can check that this solution verifes the heat equation andu(x,0) =x4.We now derive a recurrence relation that relates the solutions corresponding ton-1,n, andn+ 1. Letun=un(x, t) denote the solution with initial temperaturedistributionun(x,0) =xn. We have the Following recurrence relationun+1=xun+2ntun-1.The prooF oF this Formula is very much like the prooF oF Bonnet’s recurrence FormulaFor the Legendre polynomials (Section 5.6). BeFore we give the prooF, let us veriFythe Formula withn= 3. The Formula states thatu4u3+6tu2. Sineu4=x4tx2t2,u3=x3tx, andu2=x2t, we see that the Formula is trueForn= 3. We now prove the Formula using Leibniz rule oF diﬀerentiation. As in
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