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5. Let ao 0 and au <0 be given. Consider the following eigenvalue problem with the Robin boundary conditions, ChaeX(L)=0 Show that there is one negative eigenvalue if ao + aL-aaLL and two negat...
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)...
solve the PDE +u= at2 on 3 € (0,L), t > 0, with boundary conditions au...
solve the PDE
+u= at2 on 3 € (0,L), t > 0, with boundary conditions au 2x2 u(0,t) = 0, u(L, t) = 0 au and initial condition u(x,0) = f(x), at (x,0) = g(x) following the steps below. (a) Separate the variables and write differential equations for the functions (x) and h(t); pick the separation constant so that we recover a problem already studied. (b) Find the eigenfunctions and eigenvalues. (c) Write the general solution for this problem. (d)...
Let u be the solution to the initial boundary value problem for the Heat Equation au(t,) -48Fu(t,), te (0,oo), z (0,5); with boundary conditions u(t,0) 0, u(t,5) 0, and with initial condition 5 15 15...
Let u be the solution to the initial boundary value problem for the Heat Equation au(t,) -48Fu(t,), te (0,oo), z (0,5); with boundary conditions u(t,0) 0, u(t,5) 0, and with initial condition 5 15 15 The solution u of the problem above, with the conventions given in class, has the form with the normalization conditions vn(0)-1, u Find the functions vnwn and the constants cn n(t) wnr)
Let u be the solution to the initial boundary value problem for the...
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.
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...
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 equations (la)-(1c) model? (Hint: Give an interpretation for the PDE, boundary conditions and intial condition.) c) Use the method of separation of variables to separate the above problem into two sub- problems (one that depends on space and the other only on time) (d) What...
Problem 9. Solve the LAPLACE equation Au=0 inside the unit ball with the following boundary conditions:...
Problem 9. Solve the LAPLACE equation Au=0 inside the unit ball with the following boundary conditions: a) 4(1., )=1. b) u(1,0,0)=0. c) (1,0,6)=sino.
Consider the below wave equation with the given conditions. au 81 Ox? u(0,1) het au 0...
Consider the below wave equation with the given conditions. au 81 Ox? u(0,1) het au 0 < x < 4, t > 0, u(4,t) = 0, 1 > 0 op u(x,0) = 0, ди at = 6x(4- x) = 384 ${1 - (-1)"} sin(npox/4), 0< x < 4. n=1 The solution to the above boundary-value problem is of the form u(x,t) = 8(n, t) sin "* n=1 Find the function g(n,1).
1. Show that the eigenvalue problem (4.75-4.77) has no negative eigenvalues. Hint: Use an energy argument-multiply the ODE by y and integrate from r 0 to r R; use integration by parts and use th...
1. Show that the eigenvalue problem (4.75-4.77) has no negative eigenvalues. Hint: Use an energy argument-multiply the ODE by y and integrate from r 0 to r R; use integration by parts and use the boundedness at r0 to get the boundary term to vanish. (4.75) which is Bessel's equation. Condition (4.72) leads to the boundary condition y(R)0, (4.76) and we impose the boundedness requirement y(0) bounded (4.77)
1. Show that the eigenvalue problem (4.75-4.77) has no negative eigenvalues. Hint:...
Solve the following problems subject to the given boundary conditions. Show the formulas for any ...
please
solve #2
Solve the following problems subject to the given boundary conditions. Show the formulas for any arbitrary constants (Ao, An, Bn), but you do not need to actually calculate them tu a(0. t)=0. u(1, t) = 5 u(z,0-82-1 2 0< x<2, t0 u(0, t) = 0, u(2. t) = 0 a(x, 0) 0, tr(r,0) = 0 3 ー+-=-10, 0
Consider the following boundary value problem, y" +(+5) y = 0, y'() = 0, y(9) =...
Consider the following boundary value problem, y" +(+5) y = 0, y'() = 0, y(9) = 0 (a) Find the eigenvalues. (b) Find the eigenfunctions. Take the arbitrary constant (either cu or c) from the general solution to be 1. Consider the following boundary value problem, y" + (8 + 5) y = 0, y'(o) = 0, 9) = 0 (a) Find the eigenvalues. (b) Find the eigenfunctions. Take the arbitrary constant (either cy or c2) from the general solution...
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)...
solve the PDE
+u= at2 on 3 € (0,L), t > 0, with boundary conditions au 2x2 u(0,t) = 0, u(L, t) = 0 au and initial condition u(x,0) = f(x), at (x,0) = g(x) following the steps below. (a) Separate the variables and write differential equations for the functions (x) and h(t); pick the separation constant so that we recover a problem already studied. (b) Find the eigenfunctions and eigenvalues. (c) Write the general solution for this problem. (d)...
Let u be the solution to the initial boundary value problem for the Heat Equation au(t,) -48Fu(t,), te (0,oo), z (0,5); with boundary conditions u(t,0) 0, u(t,5) 0, and with initial condition 5 15 15 The solution u of the problem above, with the conventions given in class, has the form with the normalization conditions vn(0)-1, u Find the functions vnwn and the constants cn n(t) wnr)
Let u be the solution to the initial boundary value problem for the...
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.
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 equations (la)-(1c) model? (Hint: Give an interpretation for the PDE, boundary conditions and intial condition.) c) Use the method of separation of variables to separate the above problem into two sub- problems (one that depends on space and the other only on time) (d) What...
Problem 9. Solve the LAPLACE equation Au=0 inside the unit ball with the following boundary conditions: a) 4(1., )=1. b) u(1,0,0)=0. c) (1,0,6)=sino.
Consider the below wave equation with the given conditions. au 81 Ox? u(0,1) het au 0 < x < 4, t > 0, u(4,t) = 0, 1 > 0 op u(x,0) = 0, ди at = 6x(4- x) = 384 ${1 - (-1)"} sin(npox/4), 0< x < 4. n=1 The solution to the above boundary-value problem is of the form u(x,t) = 8(n, t) sin "* n=1 Find the function g(n,1).
1. Show that the eigenvalue problem (4.75-4.77) has no negative eigenvalues. Hint: Use an energy argument-multiply the ODE by y and integrate from r 0 to r R; use integration by parts and use the boundedness at r0 to get the boundary term to vanish. (4.75) which is Bessel's equation. Condition (4.72) leads to the boundary condition y(R)0, (4.76) and we impose the boundedness requirement y(0) bounded (4.77)
1. Show that the eigenvalue problem (4.75-4.77) has no negative eigenvalues. Hint:...
please
solve #2
Solve the following problems subject to the given boundary conditions. Show the formulas for any arbitrary constants (Ao, An, Bn), but you do not need to actually calculate them tu a(0. t)=0. u(1, t) = 5 u(z,0-82-1 2 0< x<2, t0 u(0, t) = 0, u(2. t) = 0 a(x, 0) 0, tr(r,0) = 0 3 ー+-=-10, 0
Consider the following boundary value problem, y" +(+5) y = 0, y'() = 0, y(9) = 0 (a) Find the eigenvalues. (b) Find the eigenfunctions. Take the arbitrary constant (either cu or c) from the general solution to be 1. Consider the following boundary value problem, y" + (8 + 5) y = 0, y'(o) = 0, 9) = 0 (a) Find the eigenvalues. (b) Find the eigenfunctions. Take the arbitrary constant (either cy or c2) from the general solution...
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