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Review and Introduction
The Heat Equation
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absolutely integrable apply arbitrary C function assume bnsin(n7rx/L boundary conditions characteristic curve characteristic lines compute continuous function convergence cosine D'Alembert's formula decay order density derivatives differential Dirichlet problem disk Duhamel's principle eigenfunctions eigenvalues Example fe(x Figure finite first-order formal solution Fourier coefficients Fourier series Fourier sine series Fourier transform FS f(x function defined function f(x given graph Green's formula harmonic function heat equation Hence Hint infinite initial conditions initial/boundary-value problem interval J-oo Laplace's equation Let f(x linear PDE Maximum Principle Maximum/Minimum Principle Note obtain orthogonal parametric Parseval's equality particular solution Poisson Integral Formula problem D.E. product solutions proof Proposition prove Remark result satisfies the B.C. Section 3.1 separation of variables Show side condition curve sin(x solution u(x,t solve string superposition principle temperature distribution velocity wave equation XX B.C. xx yy zero