![1D string and heat conductor Problem 1.1. (4 pts) Consider the 1D vibrating string equation ch.. (t,x) = hn(t,x) + fh(t,z), x E [0, L], f > 0 with the boundary condition h(t,0-0, hz(t, L)=0. Write the most general solution and discuss the qualitative behaviour of the solution (especially its depen- dence on t), can you give a physical interpretation to the f-term? Problem 1.2. (2 pts) Continuing from the above exercise. Now add the initial conditions ㄫㄨ 0, a) = 2 sin3 sin ht (0, z) sin 2L. 2L Now give the complete solution.](http://img.homeworklib.com/questions/6fb67140-14f4-11ec-ac21-7167048c0872.png?x-oss-process=image/resize,w_560)
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1D string and heat conductor Problem 1.1. (4 pts) Consider the 1D...
Problem 33 Solve the boundary value 1D heat problem with the given data. In each case,...
Problem 33 Solve the boundary value 1D heat problem with the given data. In each case, give a brief physical explanation of the problem. L =,a=1, u(0,t) = u(7,t) = 0, u(x,0) = f(x) = 30 sin x
4. Consider the following initial value problem of the 1D wave equation with mixed boundary condition...
4. Consider the following initial value problem of the 1D wave equation with mixed boundary condition IC: u(z, t = 0) = g(x), ut(z, t = 0) = h(z), BC: u(0, t)0, u(l,t) 0, t>0 0 < x < 1, (a)Use the energy method to show that there is at most one solution for the initial-boundary value problem. (b)Suppose u(x,t)-X()T(t) is a seperable solution. Show that X and T satisfy for some λ E R. Find all the eigenvalues An...
3. Consider the following Neumann problem for the heat equation: 14(0,t)=14(L,t)=0, t>0 u(x,0)- f...
3. Consider the following Neumann problem for the heat equation: 14(0,t)=14(L,t)=0, t>0 u(x,0)- f(x),0<x<L (a) Give a short physical interpretation of this problem. (b) Given the following initial condition, 2 *2 2 solve the initial boundary value problem for u(x,t.
3. Consider the following Neumann problem for the heat equation: 14(0,t)=14(L,t)=0, t>0 u(x,0)- f(x),0
Problem 33 Solve the boundary value 1D heat problem with the given data. In each case, give a brief physical explanation of the problem. L =,a=1, u(0,t) = u(7,t) = 0, u(x,0) = f(x) = 30 sin x
4. Consider the following initial value problem of the 1D wave equation with mixed boundary condition IC: u(z, t = 0) = g(x), ut(z, t = 0) = h(z), BC: u(0, t)0, u(l,t) 0, t>0 0 < x < 1, (a)Use the energy method to show that there is at most one solution for the initial-boundary value problem. (b)Suppose u(x,t)-X()T(t) is a seperable solution. Show that X and T satisfy for some λ E R. Find all the eigenvalues An...
3. Consider the following Neumann problem for the heat equation: 14(0,t)=14(L,t)=0, t>0 u(x,0)- f(x),0<x<L (a) Give a short physical interpretation of this problem. (b) Given the following initial condition, 2 *2 2 solve the initial boundary value problem for u(x,t.
3. Consider the following Neumann problem for the heat equation: 14(0,t)=14(L,t)=0, t>0 u(x,0)- f(x),0
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