Consider the boundary value problem X ′′ +λX=0 , X ′ (0)=0 ,
X′(π)=0 . Find...
Consider the boundary value problem X ′′ +λX=0 , X ′ (0)=0 ,
X′(π)=0 . Find all real values of λ for which there is a
non-trivial solution of the problem and find the corresponding
solution.
Find the unique solution u of the parabolic boundary value
problem
Ut −Uxx =e^(−t)*sin(3x), 0<x<π, t>0,
U(0,t) = U(π,t) = 0, t > 0,
U(x, 0) = e^(π), 0 ≤ x ≤ π.
Consider the Sturm-Liouville problem
X′′(x) + λX(x) = 0 subject toX′(0) = 0, X(l) = 0.
Are the boundary conditions symmetric?
Do these boundary conditions yield negative eigenvalues?
Determine the eigenvalues and eigenfunctions, Xn(x). (It is
enough in some cases to provide the equation that determines the
eigenvalues rather than an explicit formula.)
Are the eigenfunctions orthogonal?
Consider the second-order boundary value problem
y′′ +(2x^2 +3)y′ −y =6x, 0≤x ≤1, (4)
y(0) = 1, y(1) = 0.
(a) Rewrite the second-order equation (4) as a system of two
first-order equations
involving variables y and z. [2]
(b) Suppose that yn and zn are approximations to y(xn) and
z(xn), respectively, where xn = nh, n = 0,...,N and h = 1/N for
some positive integer N. Find the iterative formula when using the
modified Euler method to approximate...
Using MATLAB:
Consider the following Boundary Value Differential Equation:
y''+4y=0
y(0)=-2
y(π/4)=10
Which has the exact solution: y(x)= -2cos(2x)+10sin(2x)
Create a program that will allow the user to input the step size
(in x), and two guesses for y'(0). The program will then use the
Euler method along with the shooting method to solve this problem.
The program should give the true error at y(π/8). Run your code
with step sizes of π/400 and π/4000 and compare the errors. Chose...
Find the eigenvalues λn and eigenfunctions
yn(x) for the given boundary-value problem. (Give your
answers in terms of k, making sure that each value of
k corresponds to two unique eigenvalues.)
y'' + λy = 0, y(−π) = 0, y(π) = 0
λ2k − 1 =, k=1,2,3,...
y2k − 1(x) =, k=1,2,3,...
λ2k =, k=1,2,3,...
y2k(x) =, k=1,2,3,...