Monthly Archives: January 2012

Solving Heat Equation with Non-Homogeneous BCs 2: Time-Dependent BCs

Consider the heat IBVP: \begin{align*} &u_t=\alpha^2 u_{xx},\ 0<x<L,\ t>0\\ &{\rm BCs:}\ \left\{\begin{aligned} &u(0,t)=g_1(t),\ t>0\\ &u_x(L,t)+hu(L,t)=g_2(t), \end{aligned} \right.\\ &{\rm IC:}\ u(x,0)=\phi(x),\ 0<x<L. \end{align*} Again we seek a solution of the form: $$u(x,t)=S(x,t)+U(x,t)$$ where $S(x,t)$ satisfies the non-homogeneous BCs: \begin{align*} S(0,t)&=g_1(t)\ \ … Continue reading

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Solving Heat Equation with Non-Homogeneous BCs 1: Time-Indepdendent BCs

Consider the following 1-dimensional heat IBVP: \begin{align*} &u_t=\alpha^2 u_{xx},\ 0<x<L,\ t>0\\ &{\rm BCs:}\ \left\{\begin{aligned} u(0,t)&=c_1,\ t>0\\ u(L,t)&=c_2, \end{aligned} \right.\\ &{\rm IC:}\ u(x,0)=\phi(x),\ 0<x<L. \end{align*} Since the BCs are not homogeneous, the method of separation of variables cannot be used. Let … Continue reading

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1-Dimensional Heat Initial Boundary Value Problems 3: An Example of Heat IBVP with Mixed Boundary Conditions (Insulated and Specified Flux)

Let us consider the heat IBVP: \begin{align*} u_t=\alpha^2u_{xx},\ 0<x<1,\ t>0\\ u_x(0,t)=0,\ t>0\\ u_x(1,t)+u(1,t)=0,\ t>0\\ u(x,0)=1,\ 0<x<1. \end{align*} First we solve the Sturm-Liouville problem: \begin{align*} X^{\prime\prime}=kX\ \mbox{with BCs}\\ \left\{ \begin{aligned} &X’(0)=0\\ &X’(1)+X(1)=0. \end{aligned} \right. \end{align*} The two cases $k=0$ and $k=\lambda^2>0$ … Continue reading

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1-Dimensional Heat Initial Boundary Value Problems 2: Sturm-Liouville Problems and Orthogonal Functions

Sturm-Liouville Problems The homogeneous boundary conditions of 1D heat conduction problem are given by \begin{align*} -\kappa_1u_x(0,t)+h_1u(0,t)&=0,\ t>0\\ \kappa_2u_x(L,t)+h_2u(L,t)&=0,\ t>0 \end{align*} (See here) The homogeneous BCs for the second order linear differential equation $$X^{\prime\prime}=kX\ \ \ \ \ (1)$$ is then … Continue reading

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1-Dimensional Heat Initial Boundary Value Problems 1: Separation of Variables

Let us consider the following assumptions for a heat conduction problem. The region $\Omega$ is a cylinder of length $L$ centered on the $x$-axis. The lateral surface is insulated. The left end ($x=0$) and the right end ($x=L$) have boundary … Continue reading

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Rotational Symmetry and the Conservation of Angular Momentum in Quantum Mechanics

Definition. A quantum mechanical system is said to be rotationally invariant if the system still obeys Schrödinger’s equation after a rotation. Let us assume that the quantum mechanical system under consideration is rotationally invariant. Consider an infinitesimal rotation of a … Continue reading

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