§3.3.1 Separation of Cartesian variables: Orthogonal functions

Slides:

Advertisements

Similar presentations

The Infinite Square Well Now we are ready to solve the time independent Schrödinger equation (TISE) to get ψ(x) and E. (Recall Ψ(x,t) = ψ(x)e −i2  Et/h.

Advertisements

P. Venkataraman Mechanical Engineering P. Venkataraman Rochester Institute of Technology DETC2014 – 35148: Continuous Solution for Boundary Value Problems.

BVP Weak Formulation Weak Formulation ( variational formulation) where Multiply equation (1) by and then integrate over the domain Green’s theorem gives.

ECE602 BME I Partial Differential Equations in Biomedical Engineering.

EEE340Lecture 171 The E-field and surface charge density are And (4.15)

Weak Formulation ( variational formulation)

Math 015 Section 6.1 Equations. Many students like to remember the process for solving linear equations as: any algebra expressions on each side variable.

Lecture outline Support vector machines. Support Vector Machines Find a linear hyperplane (decision boundary) that will separate the data.

Linear Algebra, Principal Component Analysis and their Chemometrics Applications.

1Chapter 2. 2 Example 3Chapter 2 4 EXAMPLE 5Chapter 2.

Copy and complete using a review term from the list below.

Solving quadratic equations Factorisation Type 1: No constant term Solve x 2 – 6x = 0 x (x – 6) = 0 x = 0 or x – 6 = 0 Solutions: x = 0 or x = 6 Graph.

Chapter 2 Systems of Linear Equations and Matrices Section 2.4 Multiplication of Matrices.

Chapter 13 Section 13.5 Lines. x y z A direction vector d for the line can be found by finding the vector from the first point to the second. To get.

Math 3120 Differential Equations with Boundary Value Problems Chapter 4: Higher-Order Differential Equations Section 4-9: Solving Systems of Linear Differential.

1. Inverse of A 2. Inverse of a 2x2 Matrix 3. Matrix With No Inverse 4. Solving a Matrix Equation 1.

Goal: Solve a system of linear equations in two variables by the linear combination method.

§1.2 Differential Calculus

§3.4. 1–3 Multipole expansion Christopher Crawford PHY

§1.2 Differential Calculus Christopher Crawford PHY 416G

Solve a system of linear equations By reducing a matrix Pamela Leutwyler.

Do Now (3x + y) – (2x + y) 4(2x + 3y) – (8x – y)

§ Separation of Cartesian variables: Orthogonal functions Christopher Crawford PHY

RECOGNIZING INCONSISTENT LINEAR SYSTEMS. What is an Inconsistent Linear System?  An inconsistent linear system is a system of equations that has no solutions.

Differential Equations Linear Equations with Variable Coefficients.

Chapter 8 Systems of Linear Equations in Two Variables Section 8.3.

3.3 Solving Linear Systems by Linear Combination 10/12/12.

Solving Partial Differential Equation Numerically Pertemuan 13 Matakuliah: S0262-Analisis Numerik Tahun: 2010.

Chapter 5: Matrices and Determinants Section 5.5: Augmented Matrix Solutions.

1.6 Solving Linear Systems in Three Variables 10/23/12.

§ Linear Spaces Christopher Crawford PHY

§ Separation of spherical variables: zonal harmonics Christopher Crawford PHY

Using Matrices to Solve a 3-Variable System

Chapter 3 Linear Systems Review

CHAPTER R: Basic Concepts of Algebra

Christopher Crawford PHY

Differential Equations

Equations Quadratic in form factorable equations

Boundary-Value Problems in Rectangular Coordinates

Christopher Crawford PHY

Differential Equations

Christopher Crawford PHY

Christopher Crawford PHY

PHY 711 Classical Mechanics and Mathematical Methods

§1.5 Delta Function; Function Spaces

§1.1.4 Affine space (points)

§5.3: Magnetic Multipole Expansion

Christopher Crawford PHY

Solving Linear Systems Algebraically

§3.4.1–3 Multipole expansion

§7.2 Maxwell Equations the wave equation

Christopher Crawford PHY

Christopher Crawford PHY

Chapter 1:First order Partial Differential Equations

State Space Method.

Linear Algebra Lecture 3.

Differential Equations

Solving Equations by Combining Like Terms

Graphing Linear Equations

Introduction to Ordinary Differential Equations

Algebra EquationsJeopardy

Boundary Value Problems

Systems of Equations Solve by Graphing.

PHY 711 Classical Mechanics and Mathematical Methods

Reading: Chapter in JDJ

Bell Work Solve for “x” and check your solution

Solving a System of Linear Equations

§3.3.1 Sturm-Liouville theorem: orthogonal eigenfunctions

Solving Systems of Linear Equations by Elimination

Presentation transcript:

§3.3.1 Separation of Cartesian variables: Orthogonal functions Christopher Crawford PHY 416 2014-10-22

Outline Analogies from Chapter 1 Inner, outer products Projections, eigenstuff Example boundary value problem – square box Separation of variables Solution of boundary conditions Numerical methods for BVPs Relaxation method Finite difference method Finite element method

Analogies from Chapter 1 Inner vs. outer product – projections & components Linear operators – stretches & rotations

Separation of variables technique Goal: solve Laplace’s equation (PDE) by converting it into one ODE per variable Method: separate the equation into separate terms in x, y, z start with factored solution V(x,y,z) = X(x) Y(y) Z(z) Trick: if f (x) = g (y) then they both must be constant Endgame: form the most general solution possible as a linear combination of single-variable solutions. Solve for coefficients using the boundary conditions. Trick 2: use orthogonal functions to project out coefficients Analogy: the set of all solutions forms a vector space the basis vectors are independent individual solutions

Example: rectangular box Boundary value problem: Laplace equation

Example: rectangular box Boundary value problem: Boundary conditions 6