![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...
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...
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
Please help me with this 1D vibrating string problem. That has a Dirichlet boundary condition at...
Please help me with this 1D vibrating string problem. That has a
Dirichlet boundary condition at both ends and the string is at rest
when t=0.
Picture on the equation below
What is missing for this to be solved? Please elaborate
htt(t, x)=c2hxx(t, x) + f sin(vt), x E [0, π].
Problem 1. Consider the vibration of a string with two ends fixed. In addition, assume that the s...
parts a,b, c
Problem 1. Consider the vibration of a string with two ends fixed. In addition, assume that the string is initially at rest. The initial boundary value problem (IBVP) is written as u(0,t) -u(1,t) u(x,0) = f(x), 0 ut (z, 0-0, 0 < x < 1. The solution of this IBVP using the method of separation of variables is given by n-l a) Find the coefficients bn. b) Show that this wave function can be written as the...
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 1. Consider the nonhomogencous heat equation for u(a,t) subject to the nonhomogeneous boundary conditions u(0,t1, t)- 0, and the initial condition 1--+ sin(z) u(z,0) = e solution u(z, t)...
Problem 1. Consider the nonhomogencous heat equation for u(a,t) subject to the nonhomogeneous boundary conditions u(0,t1, t)- 0, and the initial condition 1--+ sin(z) u(z,0) = e solution u(z, t) by completing each of the following steps Find the equilibrium temperature distribution we r) Find th (b) Denote v, t)t) - ()Derive the IBVP for the function vz,t). (c) Find v(x, t) (d) Find u(x, t)
Problem 1. Consider the nonhomogencous heat equation for u(a,t) subject to the nonhomogeneous boundary...
Problem 1 (15 pts) Consider heat conduction on a slender homogeneous metal wire with constant crosssection...
Problem 1 (15 pts) Consider heat conduction on a slender homogeneous metal wire with constant crosssection as shown in Fig.1. L- 10cm Conductivity k = 100 w/m°C. Q(x)= 100.000W/㎡. At x = 0, q = 250 W/ m2. TL = 25 oC. Governing equation: _kdTeQ (0%L) Boundary condition: dT -k Figure 1 Heat conduction on a 1-D metal wire. a. Solve for T (x) with two linear elements (X1 = 0, x2-4cm, and X3 = 10cm) ; b. Compare with...
1. (10 points, part I) Consider the following initial boundary value problem lU (la) (1b) (1c) 0L, t> 0 3 cos ( a(x, 0) (a) Classify the partial differential equation (1a) (b) What do the equation...
1. (10 points, part I) Consider the following initial boundary value problem lU (la) (1b) (1c) 0L, t> 0 3 cos ( a(x, 0) (a) Classify the partial differential equation (1a) (b) What do the equations (la)-(1c) model? (Hint: Give an interpretation for the PDE, boundary conditions and intial condition.) c) Use the method of separation of variables to separate the above problem into two sub- problems (one that depends on space and the other only on time) (d) What...
Problem 1. Consider the nonhomogeneous heat equation for u,t) ut = uzz + sin(2x), 0<x<π, t>0 subject to the nonhomogeneous boundary conditions u(0, t) t > 0 u(n, t) = 0, 1, - and the...
Problem 1. Consider the nonhomogeneous heat equation for u,t) ut = uzz + sin(2x), 0<x<π, t>0 subject to the nonhomogeneous boundary conditions u(0, t) t > 0 u(n, t) = 0, 1, - and the initial condition Lee) Find the solution u(z, t) by completing each of the following steps: (a) Find the equilibrium temperature distribution ue(x). (b) Denote v(x, t) u(a, t) - e(). Derive the IBVP for the function v(x,t). (c) Find v(x, t) (d) Find u(, t)...
4. Consider the homogeneous heat-conduction problem wr =0, u(z,0)=f(x) (15) describing the temporal evolution of the...
4. Consider the homogeneous heat-conduction problem wr =0, u(z,0)=f(x) (15) describing the temporal evolution of the temperature u(r, t) along a constant-thermal-diffusivity rod of length L whose end at x = 0 is held at zero temperature and whose end at r L is insulated (a) Introduce a separable solution of the form u-d(x) G(t) in (15) and find the two ODEs that govern φ(x) and G(t) and homoge- neous boundary conditions on φ(x). Take λ as the separation constant...
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...
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
Please help me with this 1D vibrating string problem. That has a
Dirichlet boundary condition at both ends and the string is at rest
when t=0.
Picture on the equation below
What is missing for this to be solved? Please elaborate
htt(t, x)=c2hxx(t, x) + f sin(vt), x E [0, π].
parts a,b, c
Problem 1. Consider the vibration of a string with two ends fixed. In addition, assume that the string is initially at rest. The initial boundary value problem (IBVP) is written as u(0,t) -u(1,t) u(x,0) = f(x), 0 ut (z, 0-0, 0 < x < 1. The solution of this IBVP using the method of separation of variables is given by n-l a) Find the coefficients bn. b) Show that this wave function can be written as the...
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 1. Consider the nonhomogencous heat equation for u(a,t) subject to the nonhomogeneous boundary conditions u(0,t1, t)- 0, and the initial condition 1--+ sin(z) u(z,0) = e solution u(z, t) by completing each of the following steps Find the equilibrium temperature distribution we r) Find th (b) Denote v, t)t) - ()Derive the IBVP for the function vz,t). (c) Find v(x, t) (d) Find u(x, t)
Problem 1. Consider the nonhomogencous heat equation for u(a,t) subject to the nonhomogeneous boundary...
Problem 1 (15 pts) Consider heat conduction on a slender homogeneous metal wire with constant crosssection as shown in Fig.1. L- 10cm Conductivity k = 100 w/m°C. Q(x)= 100.000W/㎡. At x = 0, q = 250 W/ m2. TL = 25 oC. Governing equation: _kdTeQ (0%L) Boundary condition: dT -k Figure 1 Heat conduction on a 1-D metal wire. a. Solve for T (x) with two linear elements (X1 = 0, x2-4cm, and X3 = 10cm) ; b. Compare with...
1. (10 points, part I) Consider the following initial boundary value problem lU (la) (1b) (1c) 0L, t> 0 3 cos ( a(x, 0) (a) Classify the partial differential equation (1a) (b) What do the equations (la)-(1c) model? (Hint: Give an interpretation for the PDE, boundary conditions and intial condition.) c) Use the method of separation of variables to separate the above problem into two sub- problems (one that depends on space and the other only on time) (d) What...
Problem 1. Consider the nonhomogeneous heat equation for u,t) ut = uzz + sin(2x), 0<x<π, t>0 subject to the nonhomogeneous boundary conditions u(0, t) t > 0 u(n, t) = 0, 1, - and the initial condition Lee) Find the solution u(z, t) by completing each of the following steps: (a) Find the equilibrium temperature distribution ue(x). (b) Denote v(x, t) u(a, t) - e(). Derive the IBVP for the function v(x,t). (c) Find v(x, t) (d) Find u(, t)...
4. Consider the homogeneous heat-conduction problem wr =0, u(z,0)=f(x) (15) describing the temporal evolution of the temperature u(r, t) along a constant-thermal-diffusivity rod of length L whose end at x = 0 is held at zero temperature and whose end at r L is insulated (a) Introduce a separable solution of the form u-d(x) G(t) in (15) and find the two ODEs that govern φ(x) and G(t) and homoge- neous boundary conditions on φ(x). Take λ as the separation constant...
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