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MAT128C: Numerical Analysis

Project One: Explicit Methods for Ordinary Differential Equations

Due Date: April 22, 2019

数值分析作业代写 First, you will write a function called “forward euler” which implements the forward Euler method. It is defined via

1. Project description  数值分析作业代写

You will implement three different explicit methods for approximating the solution of the initial value problem

Here, F  is a smooth mapping R × Rn  → Rn, the desired solution ˙y  is a mapping [a, b] → Rn, and η  is a vector in Rn  specifiying the initial value of the solution.  The solution ˙y  will be represented via its values at the m equispaced points

t_j = a + h (j − 1), j = 1, 2, . . . , m, (2)

where

In other words, m is the number of discretization nodes in the interval [a, b]. The output of each of your routines will be the n × m matrix whose j^th column gives the approximation of ˙y(t_j) produced by your routine.  We will denote this approximation by ˙y_j.

First, you will write a function called “forward euler” which implements the forward Euler method. It is defined via  数值分析作业代写

The function “forward euler” must conform the calling syntax provided for you in the “forward euler.m”.

You will also write a function called “explicit trapezoid” which implements the explicit trapezoid method.  In this method, the value of ˙y₁ is given by

and for each j = 1, 2, . . . , m − 1, the value of ˙y_j+1 is approximating by first calculating

and then letting

The calling syntax for “explicit trapezoid” is given to you in the file “explicit trapezoid.m”.

The last of the methods you will implement is the fourth order Runge-Kutta scheme. In this scheme,

and for each j = 1, 2, . . . , m − 1, the value of ˙y_j+1 is approximating by first calculating

and then letting

Your implementation will consist of a function called “rk4” whose calling syntax is given in the file “rk4.m”.

2. Testing and grading  数值分析作业代写

A public test code is given in the file “project1 test1.m”. Another test code, called “project1 test2.m”, will be used to test your function as well.  Half of the project grade will come from the first test  file, and the second half will come from the second.  There are four tests of your function in each of the test codes, and each test is worth one point.

You will get a 0 on your project  if it does  not run.  Please start work on your project  early  and  come see either myself or our TA, Karry Wong, if you are having difficulties getting MATLAB to run.

Submitting your project:

You will submit your project using canvas. You should submit three files:  “forward  euler.m”, “explicit trapezoid.m” and “rk4.m” file. You must submit your file by 11:59 PM on the due data. Late assignments will not be accepted.

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