The transport of pollution in an aquifer is described by advection-reaction equation an of the form...
#2 ONLY PLEASE 1. Consider the non-Sturm-Liouville differential equation Multiply this equation by H(x). Determine H(x) such that the equation may be reduced to the standard Sturm-Liouville form: do Given a(z), 3(2), and 7(2), what are p(x), σ(x), and q(x) 2. Consider the eigenvalue problem (a) Use the result from the previous problem to put this in Sturm-Liouville form (b) Using the Rayleigh quotient, show that λ > 0. (c) Solve this equation subject to the boundary conditions and determine...
(a) Consider the following ODE ф" — 4ф + 4ф+ 0 (1) - with (0)(1)0 i. Put (1 into standard Sturm-Liouville form ii. Find the corresponding eigenvalue relation and eigenfunctions. Note that you do not have to normalise the eigenfunctions. (b) Solve the heat equation (2) 0<х<1 t>0 Ut u(0, t) u(1,t) u(x, 0) sin(2тx) + 1 (a) Consider the following ODE ф" — 4ф + 4ф+ 0 (1) - with (0)(1)0 i. Put (1 into standard Sturm-Liouville form ii....
6. Consider the eigenvalue problem 1 < x < 2, y(1) = 0, y(2) = 0. (a) Write the problem in Sturm-Liouville form, identifying p, q, and w. (b) Is the problem regular? Explain |(c) Is the operator S symmetric? Explain. (d) Find all eigenvalues and eigenfunctions. Discuss in light of Theorem 4.3 (e) Find the orthogonal expansion of f(x) = ln x, 1 < x < 2, in terms of these eigenfunctions. (f) Find the smallest N such that...
please solve all 3 Differential Equation problems 3.8.7 Question Help Consider the following eigenvalue problem for which all of its eigenvalues are nonnegative y',thy-0; y(0)-0, y(1) + y'(1)-0 (a) Show that λ =0 is not an eigenvalue (b) Show that the eigenfunctions are the functions {sin α11,o, where αη įs the nth positive root of the equation tan z -z (c) Draw a sketch indicating the roots as the points of intersection of the curves y tan z and y...
Solve part (d) 6. Consider the eigenvalue problem 2"xy3y Ay 0 y(1)0, y(2)= 0. + 1 < x< 2, (a) Write the problem in Sturm-Liouville form, identifying p, q, and w. (b) Is the problem regular? Explain (c) Is the operator S symmetric? Explain (d) Find all eigenvalues and eigenfunctions. Discuss in light of Theorem 4.3 ln x, 1 < 2, in terms of these (e) Find the orthogonal expansion of f(x) eigenfunctions _ 6. Consider the eigenvalue problem 2"xy3y...
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)...
The conductive heat transfer in a rod of length L is described by the equation au ди əraat ,0<r<L,+20 where u(x, t) is the local temperature of the rod, t is time, and a is a positive constant describing the thermal conductivity of the rod. The initial and boundary conditions are: T(r, 0) = 0, T(L, t) = 0, and T (0, 1) = 1 for > 0 (1) Find the general solution of this PDE. (11) Find the eigenvalues...
4. Advection-diffusion on a bounded domain: Suppose that a large population of ants is following a pheromone trail to a new nesting site, located at z = L. Ants originate frorn their nest at z-0, with flux q(0,t)-α-constant. Once an ant reaches the new nesting site, it stays there, and is removed from the domain. In the domain x E [0, L], the density of ants ρ(x, t) obeys the advection-diffusion equation (a) Explain why the boundary conditions are&L--:-Has-a (b)...
(4 points) This problem is concerned with solving an initial boundary value problem for the heat equation: u,(x, t)- uxx(x,), 0
3. (10 Points, part III) Consider the Sturm-Liouville differential equation where the coefficients p(z), q(z), and σ(z) are real and continous on la, b , and p(2) and σ(z) are strictly positive for all a,b (a) Derive the Rayleigh quotient λ from (2). b) What does this quotient describe? Give two examples of applications for this formula. (c) what are the neces,ary conditions for λ > 0 to be satisfied? (d) Recall that the minimum value of the Rayleigh quotient...