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PDE Question thank you 0 0, t 4. Solve for ρ(x, t) on x Hint: Sketch the characteristics and consider how they relat...
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...
2. Consider the pde 0 <а < о, w(z,0) — 0, w(0, t) - t> 0, xwf = 0, = t Wr = (a) Use separation of variables to...
2. Consider the pde 0 <а < о, w(z,0) — 0, w(0, t) - t> 0, xwf = 0, = t Wr = (a) Use separation of variables to show that w(x, t) exp(k(t where k is a constant. (b) Show that the above solution does not satisfy both the initial and boundary conditions. (c) Use Laplace Transforms to solve the above pde.
2. Consider the pde 0
PDE question Consider the one dimensional wave equation on the half line: Ut(x,0) = g(x) Utt...
PDE
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Consider the one dimensional wave equation on the half line: Ut(x,0) = g(x) Utt - Uzx= 0 0 < < u(0,t) = 0 u(x,0) = f(x) (a) What is the solution? (b) For the particular initial conditions 12 - 2 25254 f(x) = { 6- 4<r<6 otherwise g(x) = 0 sketch the solution u(x, t) for t = 0, 2, 4, 6.
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
Courses LMS Integration Documentation Homework 4 EMTH 250-Advanced Math II-Spring 2019 Q1 0 solutions submitted (max: Unlimited) 12.3 Use of Fourier Series to Solve Wave PDE Find and sketch or gr...
Courses LMS Integration Documentation Homework 4 EMTH 250-Advanced Math II-Spring 2019 Q1 0 solutions submitted (max: Unlimited) 12.3 Use of Fourier Series to Solve Wave PDE Find and sketch or graph (as in Fig. 288 in Sec. 12.3) the deflection u(x, t) of a vibrating string of length π, extending from x 0 to x T, and c2 T/p 4 starting with velocity zero and deflection: sin3r Make use of the following formulas. Remeber that the initial deflection is f(x),...
2. Consider the following 1-D wave equation with initial condition u (x, 0)- F (x) where F(x) is a given function. a) Show that u (x, t)-F (x - t) is a solution to the given PDE. b) If the function F...
2. Consider the following 1-D wave equation with initial condition u (x, 0)- F (x) where F(x) is a given function. a) Show that u (x, t)-F (x - t) is a solution to the given PDE. b) If the function F is given as 1; x< 10 x > 10 u(x, 0) = F(x) = use part (a) to write the solution u(x, t) c) Sketch u(x,0) and u(x,1) on the same u-versus-x graph d) Explain in your own...
=T 20 marks) Consider the following PDE with boundary and initial conditions: U = Upx +...
=T 20 marks) Consider the following PDE with boundary and initial conditions: U = Upx + ur, for 0<x< 1 and to with u(0,t) = 1, u(1,t) = 0, u(1,0) = (a) Find the steady state solution, us(1), for the PDE. (b) Let Uſz,t) = u(?, t) – us(T). Derive a PDE plus boundary and initial conditions for U(2,t). Show your working. (c) Use separation of variables to solve the resulting problem for U. You may leave the inner products...
Please answer question number 2, Thank you Engineering Mathematics (-) # 6 HM. olve the PDE of the vibrating string with...
Please answer question number 2, Thank you
Engineering Mathematics (-) # 6 HM. olve the PDE of the vibrating string with given initial velocity and zero initial displacement by use of Fourier sine series. 02u(x,t) = c2-211(x,t) ax2 PDE. : t>0 0<x<L 2 , Ot , BCs u(0,1) 0u(L,t) 0, t20 IC u(x,0) = 0 , 0 x L : an(x,0) =h(x) 0 L x , ot in problem (1), u(x,t)=? (2). Suppose that h(x)-x(1-cos(-))
Engineering Mathematics (-) # 6...
clear writing please and thank you Problem 2. Given the PDE ut + uy = u?...
clear writing please and thank you
Problem 2. Given the PDE ut + uy = u? u(3,0) = g(x) for IER, >0, for ER. (a) Sketch the characteristic curve that passes through P(2,3) in the xt-plane. Find u(2,3) without using the exact solution ulit, t). (b) Use the method of characteristics to find the solution u(, t).
Please show all steps and write in a legible way. Thank you beforehand! Fully solve the following PDE. Consider a compre...
Please show all steps and
write in a legible way. Thank you beforehand!
Fully solve the following PDE. Consider a compressed gas container of length L, divided in half by a baffle To the left of the porous baffle is a gas of species A, and to the right of it is a different gas of species B. Suppose they are at the same pressure, so that when the baffle is removed at time t-0 the two gases proceed to...
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...
2. Consider the pde 0 <а < о, w(z,0) — 0, w(0, t) - t> 0, xwf = 0, = t Wr = (a) Use separation of variables to show that w(x, t) exp(k(t where k is a constant. (b) Show that the above solution does not satisfy both the initial and boundary conditions. (c) Use Laplace Transforms to solve the above pde.
2. Consider the pde 0
PDE
question
Consider the one dimensional wave equation on the half line: Ut(x,0) = g(x) Utt - Uzx= 0 0 < < u(0,t) = 0 u(x,0) = f(x) (a) What is the solution? (b) For the particular initial conditions 12 - 2 25254 f(x) = { 6- 4<r<6 otherwise g(x) = 0 sketch the solution u(x, t) for t = 0, 2, 4, 6.
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
Courses LMS Integration Documentation Homework 4 EMTH 250-Advanced Math II-Spring 2019 Q1 0 solutions submitted (max: Unlimited) 12.3 Use of Fourier Series to Solve Wave PDE Find and sketch or graph (as in Fig. 288 in Sec. 12.3) the deflection u(x, t) of a vibrating string of length π, extending from x 0 to x T, and c2 T/p 4 starting with velocity zero and deflection: sin3r Make use of the following formulas. Remeber that the initial deflection is f(x),...
2. Consider the following 1-D wave equation with initial condition u (x, 0)- F (x) where F(x) is a given function. a) Show that u (x, t)-F (x - t) is a solution to the given PDE. b) If the function F is given as 1; x< 10 x > 10 u(x, 0) = F(x) = use part (a) to write the solution u(x, t) c) Sketch u(x,0) and u(x,1) on the same u-versus-x graph d) Explain in your own...
=T 20 marks) Consider the following PDE with boundary and initial conditions: U = Upx + ur, for 0<x< 1 and to with u(0,t) = 1, u(1,t) = 0, u(1,0) = (a) Find the steady state solution, us(1), for the PDE. (b) Let Uſz,t) = u(?, t) – us(T). Derive a PDE plus boundary and initial conditions for U(2,t). Show your working. (c) Use separation of variables to solve the resulting problem for U. You may leave the inner products...
Please answer question number 2, Thank you
Engineering Mathematics (-) # 6 HM. olve the PDE of the vibrating string with given initial velocity and zero initial displacement by use of Fourier sine series. 02u(x,t) = c2-211(x,t) ax2 PDE. : t>0 0<x<L 2 , Ot , BCs u(0,1) 0u(L,t) 0, t20 IC u(x,0) = 0 , 0 x L : an(x,0) =h(x) 0 L x , ot in problem (1), u(x,t)=? (2). Suppose that h(x)-x(1-cos(-))
Engineering Mathematics (-) # 6...
clear writing please and thank you
Problem 2. Given the PDE ut + uy = u? u(3,0) = g(x) for IER, >0, for ER. (a) Sketch the characteristic curve that passes through P(2,3) in the xt-plane. Find u(2,3) without using the exact solution ulit, t). (b) Use the method of characteristics to find the solution u(, t).
Please show all steps and
write in a legible way. Thank you beforehand!
Fully solve the following PDE. Consider a compressed gas container of length L, divided in half by a baffle To the left of the porous baffle is a gas of species A, and to the right of it is a different gas of species B. Suppose they are at the same pressure, so that when the baffle is removed at time t-0 the two gases proceed to...
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