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2. Consider the initial-boundary value problem 0 0, t >0 Uz uェェ, Uz(0, t) _ u(0, t) = 0. a(z, 0...
(1 point) This problem is concerned with solving an initial boundary value problem for the heat...
(1 point) This problem is concerned with solving an initial boundary value problem for the heat equation: (0,t)-0, t0 u,o)- in the form, ie where the term involving cy may be missing. Here y is the eigenfunction for Ay- 0 so if zero is not an eigenvalue then this term will be zero First find the eigenvalues and orthonormal eigenfunctions for n1.iA. Pa(x). For n 0 there may or may not be an eigenpair. Give all these as a comma...
1. Solve the boundary value problem ut =-3uzzzz + 5uzz, u(z, 0) = r(z) (-00 < z < oo, t > 0), usi...
1. Solve the boundary value problem ut =-3uzzzz + 5uzz, u(z, 0) = r(z) (-00 < z < oo, t > 0), using direct and inverse Fourier transforms U(w,t)-홅启u(z, t) ei r dr, u(z,t)-二U( ,t) e ur d . You need to explain where you use linearity of Fourier transform and how you transform derivatives in z and in t 2. Find the Fourier transform F() of the following function f(x) and determine whether F() is a continuous function (a)...
13. Find an infinite-series representation for the solution to the heat absorption- radiation problem in th uz(0,t)-aou(0, t) = 0, uz(1, t) + alu(1, t) = 0, t > 0, u(x,0) f(x), 0<a<1, wh...
13. Find an infinite-series representation for the solution to the heat absorption- radiation problem in th uz(0,t)-aou(0, t) = 0, uz(1, t) + alu(1, t) = 0, t > 0, u(x,0) f(x), 0<a<1, where ao < 0, a1 >0, and ao +a> -aoa1. State why there is radiation at x = 1, absorption at z = 0, and the radiation greatly exceeds the absorp- tion. Choose ao--0.25 and a1 4 and find the first four eigenvalues; if f(x) = 2(1-2),...
Let u be the solution to the initial boundary value problem for the Heat Equation, au(t,z) 28?u(t,z), te (0,00), z (0,3); with initial condition u(0, z)fx), where f(0) 0 and f (3) 0 and with boundary...
Let u be the solution to the initial boundary value problem for the Heat Equation, au(t,z) 28?u(t,z), te (0,00), z (0,3); with initial condition u(0, z)fx), where f(0) 0 and f (3) 0 and with boundary conditions u(t,0)-0, r 30 Using separation of variables, the solution of this problem is 4X with the normalization conditions un(m3ī)-. n@) : ї, a. (5/10) Find the functions wn with index n1. Wnlz) b. (5/10) Find the functions vn with index n 1. n(t)...
PDE. Please show all steps in detail. 2. Consider the 1D heat equation in a rod...
PDE. Please show all steps in detail.
2. Consider the 1D heat equation in a rod of length with diffusion constant Suppose the left endpoint is convecting (in obedience to Newton's Law of Cooling with proportionality constant K-1) with an outside medium which is 5000. while the right endpoint is insulated. The initial temperature distribution in the rod is given by f(a)- 2000 -0.65 300, 0<<t (a) Set up the initial-boundary value problem modeling this scenario. (b) Set up and...
(4 points) This problem is concerned with solving an initial boundary value problem for the heat ...
(4 points) This problem is concerned with solving an initial boundary value problem for the heat equation: u,(x, t)- uxx(x,), 0
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...
12. Consider the unusual eigenvalue problem ux(0) = ur(l) = v(1)-U(0) (a) Show that 2 0 is a double eigenvalue. (b) Get an equation for the positive eigenvalues a>0. 102 CHAPTER 4 BOUNDARY PROBLEM...
12. Consider the unusual eigenvalue problem ux(0) = ur(l) = v(1)-U(0) (a) Show that 2 0 is a double eigenvalue. (b) Get an equation for the positive eigenvalues a>0. 102 CHAPTER 4 BOUNDARY PROBLEMS (c) Letting γ-IVA, reduce the equation in part (b) to the equation γ sin γ cos γ = sin (d) Use part (c) to find half of the eigenvalues explicitly and half of (e) Assuming that all the eigenvalues are nonnegative, make a list of (t)...
Mark which statements below are true, using the following Consider the diffusion problem u(0,t)=0...
Mark which statements below are true, using the following Consider the diffusion problem u(0,t)=0, u(L,t)=50 where FER is a constant, forcing term Any attempt to solve this using separation of variables fails. This is because the PDE is not homogeneous. A more fruitful approach arises from splitting the solution into the sum of two u(z,t) = X(z)T(t) + us(z), where the subscript designates the function as the steady limit and does not represent a derlvative. BEWARE: MARKING A STATEMENT TRUE...
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...
(1 point) This problem is concerned with solving an initial boundary value problem for the heat equation: (0,t)-0, t0 u,o)- in the form, ie where the term involving cy may be missing. Here y is the eigenfunction for Ay- 0 so if zero is not an eigenvalue then this term will be zero First find the eigenvalues and orthonormal eigenfunctions for n1.iA. Pa(x). For n 0 there may or may not be an eigenpair. Give all these as a comma...
1. Solve the boundary value problem ut =-3uzzzz + 5uzz, u(z, 0) = r(z) (-00 < z < oo, t > 0), using direct and inverse Fourier transforms U(w,t)-홅启u(z, t) ei r dr, u(z,t)-二U( ,t) e ur d . You need to explain where you use linearity of Fourier transform and how you transform derivatives in z and in t 2. Find the Fourier transform F() of the following function f(x) and determine whether F() is a continuous function (a)...
13. Find an infinite-series representation for the solution to the heat absorption- radiation problem in th uz(0,t)-aou(0, t) = 0, uz(1, t) + alu(1, t) = 0, t > 0, u(x,0) f(x), 0<a<1, where ao < 0, a1 >0, and ao +a> -aoa1. State why there is radiation at x = 1, absorption at z = 0, and the radiation greatly exceeds the absorp- tion. Choose ao--0.25 and a1 4 and find the first four eigenvalues; if f(x) = 2(1-2),...
Let u be the solution to the initial boundary value problem for the Heat Equation, au(t,z) 28?u(t,z), te (0,00), z (0,3); with initial condition u(0, z)fx), where f(0) 0 and f (3) 0 and with boundary conditions u(t,0)-0, r 30 Using separation of variables, the solution of this problem is 4X with the normalization conditions un(m3ī)-. n@) : ї, a. (5/10) Find the functions wn with index n1. Wnlz) b. (5/10) Find the functions vn with index n 1. n(t)...
PDE. Please show all steps in detail.
2. Consider the 1D heat equation in a rod of length with diffusion constant Suppose the left endpoint is convecting (in obedience to Newton's Law of Cooling with proportionality constant K-1) with an outside medium which is 5000. while the right endpoint is insulated. The initial temperature distribution in the rod is given by f(a)- 2000 -0.65 300, 0<<t (a) Set up the initial-boundary value problem modeling this scenario. (b) Set up and...
(4 points) This problem is concerned with solving an initial boundary value problem for the heat equation: u,(x, t)- uxx(x,), 0
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...
12. Consider the unusual eigenvalue problem ux(0) = ur(l) = v(1)-U(0) (a) Show that 2 0 is a double eigenvalue. (b) Get an equation for the positive eigenvalues a>0. 102 CHAPTER 4 BOUNDARY PROBLEMS (c) Letting γ-IVA, reduce the equation in part (b) to the equation γ sin γ cos γ = sin (d) Use part (c) to find half of the eigenvalues explicitly and half of (e) Assuming that all the eigenvalues are nonnegative, make a list of (t)...
Mark which statements below are true, using the following Consider the diffusion problem u(0,t)=0, u(L,t)=50 where FER is a constant, forcing term Any attempt to solve this using separation of variables fails. This is because the PDE is not homogeneous. A more fruitful approach arises from splitting the solution into the sum of two u(z,t) = X(z)T(t) + us(z), where the subscript designates the function as the steady limit and does not represent a derlvative. BEWARE: MARKING A STATEMENT TRUE...
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...
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