Consider the heat equation with boundary conditions with initial value Uxx = Ut u (0,t) =...

 

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Consider the heat equation with boundary conditions with initial value Uxx = Ut u (0,t) = 0 (1,t) = 0 u (x, 0) = sin (). 1 += sin 4 3x 2 Here's how to solve the problem: 1. The first step is to seek a product solution u(x,t) = v(x) et that obeys the heat equation and the BCs (but not the initial conditions), where the function (x) and the constant k are to be determined. Plug this product solution into the equation to derive a boundary value problem for the function v (x). The resulting boundary value problem should be Sturm-Liouville. 2. Solve the SL problem from part 1 to find the eigenfunctions Un (x) and eigenvalues kn. 3. Now express the general solution to the heat equation (with the specified BCs) as a sum of product solutions, u(x,t) = Bon (x) e-k where the coefficients B, are to be determined from the initial conditions. Use the orthogonality of the eigenfunctions to solve for the coefficients B. Consider the heat equation with boundary conditions with initial value Uxx = Ut u (0,t) = 0 (1,t) = 0 u (x, 0) = sin (). 1 += sin 4 3x 2 Here's how to solve the problem: 1. The first step is to seek a product solution u(x,t) = v(x) et that obeys the heat equation and the BCs (but not the initial conditions), where the function (x) and the constant k are to be determined. Plug this product solution into the equation to derive a boundary value problem for the function v (x). The resulting boundary value problem should be Sturm-Liouville. 2. Solve the SL problem from part 1 to find the eigenfunctions Un (x) and eigenvalues kn. 3. Now express the general solution to the heat equation (with the specified BCs) as a sum of product solutions, u(x,t) = Bon (x) e-k where the coefficients B, are to be determined from the initial conditions. Use the orthogonality of the eigenfunctions to solve for the coefficients B.