5. Consider a diffusion problem on the interval 0 < x < 1 governed by the...
5. Consider a diffusion problem on the interval 0 < x < 1 governed by the pde ut = uxx. Assume that the solution satisfies the boundary conditions u(0) = 0.5 and u′(1) = −3(u(1) − 0.7). Find the steady state solution v(x) for this problem. Hint: The steady state solution must satisfy the pde and the boundary conditions.
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5. Consider a diffusion problem on the interval 0 < x < 1
governed by the...
3) (25 marks) Consider the following problem: u2(0,t) 3, u(2,t)u(2,t), t>0 u(,0) 0, 0
3) (25 marks) Consider the following problem: u2(0,t) 3, u(2,t)u(2,t), t>0 u(,0) 0, 0<2 (a) Find the steady state solution u,(x) of this problem. b) Write a new PDE, boundary conditions and initial conditions for U(x, t) - u(x, t)- Cox) (c) Use separation of variables to find a solution to the PDE, boundary conditions and initial conditions. You must justify each step of your solution carefully to get full marks. (Hint: if you are unable to write the eigenvalues...
Consider the following second order PDE Uit – 9Uxx = 0, 0<x< < t > 0,...
Consider the following second order PDE Uit – 9Uxx = 0, 0<x< < t > 0, (A) and the following boundary value/initial conditions: Ux(t,0) = uſt, 5) = 0, t>0, u(0, x) = 44(0, x) = 4 cos’ x, 0<x< (BC) (IC) for the function u= u(t, x). a. (5 points) Find ordinary differential equations for functions T = T(t) and X = X(x) such that the function u(t, x) = T(t)X(x) satisfies the PDE (A). b. (5 points) Find...
Find the steady-state solution of the heat conduction equation α2uxx-ut that satisfies the given ...
Write out the solution
please
Find the steady-state solution of the heat conduction equation α2uxx-ut that satisfies the given set of boundary conditions. ux(0, t)-u(0, t) = 0, u(L, t)-T v(x) =
Find the steady-state solution of the heat conduction equation α2uxx-ut that satisfies the given set of boundary conditions. ux(0, t)-u(0, t) = 0, u(L, t)-T v(x) =
1 point) Solve the nonhomogeneous heat problem ut=uxx+4sin(2x), 0<x<π,ut=uxx+4sin(2x), 0<x<π, u(0,t)=0, u(π,t)=0u(0,t)=0, u(π,t)=0 u(x,0)=5sin(5x)u(x,0)=5sin(5x) u(x,t)=u(x,t)= Ste
1 point) Solve the nonhomogeneous heat problem
ut=uxx+4sin(2x), 0<x<π,ut=uxx+4sin(2x), 0<x<π,
u(0,t)=0, u(π,t)=0u(0,t)=0, u(π,t)=0
u(x,0)=5sin(5x)u(x,0)=5sin(5x)
u(x,t)=u(x,t)=
Steady State Solution limt→∞u(x,t)=limt→∞u(x,t)=
Please show all work.
(1 point) Solve the nonhomogeneous heat problem Ut = Uxx + 4 sin(2x), 0< x < , u(0,1) = 0, tu(T, t) = 0 u(x,0) = 5 sin(52) u(a,t) Steady State Solution limt u(x, t) = Note: You can earn partial credit on this problem. Preview My Answers Submit Answers You have attempted this problem 0 times. You have unlimited attempts...
2. In lectures we solved the heat PDE in 1 +1 dimensions with constant-temperature boundary conditions u(0,t)u(L,t) -0. If these boundary conditions change from zero temperature, we need to do a litt...
2. In lectures we solved the heat PDE in 1 +1 dimensions with constant-temperature boundary conditions u(0,t)u(L,t) -0. If these boundary conditions change from zero temperature, we need to do a little bit more work. Consider the following initial/boundary-value problem (IBVP) 2 (PDE) (BCs) (IC) u(0,t) = a, u(x,00, u(L, t)=b, st. and let's take L = 1, a = 1, b = 2 throughout for simplicity. Solve this problem using the following tricks b and A"(x)-0 (a) Find a...
Consider the diffusion problem au Ou u(0, t) = 0, u(L, t) = 50 u(x,0-fx where FER is a constant, ...
Mark which statements below are true, using the following:
Consider the diffusion problem au Ou u(0, t) = 0, u(L, t) = 50 u(x,0-fx 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 parts, taking into account that all change eventually dies out. That is there is a transient part...
Problem # 1 [15 Points] Consider the following PDE which describes a typical heat-flow problem PDE: ut = ↵2uxx, 0 < x < 1, 0 < t < 1 BCs: ux(0, t)=0 ux(1, t)=0 0 < t < 1 IC...
Problem # 1 [15 Points] Consider the following PDE which describes a typical heat-flow problem PDE: ut = ↵2uxx, 0 < x < 1, 0 < t < 1 BCs: ux(0, t)=0 ux(1, t)=0 0 < t < 1 IC: u(x, 0) = sin(⇡x), 0 x 1 (a) What is your physical interpretation of the above problem? (b) Can you draw rough sketches of the solution for various values of time? (c) What about the steady-state temperature?
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...
Problem 1 (20 points) Consider the PDE for the function u(x, t) e 0<x<T, t> 0...
Problem 1 (20 points) Consider the PDE for the function u(x, t) e 0<x<T, t> 0 with the boundary conditions n(0, t) 0, u(T, t) 0, t> 0 and the initial condition 0 u(x, 0) 1+cos(2a), (a) Give a one-sentence physical interpretation of this problem. (b) Find the solution u(x, t) using a Fourier cosine series representation An (t) cos(nax) u(x,t)= Ao(t) + n=1
1. Wave equation. Consider the wave equation on the finite interval (0, L) PDE BC where Neumann b...
1. Wave equation. Consider the wave equation on the finite interval (0, L) PDE BC where Neumann boundary conditions are specified Physically, with Neumann boundary conditions, u(r, t) could represent the height of a fluid that sloshes between two walls. (a) Find the general Fourier series solution by repeating the derivation from class now considering Neumann instead of Dirichlet boundary conditions. Your final solution should be (b) Consider the following general initial conditions u(x, 0)x) IC IC Derive formulas that...
3) (25 marks) Consider the following problem: u2(0,t) 3, u(2,t)u(2,t), t>0 u(,0) 0, 0<2 (a) Find the steady state solution u,(x) of this problem. b) Write a new PDE, boundary conditions and initial conditions for U(x, t) - u(x, t)- Cox) (c) Use separation of variables to find a solution to the PDE, boundary conditions and initial conditions. You must justify each step of your solution carefully to get full marks. (Hint: if you are unable to write the eigenvalues...
Consider the following second order PDE Uit – 9Uxx = 0, 0<x< < t > 0, (A) and the following boundary value/initial conditions: Ux(t,0) = uſt, 5) = 0, t>0, u(0, x) = 44(0, x) = 4 cos’ x, 0<x< (BC) (IC) for the function u= u(t, x). a. (5 points) Find ordinary differential equations for functions T = T(t) and X = X(x) such that the function u(t, x) = T(t)X(x) satisfies the PDE (A). b. (5 points) Find...
Write out the solution
please
Find the steady-state solution of the heat conduction equation α2uxx-ut that satisfies the given set of boundary conditions. ux(0, t)-u(0, t) = 0, u(L, t)-T v(x) =
Find the steady-state solution of the heat conduction equation α2uxx-ut that satisfies the given set of boundary conditions. ux(0, t)-u(0, t) = 0, u(L, t)-T v(x) =
1 point) Solve the nonhomogeneous heat problem
ut=uxx+4sin(2x), 0<x<π,ut=uxx+4sin(2x), 0<x<π,
u(0,t)=0, u(π,t)=0u(0,t)=0, u(π,t)=0
u(x,0)=5sin(5x)u(x,0)=5sin(5x)
u(x,t)=u(x,t)=
Steady State Solution limt→∞u(x,t)=limt→∞u(x,t)=
Please show all work.
(1 point) Solve the nonhomogeneous heat problem Ut = Uxx + 4 sin(2x), 0< x < , u(0,1) = 0, tu(T, t) = 0 u(x,0) = 5 sin(52) u(a,t) Steady State Solution limt u(x, t) = Note: You can earn partial credit on this problem. Preview My Answers Submit Answers You have attempted this problem 0 times. You have unlimited attempts...
2. In lectures we solved the heat PDE in 1 +1 dimensions with constant-temperature boundary conditions u(0,t)u(L,t) -0. If these boundary conditions change from zero temperature, we need to do a little bit more work. Consider the following initial/boundary-value problem (IBVP) 2 (PDE) (BCs) (IC) u(0,t) = a, u(x,00, u(L, t)=b, st. and let's take L = 1, a = 1, b = 2 throughout for simplicity. Solve this problem using the following tricks b and A"(x)-0 (a) Find a...
Mark which statements below are true, using the following:
Consider the diffusion problem au Ou u(0, t) = 0, u(L, t) = 50 u(x,0-fx 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 parts, taking into account that all change eventually dies out. That is there is a transient part...
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...
Problem 1 (20 points) Consider the PDE for the function u(x, t) e 0<x<T, t> 0 with the boundary conditions n(0, t) 0, u(T, t) 0, t> 0 and the initial condition 0 u(x, 0) 1+cos(2a), (a) Give a one-sentence physical interpretation of this problem. (b) Find the solution u(x, t) using a Fourier cosine series representation An (t) cos(nax) u(x,t)= Ao(t) + n=1
1. Wave equation. Consider the wave equation on the finite interval (0, L) PDE BC where Neumann boundary conditions are specified Physically, with Neumann boundary conditions, u(r, t) could represent the height of a fluid that sloshes between two walls. (a) Find the general Fourier series solution by repeating the derivation from class now considering Neumann instead of Dirichlet boundary conditions. Your final solution should be (b) Consider the following general initial conditions u(x, 0)x) IC IC Derive formulas that...
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