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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...
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...
PDE questions. Please show all
steps in detail.
2. Consider the initial-boundary value problem 0
4. Consider the boundary value problem defined by the partial differential equation д?и д?и = 0,...
4. Consider the boundary value problem defined by the partial differential equation д?и д?и = 0, ду? y > 0, да? with boundary conditions u(0, y) = u(T,y) = 0, u(x, 0) = 1 and limy-v00 |u(x, y)|< 0o. (a) Use separation of variables to find the eigenvalues and general series solution in terms of the normal modes. (b) Impose the inhomogeneous boundary condition u(x,0) = 1 to find the constants in the general series solution and hence the solution...
3. Consider the follo 2 lu The boundary conditions are: u(0,y, t) - u(x, 0,t) - 0, ou (a, y, t) =...
*Note: Please answer all parts, and explain all workings. Thank
you!
3. Consider the follo 2 lu The boundary conditions are: u(0,y, t) - u(x, 0,t) - 0, ou (a, y, t) = (x, b, t) = 0 ay The initial conditions are: at t-0,11-4 (x,y)--Yo(x,y) . ot a) Assume u(x,y,t) - X(x)Y(y)T(t), derive the eigenvalue problems: a) Apply the boundary conditions and derive all the possible eigenvalues for λι, λ2 and corresponding eigen-functions, Xm,Yn b) for any combination of...
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)...
Consider the partial differential equation together with the boundary conditions u(0, t) 0 and u(1,t)0 for t20 and the initial condition u(z,0) = z(1-2) for 0 < x < 1. (a) If n is a positive in...
Consider the partial differential equation together with the boundary conditions u(0, t) 0 and u(1,t)0 for t20 and the initial condition u(z,0) = z(1-2) for 0 < x < 1. (a) If n is a positive integer, show that the function , sin(x), satisfies the given partial differential equation and boundary conditions. (b) The general solution of the partial differential equation that satisfies the boundary conditions is Write down (but do not evaluate) an integral that can be used to...
Question 1 - 16 Consider the following intial-boundary value problem. au au 0<x< 1, 10, at2...
Question 1 - 16 Consider the following intial-boundary value problem. au au 0<x< 1, 10, at2 ax?' u(0,t) = u(11,t) = 0, 7>0, u(x,0) = 1, 34(x,0) = sin10x + 7sin50x. (show all your works). A) Find the two ordinary differential equations (ODES). B) Solve these two ODES. Show all cases 1 <0, 1 = 0, and > 0 C) Write the complete solution of this initial - boundary value problem.
Partial Differential Equation - Wave equation : Vibrating spring Question 2 A plucked string, Figure 2 shows the initial position function f (x) for a stretched string (of length L) that is set in mot...
Partial Differential
Equation
- Wave equation : Vibrating spring
Question 2 A plucked string, Figure 2 shows the initial position function f (x) for a stretched string (of length L) that is set in motion by moving t at midpoint x =-aside the distance-bL and releasing it from rest timet- 0. f (x) bL Figure 2 (a) If the length of string is 10cm with amplitude 5cm was set initially, state the initial condition and the boundary conditions for the...
6. a) For a thin conducting rod of length L = π, the temperature U(x, t) at a point 0 Sx S L at timet>0 is determined by the differential equation U, Uxx with boundary data U(x, 0) fx) and U(0...
6. a) For a thin conducting rod of length L = π, the temperature U(x, t) at a point 0 Sx S L at timet>0 is determined by the differential equation U, Uxx with boundary data U(x, 0) fx) and U(0,) UL, t)- 0 for all0. Show that for any positive integer k, the function U(x, t)- exp (-ak21) sin kx is a solution. It follows that Σ exp (-ak2 t) Bk sin kx is the general solution where Σ...
Consider the second order partial differential equation du/dt= d^2u/dx^2 +2du/dx+u over the domain x in [0,l) and t>=...
Consider the second order partial differential equation du/dt=
d^2u/dx^2 +2du/dx+u over the domain x in [0,l) and t>=0. It is
given that u(0,t)=u(l,t)=0. Use the method of separation of
variables to prove that the general solution with the given
boundary condition is u(x,t)= infinity series n=1
bnsin(npix/l)exp(-x-((npi/l)^2)t) where bn is
a constant for every n N
Hint u(x,t)=X(x)T(t)
tnsit te Seind ond partial difertinl cuatan +2n St the dowain e To,e) an Use metod o Separet ion Vaiades to rore...
PDE questions. Please show all
steps in detail.
2. Consider the initial-boundary value problem 0
4. Consider the boundary value problem defined by the partial differential equation д?и д?и = 0, ду? y > 0, да? with boundary conditions u(0, y) = u(T,y) = 0, u(x, 0) = 1 and limy-v00 |u(x, y)|< 0o. (a) Use separation of variables to find the eigenvalues and general series solution in terms of the normal modes. (b) Impose the inhomogeneous boundary condition u(x,0) = 1 to find the constants in the general series solution and hence the solution...
*Note: Please answer all parts, and explain all workings. Thank
you!
3. Consider the follo 2 lu The boundary conditions are: u(0,y, t) - u(x, 0,t) - 0, ou (a, y, t) = (x, b, t) = 0 ay The initial conditions are: at t-0,11-4 (x,y)--Yo(x,y) . ot a) Assume u(x,y,t) - X(x)Y(y)T(t), derive the eigenvalue problems: a) Apply the boundary conditions and derive all the possible eigenvalues for λι, λ2 and corresponding eigen-functions, Xm,Yn b) for any combination of...
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)...
Consider the partial differential equation together with the boundary conditions u(0, t) 0 and u(1,t)0 for t20 and the initial condition u(z,0) = z(1-2) for 0 < x < 1. (a) If n is a positive integer, show that the function , sin(x), satisfies the given partial differential equation and boundary conditions. (b) The general solution of the partial differential equation that satisfies the boundary conditions is Write down (but do not evaluate) an integral that can be used to...
Question 1 - 16 Consider the following intial-boundary value problem. au au 0<x< 1, 10, at2 ax?' u(0,t) = u(11,t) = 0, 7>0, u(x,0) = 1, 34(x,0) = sin10x + 7sin50x. (show all your works). A) Find the two ordinary differential equations (ODES). B) Solve these two ODES. Show all cases 1 <0, 1 = 0, and > 0 C) Write the complete solution of this initial - boundary value problem.
Partial Differential
Equation
- Wave equation : Vibrating spring
Question 2 A plucked string, Figure 2 shows the initial position function f (x) for a stretched string (of length L) that is set in motion by moving t at midpoint x =-aside the distance-bL and releasing it from rest timet- 0. f (x) bL Figure 2 (a) If the length of string is 10cm with amplitude 5cm was set initially, state the initial condition and the boundary conditions for the...
6. a) For a thin conducting rod of length L = π, the temperature U(x, t) at a point 0 Sx S L at timet>0 is determined by the differential equation U, Uxx with boundary data U(x, 0) fx) and U(0,) UL, t)- 0 for all0. Show that for any positive integer k, the function U(x, t)- exp (-ak21) sin kx is a solution. It follows that Σ exp (-ak2 t) Bk sin kx is the general solution where Σ...
Consider the second order partial differential equation du/dt=
d^2u/dx^2 +2du/dx+u over the domain x in [0,l) and t>=0. It is
given that u(0,t)=u(l,t)=0. Use the method of separation of
variables to prove that the general solution with the given
boundary condition is u(x,t)= infinity series n=1
bnsin(npix/l)exp(-x-((npi/l)^2)t) where bn is
a constant for every n N
Hint u(x,t)=X(x)T(t)
tnsit te Seind ond partial difertinl cuatan +2n St the dowain e To,e) an Use metod o Separet ion Vaiades to rore...
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