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Show all steps Problem 1. Consider the solution ur, t) 1-12 -2t of the heat equation ut the location of its maximum and minimum on the closed rectangle {0 x 1,0 Find t 12-2t of the heat equation ut-1...
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)...
7. Find the solution of the heat conduction problem 100uzz = ut, 0 < x <...
7. Find the solution of the heat conduction problem 100uzz = ut, 0 < x < 1, t > 0; u(0,t) 0, u1,t 0, t>0; In Problem 10, consider the conduction of heat in a rod 40 cm in length whose ends are maintained at 0°C for all t0. Find an expression for the temperature u(,t) if the initial temperature distribution in the rod is the given function. Suppose that a
Heat and Laplace equation problem 3. Solve ut – Uz = 0 with u(1,0) = 1,...
Heat and Laplace equation problem
3. Solve ut – Uz = 0 with u(1,0) = 1, and u (0,t) = U,(2,t) = 0.
Problem 4: Consider the following problem for the heat equation (1) (2) (3) ut= Uxa + s(t), xE (0,1), t > 0 u(0, t)...
Problem 4: Consider the following problem for the heat equation (1) (2) (3) ut= Uxa + s(t), xE (0,1), t > 0 u(0, t) 2, u(1, t) = 4 и (х, 0) — 2(1 — х). where s(t) describes the source term (a) Find a series solution for u(x, t) with s(t) = e"1. (b) What is the convergence criteria for the transient extension function if s(t) = 0.
Problem 4: Consider the following problem for the heat equation (1)...
Problem 4: Consider the following problem for the heat equation (1) (2) (3) ut= Uxa + s(t), xE (0,1), t > 0 u(0, t)...
Problem 4: Consider the following problem for the heat equation (1) (2) (3) ut= Uxa + s(t), xE (0,1), t > 0 u(0, t) 2, u(1, t) = 4 и (х, 0) — 2(1 — х). where s(t) describes the source term (a) Find a series solution for u(x, t) with s(t) = e"1. (b) What is the convergence criteria for the transient extension function if s(t) = 0.
Problem 4: Consider the following problem for the heat equation (1)...
6. Find the solution of the 1-dimensional heat equation on the interval 0, : Uxx, Ur...
6. Find the solution of the 1-dimensional heat equation on the interval 0, : Uxx, Ur (t, 0) U1(t, T) = 0, u(0, x) = 100 cos 2x
6. Find the solution of the 1-dimensional heat equation on the interval 0, : Uxx, Ur (t, 0) U1(t, T) = 0, u(0, x) = 100 cos 2x
Q5. Consider the Heat Equation as the following boundary-value problem, find the solution u(x, t) by...
Q5. Consider the Heat Equation as the following boundary-value problem, find the solution u(x, t) by using separation-variables methods. (25 Points) (Boundary Condition : ux0,t) = 0 ux(10,t) = 0 Heat Equation ut = 9uxx (Hint: u(xt) = X(X)T(t)) Initial Condition : u(x,0) = 0.01x(10-x)
PROBLEM 1 IS SUPPOSED TO BE A WAVE EQUATION NOT HEAT EQUATION 1. Find the solution...
PROBLEM 1 IS SUPPOSED TO BE A WAVE EQUATION NOT HEAT
EQUATION
1. Find the solution to the following boundary value initial value problem for the Heat Equation au 22u 22 = 22+ 2 0<x<1, c=1 <3 <1, C u(0,t) = 0 u(1,t) = 0 (L = 1) u(x,0) = f(x) = 3 sin(7x) + 2 sin (3x) (initial conditions) (2,0) = g(x) = sin(2x) 2. Find the solution to the following boundary value problem on the rectangle 0 <...
3. Consider the non homogeneous heat equation ut- urr+ 1 with non homogeneous boundary conditions u(0. t) 1, u(1t) (a) Find the equilibrium solution ueqx) to the non homogeneous equation. (b) The sol...
3. Consider the non homogeneous heat equation ut- urr+ 1 with non homogeneous boundary conditions u(0. t) 1, u(1t) (a) Find the equilibrium solution ueqx) to the non homogeneous equation. (b) The solution w(r, t) to the homogenized PDE wt-Wra, with w(0,t,t)0 1S -1 Verify that ugen(x, t)Ue(x) +w(x, t) solves the full PDE and BCs (c) Let u(x,0)- f(x) - 2 - ^2 be the initial condition. Find the particular solution by specifying all Fourier coefficients
3. Consider the...
Problem 1. Consider the nonhomogeneous heat equation for u(,) subject to the nonhomogeneous boundary conditions 14(0,t) 1, u(r,t)-0,t> and the initial condition the solution u(x, t) by complet...
Problem 1. Consider the nonhomogeneous heat equation for u(,) subject to the nonhomogeneous boundary conditions 14(0,t) 1, u(r,t)-0,t> and the initial condition the solution u(x, t) by completing each of the following steps (a) Find the equilibrium temperature distribution u ( (b) Denote v, t)t) - u(). Derive the IBVP for the function vz,t). (c) Find v(x, t) (d) Find u(x, t)
Problem 1. Consider the nonhomogeneous heat equation for u(,) subject to the nonhomogeneous boundary conditions 14(0,t) 1, u(r,t)-0,t>...
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)...
7. Find the solution of the heat conduction problem 100uzz = ut, 0 < x < 1, t > 0; u(0,t) 0, u1,t 0, t>0; In Problem 10, consider the conduction of heat in a rod 40 cm in length whose ends are maintained at 0°C for all t0. Find an expression for the temperature u(,t) if the initial temperature distribution in the rod is the given function. Suppose that a
Heat and Laplace equation problem
3. Solve ut – Uz = 0 with u(1,0) = 1, and u (0,t) = U,(2,t) = 0.
Problem 4: Consider the following problem for the heat equation (1) (2) (3) ut= Uxa + s(t), xE (0,1), t > 0 u(0, t) 2, u(1, t) = 4 и (х, 0) — 2(1 — х). where s(t) describes the source term (a) Find a series solution for u(x, t) with s(t) = e"1. (b) What is the convergence criteria for the transient extension function if s(t) = 0.
Problem 4: Consider the following problem for the heat equation (1)...
Problem 4: Consider the following problem for the heat equation (1) (2) (3) ut= Uxa + s(t), xE (0,1), t > 0 u(0, t) 2, u(1, t) = 4 и (х, 0) — 2(1 — х). where s(t) describes the source term (a) Find a series solution for u(x, t) with s(t) = e"1. (b) What is the convergence criteria for the transient extension function if s(t) = 0.
Problem 4: Consider the following problem for the heat equation (1)...
6. Find the solution of the 1-dimensional heat equation on the interval 0, : Uxx, Ur (t, 0) U1(t, T) = 0, u(0, x) = 100 cos 2x
6. Find the solution of the 1-dimensional heat equation on the interval 0, : Uxx, Ur (t, 0) U1(t, T) = 0, u(0, x) = 100 cos 2x
Q5. Consider the Heat Equation as the following boundary-value problem, find the solution u(x, t) by using separation-variables methods. (25 Points) (Boundary Condition : ux0,t) = 0 ux(10,t) = 0 Heat Equation ut = 9uxx (Hint: u(xt) = X(X)T(t)) Initial Condition : u(x,0) = 0.01x(10-x)
PROBLEM 1 IS SUPPOSED TO BE A WAVE EQUATION NOT HEAT
EQUATION
1. Find the solution to the following boundary value initial value problem for the Heat Equation au 22u 22 = 22+ 2 0<x<1, c=1 <3 <1, C u(0,t) = 0 u(1,t) = 0 (L = 1) u(x,0) = f(x) = 3 sin(7x) + 2 sin (3x) (initial conditions) (2,0) = g(x) = sin(2x) 2. Find the solution to the following boundary value problem on the rectangle 0 <...
3. Consider the non homogeneous heat equation ut- urr+ 1 with non homogeneous boundary conditions u(0. t) 1, u(1t) (a) Find the equilibrium solution ueqx) to the non homogeneous equation. (b) The solution w(r, t) to the homogenized PDE wt-Wra, with w(0,t,t)0 1S -1 Verify that ugen(x, t)Ue(x) +w(x, t) solves the full PDE and BCs (c) Let u(x,0)- f(x) - 2 - ^2 be the initial condition. Find the particular solution by specifying all Fourier coefficients
3. Consider the...
Problem 1. Consider the nonhomogeneous heat equation for u(,) subject to the nonhomogeneous boundary conditions 14(0,t) 1, u(r,t)-0,t> and the initial condition the solution u(x, t) by completing each of the following steps (a) Find the equilibrium temperature distribution u ( (b) Denote v, t)t) - u(). Derive the IBVP for the function vz,t). (c) Find v(x, t) (d) Find u(x, t)
Problem 1. Consider the nonhomogeneous heat equation for u(,) subject to the nonhomogeneous boundary conditions 14(0,t) 1, u(r,t)-0,t>...
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