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### assignment_4_C

Course: MATH 5520, Fall 2009
School: UConn
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5520 MATH Finite Element Methods. April 14, 2009 Assignment 4 (C) Consider the nonhomogeneous Heat equation u(x, t) t x (x) u(x, t) x = f (x, t), x (0, 1), t &gt; 0, u(0, t) = u(1, t) = 0, t &gt; 0, x (0, 1), u(x, 0) = g(x), where (x), f (x, t), and g(x) are some given functions. 1. Write down the weak form of the equation. What are the appropriate spaces of function? Explain. 2. Let Th be the...

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5520 MATH Finite Element Methods. April 14, 2009 Assignment 4 (C) Consider the nonhomogeneous Heat equation u(x, t) t x (x) u(x, t) x = f (x, t), x (0, 1), t > 0, u(0, t) = u(1, t) = 0, t > 0, x (0, 1), u(x, 0) = g(x), where (x), f (x, t), and g(x) are some given functions. 1. Write down the weak form of the equation. What are the appropriate spaces of function? Explain. 2. Let Th be the partition 0 = x0 < x1 < x2 < < xN = 1 and Vh be the space of continuous piecewise linear functions on the partition Th . Dene the semidiscrete Galerkin solution uh (t) : [0, T ] Vh . 3. Dene fully discrete Backward Euler method. 4. Modify your code from HW 2 to solve the heat equation by backward Euler method. 5. Once you it, accomplish run the convergence tests with the exact solution u(x, t) = t2 sin (x), ( 1), by successive rening the mesh in space and time. Remember that now the error e(x, t) between the exact and backward Euler solution is a function of space and time. The theory says that if the solution is smooth the error at the nal time T should obey the estimate e(T ) L2 (0,1) C h2 + k , where h is the mesh size in space and k is the time step. Si...

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DePaul - DB - 394
# This file is autogenerated. Instead of editing this file, please use the# migrations feature of ActiveRecord to incrementally modify your database, and# then regenerate this schema definition.ActiveRecord:Schema.define(:version =&gt; 19) do cre
UConn - MATH - 5520
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# Methods added to this helper will be available to all templates in the application.module ApplicationHelperend
DePaul - APP - 394
module CategoryHelperend
DePaul - APP - 394
module IngredientControllerHelperend
DePaul - APP - 394
module IngredientHelperend
DePaul - APP - 394
DePaul - APP - 394
module MeasuringUnitHelperend
DePaul - APP - 394
module RecipeHelperend
DePaul - APP - 394
module RecipesHelperend
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module SizeHelperend
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module UsersHelperend
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class CreateIngredients &lt; ActiveRecord:Migration def self.up create_table :ingredients do |t| t.column :name, :string, :null =&gt; false end end def self.down drop_table :ingredients endend
DePaul - DB - 394
class CreateMeasuringUnits &lt; ActiveRecord:Migration def self.up create_table :measuring_units do |t| t.column :name, :string, :null =&gt; false end end def self.down drop_table :measuring_units endend
DePaul - DB - 394
require 'active_record/fixtures' class CreateFractions &lt; ActiveRecord:Migration # The self.up method runs when committing the changes. def self.up create_table :fractions do |t| t.column :numerator, :integer, :null =&gt; false t.c
DePaul - DB - 394
# This code runs when commiting a migration via 'rake db:migrate'require 'active_record/fixtures'class CreateSizes &lt; ActiveRecord:Migration # The self.up method runs when committing the changes. def self.up create_table :sizes do |t|
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# This code runs when commiting a migration via 'rake db:migrate'require 'active_record/fixtures'class CreateUsers &lt; ActiveRecord:Migration # The self.up method runs when committing the changes. def self.up create_table :users do |t|
DePaul - DB - 394
class CreateRecipes &lt; ActiveRecord:Migration def self.up create_table :recipes do |t| t.column :name, :string, :null =&gt; false t.column :user_id, :integer, :null =&gt; false t.column :instructions, :text, :null =&gt; true end e
DePaul - DB - 394
class AddRecipeCategory &lt; ActiveRecord:Migration def self.up add_column :recipes, :category_id, :integer, :null =&gt; false end def self.down remove_column :recipes, :category_id endend
DePaul - DB - 394
class CreateCategories &lt; ActiveRecord:Migration def self.up create_table :categories do |t| t.column :name, :string, :null =&gt; false end end def self.down drop_table :categories endend
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class CreateRecipeLineItems &lt; ActiveRecord:Migration def self.up create_table :recipe_line_items do |t| t.column :quantity, :integer, :null =&gt; true t.column :fraction_id, :integer, :null =&gt; true t.column :measuring_unit_id, :i
DePaul - DB - 394
class SetForeignKeys &lt; ActiveRecord:Migration def self.up execute &quot;alter table recipes add constraint fk_recipe_users foreign key (user_id) references users(id)&quot; execute &quot;alter table recipes add constraint fk_recipe_categorie
DePaul - DB - 394
DePaul - DB - 394
class AddSessions &lt; ActiveRecord:Migration def self.up create_table :sessions do |t| t.column :session_id, :string t.column :data, :text t.column :updated_at, :datetime end add_index :sessions, :session_id add_ind
DePaul - DB - 394
class ModifyRecipeUsersForeignKey &lt; ActiveRecord:Migration def self.up down execute &quot;alter table recipes add constraint fk_recipe_users foreign key (user_id) references users(id) on delete cascade&quot; execute &quot;alter table rec
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class ModifyRecipeUsersLineItemsForeignKey &lt; ActiveRecord:Migration def self.up down execute &quot;alter table recipe_line_items add constraint fk_recipe_line_items_recipes foreign key (recipe_id) references recipes(id) on delete c
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# = Schema Information# Schema version: 19## Table name: categories## id :integer(11) not null, primary key# name :string(255) default(&quot;), not null#class Category &lt; ActiveRecord:Base has_many :recipes validates_presence_of :n
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# = Schema Information# Schema version: 19## Table name: ingredients## id :integer(11) not null, primary key# name :string(255) default(&quot;), not null## This class represents ingredients in the data model.class Ingredient &lt; ActiveRec
DePaul - APP - 394
# = Schema Information# Schema version: 19## Table name: measuring_units## id :integer(11) not null, primary key# name :string(255) default(&quot;), not null#class MeasuringUnit &lt; ActiveRecord:Base has_many :recipe_line_items has_many
DePaul - APP - 394
# = Schema Information# Schema version: 19## Table name: sizes## id :integer(11) not null, primary key# name :string(255) default(&quot;), not null## This class represents sizes, i.e. small, medium, large, etc. class Size &lt; ActiveRecord
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# = Schema Information# Schema version: 19## Table name: users## id :integer(11) not null, primary key# name :string(255) default(&quot;), not null# logon :string(255) default(&quot;), not null# hashed_passwo
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