Boundary value problem Definition and 79 Threads

Homework Statement Use a Green's function to solve: u" + 2u' + u = e-x with u(0) = 0 and u(1) = 1 on 0\leqx\leq1 Homework Equations This from the lecture notes in my course: The Attempt at a Solution Solving for the homogeneous equation first: u" + 2u' + u = 0...

Homework Statement Find the general solution to the boundary value problem. Homework Equations (xy')' + \lambda x^{-1}y = 0 y(1) = 0 y(e) = 0 use x = e^t The Attempt at a Solution x = e^t so \frac{dx}{dt} = e^t using chain rule: y' = e^{-t}\frac{dy}{dt} Substituting...

Boundary Value Problem + Green's Function Consider the BVP y''+4y=e^x y(0)=0 y'(1)=0 Find the Green's function for this problem. I am completely lost can someone help me out?

let be the two boundary value problem -D^{2}y(x)+f(x)y(x)= \lambda _{n} y(x) with y(0)=0=y(\infty) and the same problem -D^{2}y(x)+f(x)y(x)= \beta _{n} y(x) with y(-\infty)=0=y(\infty) i assume that in both cases the problem is SOLVABLE , so my question is , are the eigenvalues in...

I have dealt quite a lot with the boundary value electrostatics problem with a plane or spherical conducting surface in an electric field due to a single electric charge or dipole. This can be conveniently done using the method of images. Method of images simplifies a lot of things. Jackson's...

Consider the BVP y''+4y=f(x) (0\leqx\leq1) y(0)=0 y'(1)=0 Find the Green's function (two-sided) for this problem. Working: So firstly, I let y(x)=Asin2x+Bcos2x Then using the boundary conditions, Asin(2.0)+Bcos(2.0)=0 => B=0 y'(x)=2Acos(2x)-2Asin(2x) y'(0)=2A=0...

Homework Statement y''+\lambday=0 y'(0)=0 y'(pi)=0 Homework Equations The Attempt at a Solution What's puzzling me is the case when we check if the eigenvalue is zero. y''=0 y'=C1 y=C1x+C2 Now when I check the first boundary value I get C1=0 now How do I check the second one ? with the...

Homework Statement y'' + ßy = 0, y'(0)=0, y'(L)=0 Homework Equations Meh The Attempt at a Solution I so already did the ß>1 and ß<1; I'm stuck on the ß=0. It seems easy enough. y'' = 0 -----> y' = A -----> 0=A, 0=A (from the two initial conditions) ------> No...

So here's the problem: I'm asked to find the solutions to the 1-D Wave equation u_{tt} = u_{xx} subject to u(x,0) = g(x), u_t(x,0) = h(x) but also u_t(0,t) = A*u_x(0,t) and discuss why A = -1 does not allow valid solutions. I can't figure it out at all. The solutions to...

I want to solve: y(x)''-(\frac{m\pi}{a})^2y(x)=0 With boundary condition y(0)=y(a)=0. First part is very easy using constant coef. which give: y(x)=c_1 cosh(\frac{m\pi}{a}x) + c_2 sinh(\frac{m\pi}{a}x) y(0)=0 \;\Rightarrow\; c_1=0 \;\Rightarrow\; y(x) = c_2 \; sinh(\frac{m\pi}{a}...

Homework Statement A hot fluid is flowing through a thick-walled cylindrical metal tube at a constant temperature of 450C. The cylinder wall has an inner radius of 1 cm and an outer radius of 2 cm and the surrounding temperature is 20C. The temperature distribution u(r) in the metal is defined...

Hi, I am in Purcell's E&M book at the section explaining why the field is zero inside a hollow conductor of any shape. The proof given is that the potential function inside the conductor must obey Laplace's equation, and that the boundary of the region (in this case a rectangular metal box) is...

Homework Statement Given a BVP: \Delta(u)+u=1 in \Omega u=0 on \partial\Omega using linear piecewise functions, calculate the corresponding local stiffness matrix on the reference triangle : {(x,y); 0<=x<=1, 0<=y<=1-x}. The domain is a square with one point in the middle (at...

Homework Statement a) Solve for the BVP. Where A is a real number. b) For what values of A does there exist a unique solutions? What is the solution? c) For what values of A do there exist infinitely many solutions? d) For what values of A do there exist no solutions? Homework...

{\frac {{\it du}}{{\it dx}}}=998\,u+1998\,v {\frac {{\it dv}}{{\it dx}}}=-999\,u-1999\,v u \left( 0 \right) =1 v \left( 0 \right) =0 0<x<10 Second Order Backward Difference formula {\frac {f_{{k-2}}-4\,f_{{k-1}}+3f_{{k}}}{h}} I'm trying solve this numerically in matlab, but can't seem to...

Dear all, I have system(4 equations) of BVPs. Could anybody recommend me, how to solve this system(whatever numericaly or analytical): x'=-y/sqrt(x^2+y^2) + u y'=x/sqrt(x^2+y^2) + v u' = -xy/(x^2+y^2)^3/2 u - [1/sqrt(x^2+y^2) - x^2 /(x^2+y^2)^3/2] v v' = xy/(x^2+y^2)^3/2 v -...

Homework Statement Formally solve the following boundary value problem using Fourier Transforms. Homework Equations (\partial^{2}u/\partialx^{2})+(\partial^{2}u/\partialy^{2}) = 0 (-\infty<x<\infty,0<y<1) u(x,0)= exp^{-2|x|} (-\infty<x<\infty) u(x,1)=0...

This exercise deal with the temperature u(x,y,t) in a homogeneous and thin plate. We assume that the top and bottom of the plate are insulated and the material has diffusivity k. Write the BVP . Problem: The plate is triangular , picture this as a right triangle with this coordinates, (0,0) ...

Homework Statement A pair of infinite, parallel planes are equipotential surfaces. The plane at z = 0 has an electric potential of 0 and the plane at z = b also has a potential of zero. The electric field at b is 0 at time t at which there is a constant, positive charge density between the...

Trying to solve the following boundary value problems. y'' + 4y = cos x; y(0) = 0, y(pi) = 0 y'' + 4y = sin x; y(0) = 0, y(pi) = 0 The answer key says that there's no solution to the first part, but there is a solution to the 2nd part. I'm really lost and am not sure how to go...

Homework Statement I have the solution to the differential equation : Phi = A*sin(x) + B*cos(x) and need to apply the boundary conditions Phi(-a/2) = Phi(a/2) = 0. Homework Equations The Attempt at a Solution I am confused. If I plug these in, then I get A*sin(-ka/2) = -...

i've some problem to solve this kind of differential equation, i 've put the link where is the file of my equation, i search a methods that solve it iterativly. the file is there diablo221 .altervista.org / risultato. nb

Homework Statement A solid sphere is placed in an otherwise uniform electric field. Its upper half is made up from a material with dielectric constant e_1; the other half has dielectric constant e_2. The plane at which the parts of the sphere intersect is parallel to the uniform field at...

A boundary value problem "discussion" So, let's say we are given a function f : [0, 1] --> R and constants a, b, and we want to find u : [0, 1] --> R such that u''(x) + f = 0 on <0, 1> with u(1) = a and u'(0) = -b. One can easily obtain the exact solution to this problem merely by using...

Schodinger's equation for one-dimensional motion of a particle whose potential energy is zero is \frac{d^2}{dx^2}\psi +(2mE/h^2)^\frac{1}{2}\psi = 0 where \psi is the wave function, m the mass of the particle, E its kinetic energy and h is Planck's constant. Show that \psi = Asin(kx) +...

I've got a nonhomogeneous BVP I'm trying to solve. Both my book and my professor tend to focus on the really hard cases and completely skipp over the easier ones like this, so I'm not really sure how to solve it. It's the heat equation in a disk (polar coordinates) with no angle dependence...

I need a bit of help with these boundary value problems. I'm trying to find their eigenvalues and eigenfunctions and although I pretty much know how to do it, I want to exactly WHY I'm doing each step. I attached part of my work, and on it I have a little question next to the steps I need...

Let D be a domain inside a simple closed curve C in R2. Consider the boundary value problem (\Delta u)(x,y) = 0, \ (x,y) \in D, \\ \frac{\partial u}{\partial n} (x,y) = 0 , \ (x,y) \in C. where n is the outward unit normal on C. Use Green's Theorem to prove taht every solution u...

I need some help starting off on this question. Electrostatic potential V (x,y) in the channel - \infty < x < \infty, 0 \leq y \leq a satisfies the Laplace Equation \frac{\partial^2 V}{\partial x^2} + \frac{\partial^2 V}{\partial y^2}= 0 the wall y = 0 is earthed so that V (x,0) = 0...