# Focus on the homogeneous heat equation f t (x, t) = f xx (x, t), -8 < t 0 subjected to...

Focus on the homogeneous heat equation f t (x, t) = f xx (x, t), -8 < t 0 subjected to the following initial data on time at t = s f(x, s; s) = f(x, s), -8 the solution of the problem is given by Establish Duhamels principle which states that in an infinite domain the solution of the corresponding inhomogeneous problem namely, f t (x, t) = f xx (x, t) + f(t, x), -8 < t 0 subject to the vanishing initial condition f(x, 0) = 0 is given b i.e. f(x, t) has the form which translates to A problem along similar lines can be set up and solved for the heat equation defined Focus on the homogeneous heat equation f t (x, t) = f xx (x, t), -8 < t 0 subjected to the following initial data on time at t = s f(x, s; s) = f(x, s), -8 the solution of the problem is given by Establish Duhamels principle which states that in an infinite domain the solution of the corresponding inhomogeneous problem namely, f t (x, t) = f xx (x, t) + f(t, x), -8 < t 0 subject to the vanishing initial condition f(x, 0) = 0 is given b i.e. f(x, t) has the form which translates to A problem along similar lines can be set up and solved for the heat equation defined over a finite interval. Duhamels principle also holds for the solvability of the inhomogeneous wave equation where the source term is transferred to the initial velocity for the corresponding homogeneous problem.

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