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1. Consider the insulated heat equation up = cum, 0 <r<L, t > 0 u (0,t)...
3. (20 points) Denote u(ar, y) the steady-state temperature in a rectangle area 0 z 10,...
3. (20 points) Denote u(ar, y) the steady-state temperature in a rectangle area 0 z 10, 0yS 1. Find the temperature in the rectangle if the temperature on the up side is kept at 0°, the lower side at 10° while the temperature on the left side is S0)= sin(y) and the right side is insulated. Answer the following questions. (a) (10 points) Write the Dirichlet problem including the Laplace's equation in two dimensions and the boundary conditions. (b) (10...
츨…<L. t-o(see (1) in Section i2 4) s bject to the given conditions. Solve the wave equation, .a a r u(0, t)=0, u(T, t)=0, t> 0 ux, o) 0.01 sin(5tx), 0u t=0 u(x, t) = n=1 Need Help? LRead it ....
츨…<L. t-o(see (1) in Section i2 4) s bject to the given conditions. Solve the wave equation, .a a r u(0, t)=0, u(T, t)=0, t> 0 ux, o) 0.01 sin(5tx), 0u t=0 u(x, t) = n=1 Need Help? LRead it . Talk to a Tutor l
츨… 0 ux, o) 0.01 sin(5tx), 0u t=0 u(x, t) = n=1 Need Help? LRead it . Talk to a Tutor l
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...
14 points Consider the following equation : PDE: u+ 0 ,0<x <1, 0<y <1 BCs: u(0, y)= 0, u (1, y ) = 0 ,0...
14 points Consider the following equation : PDE: u+ 0 ,0<x <1, 0<y <1 BCs: u(0, y)= 0, u (1, y ) = 0 ,0<y <1 ICs: u (x,0)=0, u (x,1)=2 ,0<x <1 a) Using the PDE and the boundary conditions write the form of the solution u (x ,t) b) Now apply the initial condition to solve for the unknown coefficients in the solution from part (a)
14 points Consider the following equation : PDE: u+ 0 ,0
Solve the Dirichlet problem in an infinite strip uxx + uyy=0 for x ϵ R and 0 <y <b , u(x,0)=f(x) , u(x,b)=g(x). (...
Solve the Dirichlet problem in an infinite strip
uxx + uyy=0
for x ϵ R and 0 <y <b ,
u(x,0)=f(x) ,
u(x,b)=g(x). (Hint: first
do the case f=0. The case g=0 reduces to this one
by the substitution y→ b-y , and the case general is
obtained by superposition)
4. Solve the Dirichlet problem in an infinite strip: uxx + Uyy 0 <у<b, u(x, 0) — S(x), и(х, b) — g(x). (Hint: First do the case The case g...
a) Find the solution to the following interior Dirichlet problem with radius R=1 1 PDE Urr...
a) Find the solution to the following interior Dirichlet problem with radius R=1 1 PDE Urr + Up t 0 0 <r <1 wee p2 r BC u (1,0) = 10 + 3 sin(0) 10 cos(20) 0 <0 < 27 b) Consider the above problem on the unit square (x,y) domain PDE Urr + Uyy = 0 0<x<1 0<y <1 Transform the solution u(r, 0) from "a)" to the solution u(x, y) for "b)" Use the solution u(x,y) to calculate...
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. Using separation of variables to solve the heat equation, u- kuxx on the interval 0 < x< 1 with boundary conditions ux(0, t) = 0 and ux(1, t) yields the general solution, 1, 0<x < 1/2...
3. Using separation of variables to solve the heat equation, u- kuxx on the interval 0 < x< 1 with boundary conditions ux(0, t) = 0 and ux(1, t) yields the general solution, 1, 0<x < 1/2 0, 1/2 x<1 Determine the coefficients An (n = 0, 1, 2, . . .) when u(x,0) = f(x) =
3. Using separation of variables to solve the heat equation, u- kuxx on the interval 0
Q , Solve the heat equation in one dimension: subject to the conditions u (0,t)-u (π ,t )-0 and V (x,0) sin 3x Q , Solve the heat equation in one dimension: subject to the conditions u (0,t)-u (...
Q , Solve the heat equation in one dimension: subject to the conditions u (0,t)-u (π ,t )-0 and V (x,0) sin 3x
Q , Solve the heat equation in one dimension: subject to the conditions u (0,t)-u (π ,t )-0 and V (x,0) sin 3x
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...
3. (20 points) Denote u(ar, y) the steady-state temperature in a rectangle area 0 z 10, 0yS 1. Find the temperature in the rectangle if the temperature on the up side is kept at 0°, the lower side at 10° while the temperature on the left side is S0)= sin(y) and the right side is insulated. Answer the following questions. (a) (10 points) Write the Dirichlet problem including the Laplace's equation in two dimensions and the boundary conditions. (b) (10...
츨…<L. t-o(see (1) in Section i2 4) s bject to the given conditions. Solve the wave equation, .a a r u(0, t)=0, u(T, t)=0, t> 0 ux, o) 0.01 sin(5tx), 0u t=0 u(x, t) = n=1 Need Help? LRead it . Talk to a Tutor l
츨… 0 ux, o) 0.01 sin(5tx), 0u t=0 u(x, t) = n=1 Need Help? LRead it . Talk to a Tutor l
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...
14 points Consider the following equation : PDE: u+ 0 ,0<x <1, 0<y <1 BCs: u(0, y)= 0, u (1, y ) = 0 ,0<y <1 ICs: u (x,0)=0, u (x,1)=2 ,0<x <1 a) Using the PDE and the boundary conditions write the form of the solution u (x ,t) b) Now apply the initial condition to solve for the unknown coefficients in the solution from part (a)
14 points Consider the following equation : PDE: u+ 0 ,0
Solve the Dirichlet problem in an infinite strip
uxx + uyy=0
for x ϵ R and 0 <y <b ,
u(x,0)=f(x) ,
u(x,b)=g(x). (Hint: first
do the case f=0. The case g=0 reduces to this one
by the substitution y→ b-y , and the case general is
obtained by superposition)
4. Solve the Dirichlet problem in an infinite strip: uxx + Uyy 0 <у<b, u(x, 0) — S(x), и(х, b) — g(x). (Hint: First do the case The case g...
a) Find the solution to the following interior Dirichlet problem with radius R=1 1 PDE Urr + Up t 0 0 <r <1 wee p2 r BC u (1,0) = 10 + 3 sin(0) 10 cos(20) 0 <0 < 27 b) Consider the above problem on the unit square (x,y) domain PDE Urr + Uyy = 0 0<x<1 0<y <1 Transform the solution u(r, 0) from "a)" to the solution u(x, y) for "b)" Use the solution u(x,y) to calculate...
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. Using separation of variables to solve the heat equation, u- kuxx on the interval 0 < x< 1 with boundary conditions ux(0, t) = 0 and ux(1, t) yields the general solution, 1, 0<x < 1/2 0, 1/2 x<1 Determine the coefficients An (n = 0, 1, 2, . . .) when u(x,0) = f(x) =
3. Using separation of variables to solve the heat equation, u- kuxx on the interval 0
Q , Solve the heat equation in one dimension: subject to the conditions u (0,t)-u (π ,t )-0 and V (x,0) sin 3x
Q , Solve the heat equation in one dimension: subject to the conditions u (0,t)-u (π ,t )-0 and V (x,0) sin 3x
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...
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